27815722 Department of the Army Technical Manual Tm 5 237 Surveying Computer s Manual October 1964 (1)
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TM 5237 DEPARTMENT
OF THE ARMY TECHNICAL
MANUAL
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SURVEYING COMPUTER'S MANUAL
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HEADQUARTERS

DEPARTMENT OCTOBER 1964
OF THE ARMY
This manual contains copyrighted material.
*TM 5237 TECHNICAL MANUAL
HEADQUARTERS DEPARTMENT OF THE ARMY WASHINGTON, D.C., 30 October 1964
No. 5237
SURVEYING COMPUTER'S MANUAL
CHAPTER
Paragraphs 1. INTRODUCTION_________14
2. 3. Section I. II. III. IV. V. VI. CHAPTER
4.
Section I. II. III. IV. CHAPTER
5.
ASTRONOMIC TABLES______________________ 59 ASTRONOMIC OBSERVATION COMPUTATIONS Conversion of time10,11 Computation of azimuth1216 Determination of latitude1726 Determination of longitude_ 2731 Computation of latitude and longitude from observations made with the astrolabe________________________ 3235 Astronomic results36,37 DISTANCE MEASUREMENTS Tape measurements3842 Tachymetry measurements4346 Measurements using light waves____  _ 4749 Measurements using electromagnetic waves5053 TRIANGULATION Preparation of data for adjustment5461 Quadrilateral adjustment (least squares method)6265 Geographic position6672
Section I. II. III. IV. Adjustment of triangulation net7376 V. Special problems7781 VI. Shoreship triangulation_________ _____ _________ 82, 83 CHAPTER
6.
7.
Section I. II. CHAPTER
8.
Section I. II. III. CHAPTER
9.
Section I. II. CHAPTER 10.
Section I. II. III. IV. CHAPTER
11.
Section I. II. III. IV.
3 4 10 15 36 65 80 92 95 105 106 118 131 167 175 188 221 232
TRILATERATION
Section I. Preparing data for adjustmentII. Trilateration adjustment by the method of least squares__ CHAPTER
Page
84, 85 8690
TRIANGULATIONTRILATERATION COMBINATION Preparing data for adjustment9194 Adjustment using combined measurements9597 GEOGRAPHIC TRAVERSE Introduction  98100 Adjustment of traverse (least squares method)  101105 Adjustment of traverse (approximate method)  106108 RESULTS OF HORIZONTAL CONTROL SURVEYS Tabulation of results  109112 Description of horizontal control station 113, 114 DIFFERENTIAL LEVELING Differential level line115118 Adjustment of a level net119121 122 123 Description of vertical control station______ Computation of tide observations124126 TRIGONOMETRIC LEVELING Abstract of zenith distances127 128 Trigonometric elevations from reciprocal observations 129131 Trigonometric elevations from nonreciprocal observations 132134 Adjustment of trigonometric elevations_135 136
*This manual supersedes TM 5237, 21 May 1957.
236 236 238 239 245 245 255 263 266 270 271 279 282 287 291 295 298
CHAPTER 12. ALTIMETER LEVELING
Paragraphs
Page
Singlebase and leapfrog methods __ 137140 Twobase method .141143 THE UNIVERSAL TRANSVERSE MERCATOR GRID 144151 Mathematics and construction of the UTM grid_ Conversion of geographic coordinates to UTM coordinates_ 152, 153 Conversion of UTM coordinates to geographic coordinates__ 154, 155 UTM grid azimuths 156, 157 UTM scale fact.r  158, 159 Zone to zone transformation on the UTM grid160163 HORIZONTAL CONTROL USING UTM GRID Position computation on the UTM grid 164166 Triangulation on the UTM grid_    _ 167, 168 Traverse adjustments on the UTM grid169, 170 COMPUTATIONS ON THE UNIVERSAL POLAR STEREOGRAPHIC GRID Section I. Universal polar stereographic transformations  171173 II. UPS scale factor and convergence_ 174, 175 CHAPTER 16. OTHER GRID SYSTEMS Section I. Transverse mercator projection . _ 176178 II. Lambert conical conformal projection_ _ _ 179181 III. State plane coordinate systems in Alaska _ _ 182, 183 IV. World polyconic projection_ 184186 CHAPTER 17. GRIDS AND DECLINATIONS FOR MAPS 187, 188 Section I. Dimensions of a grid__ II. Grid and magnetic declination189, 190 CHAPTER 18. DATUM OR SPHEROID SHIFT BY TRANSFORMATION OF GRID COORDINATES191,192 19. REDUCTION OF GRAVITY OBSERVATIONS __193197 APPENDIX I. REFERENCESII. CHARTS AND GRAPHSIII. TABLES
300 304
Section I. II. CHAPTER 13. Section I. II. III. IV. V. VI. CHAPTER 14. Section I. II. III. CHAPTER 15.
GLOSSARY.INDEX

312 316 319 321 325 327 333 336 354
360 362 363 366 368 369 372 376 378 383 394 397 403 450 457
CHAPTER 1 INTRODUCTION _
__
I
1. Purpose
3. References
This manual is published to serve as a reference and guide for the survey computer in accomplishing geodetic and topographic survey computations; to standardize the methods and procedures for completing these computations; to standardize tabulation procedures for numerical figures and results for use in records or for dissemination; and to familiarize survey computers with the accepted methods of performing survey computations.
TM 5236 contains many of the tables used in performing the computations included in this manual. Basic topographic surveying methods are discussed in TM 5441. Other references are included in appendix I.
2. Scope This manual contains descriptive material, references, sample solutions, and sample tabulations for all types of computations that are not usually completed in the field notebooks and which may be encountered in military topographic surveys. The instruction for completion of each of the computations contains both descriptive material and a detailed solution. Unless otherwise stated, all grid coordinates will be Universal Transverse Mercator Grid coordinates.
4. Accuracy a. The solutions found within this manual are designed to meet any foreseeable need of the military survey computer. Most of the included computations, unless otherwise stated, will meet the requirements for first order. Computations designed for first order accuracy may be adapted to lower order surveys by relaxing some of the refinements. b. Users of this manual are encouraged to submit recommended changes or comments to improve the manual. Comments should be keyed to the specific page, paragraph, and line of the text in which change is recommended. Reasons should be provided for each comment to insure understanding and complete evaluation. Comments should be forwarded directly to the Commandant, U.S. Army Engineer School, Fort Belvoir, Va., 22060.
CHAPTER 2 ASTRONOMIC TABLES 5. Introduction The astronomic determination of direction, latitude, longitude, and time depends upon the apparent movement of stellar bodies. Astronomic tables, in general, list the directions and positions of planets and stars, their rates of apparent movement, and certain numerical quantities necessary to convert their apparent motions to more usable information for any given instant. The most commonly used astronomic tables are the American Ephemeris and Nautical Almanac (AE&NA), Apparent Places of the Fundamental Stars (APFS), and the General Catalogue of 33342 Stars by Benjamin Boss. The first two sets of tables are published annually while the latter has been published for the epoch of 1950.
6. ,The American Ephemeris and Nautical Almanac (AE&NA) The American Ephemeris and Nautical Almanac is published annually by the U.S. Naval Observatory and is distributed by the Army as a Technical Manual, TM 5236XX, with the third number indicating the year for which the data is applicable. The volume has undergone two important changes during recent years. One is the deletion of the tenday stars while the other was the introduction of Ephemeris time which is a more precise time. Corrections, necessary to convert Universal time to Ephemeris time, are included in the Ephemeris. a. Tables Included in the AE&NA. (1) AT, reduction from Universal time to Ephemeris time. (2) Universal and Sidereal Time for Oh UT. (3) Sun (year)for Oh Ephemeris Time. (4) Besselian and Independent Day Numbers. (5) Mean places of stars. (6) Table IIfor finding the latitude by the observed altitude of Polaris and azimuth of Polaris at all hour angles.
(7) Table VIIISidereal time to mean solar time. (8) Table IXMean solar time to sidereal time. (9) Table XConversion of hours, minutes and seconds to decimals of a day. (10) Table XIConversion of time to arc. (11) Table XIIConversion of arc to time. (12) Table XIIIInterpolation constants. (13) Table XIVSecondDifference corrections. b. Use of Ephemeris Time. Starting with the AE&NA for 1960, the tabular argument in the fundamental ephemerides of the sun, moon, and the planets is Ephemeris time. Ephemeris time is the uniform measure of time as defined by the laws of dynamics and determined in principle from the orbital motions of the planets; specifically the orbital motions of the earth as represented by Newcomb's "Tables of the Sun". Universal time is defined by the rotational motion of the earth, and is determined from the apparent diurnal motions which reflect this rotation. Because of variations in the rate of rotation, Universal time is not rigorously uniform. c. Universal and Sidereal Times. Beginning with the 1960 Ephemeris the Sidereal time at Oh Universal Time and the Universal Time at Oh
Sidereal time, which formerly were included in the Ephemeris of the Sun, are tabulated both for the mean equinox of the date, and for true equinox, with the short period terms of nutation included. d. Tables of the Sun. (1) The date column. The dates found in this column represent the instant at 0 h Ephemeris Time on the date indicated. (2) The apparentright ascension column. The apparent right ascension (RA) is given in units of time for each day. The lightfaced type, to the right and between the lines, gives the tabular differences in
seconds. Linear interpolation is made by multiplying these terms by the fraction of a day elapsed since Oh, and adding the result to the right ascension given for Oh of the date. Note that the sun always increases in right ascension and the tabular differences are positive.
(3) The apparent declination column.
The
apparent declination (S) is given in the third column. Interpolated values are determined in the case of right ascensions as in (2) above. However, both the declination and tabular differences may be either positive or negative. Care must be exercised in preserving the proper sign. Both the apparent right ascension and declination of the sun contain the effects of longperiod terms of nutation and aberration, and define the apparent place (as the observer sees it) of the true sun.
(4) The horizontalparallax column.
Parallax
is the displacement in the position of a heavenly body due to observations being made from the surface of the earth instead of at its center. Horizontal parallax is the angle at the center of the sun subtended by the earth's equatorial radius. For solar observations, the true parallax is equal to the product of the horizontal parallax and the cosine of the altitude, or the sine of the zenith distance. (5) The semidiameter column. The amounts shown in this column are correct for horizontal semidiameter but, due to the effects of refractions, they will vary slightly for vertical semidiameter.
(6) The equation of time column. The sixth column gives the equation of time stated as apparent minus mean. Tabular differences are also furnished. Note that both the equation and the difference may have either algebraic sign.
e. Table VIII. Sidereal into Mean Solar Time. Table VIII of the Ephemeris gives the quantities that must be subtracted from a given time interval expressed in sidereal units in order to obtain the same interval expressed in meantime units. Or it represents the amount a meantime clock would lose as compared to a sidereal clock over a given sidereal interval. The main table is given at intervals of 1 sidereal minute for the entire day, the hour being given at the head of the column
and the minutes down the lefthand side. The corrections for seconds are given at the extreme right. Interpolation is made as in table IX (J below). The sum of the quantities from the main and seconds table is then subtracted from the sidereal interval to obtain the corresponding meantime interval. J. Table IX. Mean Solar into Sidereal Time. This table of the Ephemeris is more frequently used than table VIII. It gives the quantities that must be added to a given interval expressed in meantime units in order to obtain the same interval in sidereal units. It is the amount a sidereal clock gains with respect to a meantime clock over a given meantime interval. Table IX has the same form as table VIII. In order to convert a meantime interval to a sidereal interval, first enter the main table in the column for the hour, and on the line corresponding to the last full meantime minute. Next, the table to the right is entered using the seconds of mean time over the last full minute. Any interpolation for desired fractions of a second is made mentally. The final quantity from the seconds table is then added to the quantity taken from the main table. This gives the correction to be added to the meantime interval. Example: Find the siderealtime interval corresponding to a meantime interval of: 1 7h
14 m
12.7
From main table, correction for
1 7h
14 m =
2m
From second table, correction for 1237 = Sidereal interval

495860 03035
17
17m
025595
g. Mean Places of Stars. Mean places of 1078 stars are given in this table for the instant of the beginning of the Besselian year. This date occurs when the sun's mean longitude is 2800 and falls very close to the beginning of the calendar year. The civil date is given in decimal days at the top of the page. The mean place does not coincide with the apparent place for the same date, but constitutes a base for the application of corrections in order to find the apparent place at any given date. The formulas for this reduction mean to apparent place, are given in the section of the American Ephemeris devoted to the use of the tables, under Stars. This reduction is seldom required of the computer. The meanplace tables are useful in preparing observing lists, and for
any purpose where a close value of the star's position is not required. h. Latitude From the Altitude of Polaris. This table affords a means of determining the latitude when the altitude (h) of Polaris and the local sidereal time (LST) are known. (1) Three corrections; ao, al and a2 are extracted from table II. The arguments are the local sidereal time, the approximate latitude, and the month of observation respectively. (2) The observed altitude is corrected for refraction. The latitude is determined by taking the algebraic sum of the corrected altitude and the three corrections from table II. i. Azimuth of Polaris. Table II is also used to determine the azimuth of Polaris. The procedure is as follows: (1) Extract the values for bo, bl and b2 from table II, using the local sidereal time (LST) of observation, the latitude and the month of observation respectively, as the arguments. These corrections are added algebraically. (2) Multiply the quantity (bo+bl+b 2) by the secant of the latitude to obtain the azimuth of Polaris, as referenced to the Pole. j. Besselian and Independent Day Numbers. These day numbers, used for reducing mean to apparent place, are coefficients of the effects .on the stars position caused by the processes of precession, nutation and aberration. . They are computed by trigonometric means from the mean coordinates of the star and are dependent only on the Greenwich Ephemeric Time Date. Factors for proper motion are abstracted directly from the star catalogue. Either the Besselian or the Independent system may be used in the reductions although the Besselian system is more convenient for mass production. k. Table XII. Conversion of Arc to Time. This table is often useful in avoiding division by 15. In successive columns, the time equivalents of degrees, minutes, seconds, and decimal seconds of are are given. Examnple: Convert 32044'42'.'15 to units of time. 320 44'
=
42" 0'.'15
= =
Sum
=2h
2h
08m 2m
56
2.8 001 10m
58f81
1. Table XI. Conversion of Time to Arc. This table is the inverse of table XII. The first part is a table in arguments of hours and minutes, from which is taken the are equivalent of the even minute. The right hand section gives the are equivalents of seconds and 100ths of seconds of time.
Example: Convert 2 h 10m 58.81 to are units. 2h
10m
=320
30'
=
14'
0.81= Sum=320
44'
58
30" 12'.'15 42'.'15
m. Table X. Conversion of Hours, Minutes, and Seconds to Decimals of a Day. This table is useful in finding the decimal day equivalent to the UT for the purpose of entering the tables for the apparent places of stars. The equivalent of the hours and minutes is taken from the main table. That of the remaining seconds is given in the right hand column. This latter refinement is rarely necessary. n. Table XIII. Besselian Interpolation Constants. The use of this table is explained in paragraph 9d. 7. The Apparent Places of the Fundamental
Stars (APFS) a. Introduction. The tables in the Apparent Places of the Fundamental Stars are the result of international cooperation which has reduced the duplication of certain parts of the ephemerides published by contributing nations. The data contained in this publication is limited to tables of the stars position and certain auxiliary tables. b. The Main Table. The main table in the APFS lists the apparent positions at upper transit of 1483 stars, between 810 north and 810 south declination, at 10day intervals throughout the year. The column headings contain the catalog number, name, magnitude and type of spectrum of each star. Right ascensions are listed in units of time to thousandths of seconds. Declinations are in units of arc to the hundredths of seconds. In both cases, the seconds values are followed by the tabular differences. At the foot of each page are found the mean place, secant, and tangent of 6, and factors for computing short period terms of nutations for each star. The dates when two transits of the Greenwich meridian occur, during the same meantime day, are also given.
c. The Table of CircumpolarStars. Immediately following the main table is the table of Circumpolar Stars, which lists the apparent positions of 52 circumpolar stars for every upper star transit at Greenwich. The date refers to the civil day. Interpolation is made from the Greenwich Hour Angle of the star at the time of observation. The oneday interval, between tabulations, permits the inclusions of short period terms of nutation within the tabulated values. Right ascensions are given to two decimal places only, this being in the order of the uncertainty of circumpolar star positions. Otherwise the tables are similar to the 10day star tables. d. The FK 4 System. The "Apparent Places of Fundamental Stars" for 1964, and subsequent years, contains the 1535 stars in the Fourth Fundamental Catalogue (FK4). This volume provides the mean and apparent places of 10day and Circumpolar stars together with tables for their reduction. e. Table I. Table I in the APFS furnishes factors for computing the shortperiod terms for the 10day stars. The equations for accomplishing this are found at the foot of the pages of the table. The other necessary coefficients are tabulated under each star in the apparentplace table. j. Table II. The sidereal time of Oh, Universal Time and the long and shortperiod terms of the Equation of the Equinoxes are given. The apparent sidereal time is the sum of the mean sidereal time plus period terms. g. Table III. Table III provides the conversion factors, mean solar to sidereal time and is identical to table IX of the AE&NA. h. Table IV. Table IV is used for converting intervals of sidereal time to mean solar time and is identical to table VIII of the AE&NA. i. Table V. Table V is used for reducing hours, minutes, and seconds to decimals of days and is similar to table X of the AE&NA with the exception that table X is a sixplace table while table V is a fiveplace table. j. Table VI. Table VI lists second difference corrections for use with linearly interpolated values. This table is somewhat different from the table of Besselian coefficients. The quantities given are for the term B" (A'+ A'), the symbols being explained in paragraph 9d. The arguments are the fractional part n, and the double second difference (a' A'+a'). The latter factor is at a tabular interval of 5 units of the last place of the
fraction. The correction is always of opposite sign to (A'0o' A'). k. Table VII. Table VII is used for correcting the time of transit for the effect of diurnal aberration. The correction is rarely needed.
8. General Catalogue of 33342 Stars By Benjamin Boss This catalogue consists of five volumes, the first being used for instructions and appendixes and the other four for mean places of stars. This catalogue is principally used by the geodesist in the determination of latitude. Mean to Apparent Place reduction is accomplished by means of the Besselian or Independent Day numbers from the AE&NA. Statistical and historical information contained in these tables is explained in the first volume. 9. Interpolation a. Introduction. As the computer will have an almost continuous need for interpolation in the use of various tables, this paragraph will review linear and double interpolation. b. Linear Interpolation. Geodetic surveys and astronomy will rarely have need for more than simple linear interpolation. This assumes that the function varies as a constant ratio, that is, as a straight line, between tabular values. Most functions are actually curves when plotted on the coordinate axis. Hence, a linear interpolation is subject to some error. The amount of the error depends on the sharpness of the curve and the spacing of the tabular values. All good tables are so arranged that the errors are nearly always negligible. (1) When the interpolated value is taken as the proportional part of the difference between successive values given in the table, it is said to be interpolation on the. chord. The tables of the American Ephemeris which give these tabular differences in smaller type are so arranged that the interpolation is along the chord. (2) Another form of linear interpolation is said to be on the tangent. In this case, the small type figure is the rate of change, or slope of the tangent at the value of the function as given in the table. This form is more accurate than interpolation along the chord, provided not more than a halfinterval is taken. These forms of
linear interpolation are shown in figure 1. The function is represented in the figuIre by the curve PQ, at which point its y values are given in the table. It is cdesired to interpolate for the y value wh hen x=0.8. Point A represents the tr 'ue value on the curve. Point B represenIts the value found by interpolation on the chord, the error being BA. Interp)olation along the tangent from the near est tabular value Q gives a result at C. The error CA is less than B A. Point D is found by interpolating alo ng the tangent from P, and the error is greater than that obtained on the cho:rd
Q
(3) Step 1Interpolate for parallax on June 10 at 360 altitude. Use nearest tabulated value. p=07"'02 1/3 (07'.'0207'01) =07"017 (4) Step 2Interpolate for parallax on June 10 at 330 altitude. p=07281/3 (07'2807.26)=7'273 (5) Step 3Interpolate for parallax at altitude of 33031'19
'.
p=07.2731/6
(07"27307'.017)= 07"230= 07'"23 d. Besselian Interpolation. This method is required only in the highest class of computations. Interpolation to second and higher differences brings successively closer approximations of the true value on the curve. The tabular differences taken from the table are known as the first differences. The successive differences between the first differences are called second differences, and so on. The customary designation of these differences, to second differences only, appears below. (Function) Tabular value
lst diff.
2d dif
F1
Fo
0134
AO'
Fa
The desired value F,, lies at a fraction n, between F0 and F 1. Bessel's formula is generally preferred. It is written as: F,Fo+n1/2,B (Ao'+A')[B" ', Figure 1.
Forms of linear interpolation.
c. Double Interpolation. Double interpolation becomes necessary when a function is subject to two variables instead of one. The requirements of most double interpolations are met by a series of three chord interpolations; and the use of chord interpolations is recommended for this purpose. For example, the parallax of the sun varies during different dates of the year, and varies with the altitude of the various observations. The process of interpolation is as follows: (1) Example: Determine the parallax for June 10, 1964 when the observed altitude
2']
+
....
The quantity in the brackets represents the third difference, given for the sake of completeness. Only the first three terms are needed for second differences. The terms B", B" ', and so on, are known as the Besselian interpolation constants. B" and B" ' are given in table XV of the American Ephemeris for values of from 0 to 1. Their values are:
B" =n(n1) 2(2!)
B" ,
n1)(n1)() 3!
and may be so derived in the absence of the tables. The terms A'/2, A ', and A"', are taken from a tabulation as shown above.
is 33031'19".
(2) Parallaxfrom table VIII (app. III) Date
Altitude
Parallax
Date
Altitude
June 1
360
7"02
July 1
360
June 1
330
7"28
July 1
330
Parallax
7:01 7"26
Example: Find, with respect to second differences, the apparent declination of the sun at UT June 1.67, 1963.
Date
May 31
Decl.
(F_1)+210 46'44"6
June 1
(Fo) + 2155'26 8
June 2
(Fl) + 22003'46"2
June 3
(F 2) 22o11'42:4
Ist dif.
2d dif.
x499.4(=B')
The fraction n==0.67. Hence, from table XV, B" is 0.0553. B"' if required would be 0.0063. Applying the formula: F,= +21055'26.8± 0.67(+499.4) 0.0553(46.0) = ±22103'9. This interpolation may be simplified with little loss of accuracy by taking the term, B"(' +A') ± from table VI of the Apparent Places of Funda
23.2(=oz
'
+ 476.2(=Ai')
mental Stars. For the above example, n=.67 and ('" +±A')46.0.The correction taken from table VI is ±25 in units of the last decimal place. This, added to n®1,, which is equal to 334:'6, gives 337.1 as the interpolated difference. The final value is +2201'03'9, as by the first solution. Care should be taken that the double second difference and the figure taken from the table be in the proper decimal place.
CHAPTER 3 ASTRONOMIC OBSERVATION COMPUTATIONS Section I. CONVERSION OF TIME 10. Kinds of Time The geodetic computer will be concerned with three kinds of time in astronomical and solar computations. They are apparent sidereal, mean solar, and apparent solar. a. Apparent Sidereal. Apparent sidereal time is generally used in astronomical computations. Various expressions of sidereal time will confront the computer and the most common are listed below. (1) Greenwich Sidereal Time (GST) is apparent sidereal time at the zero meridian of longitude near Greenwich, England. Greenwich Sidereal Time is zero hours at the instant of upper transit of the Greenwich meridian (0°X) by the apparent motion of the vernal equinox. (2) Local Sidereal Time is sidereal time at the local meridian, e.g., the meridian of a survey station, and is zero hours at the instant of upper transit of the local meridian by the apparent motion of the vernal equinox. (3) Mean Sidereal Time is not used in astronomical computations. b. Mean Solar. The mean solar day is measured by the fictitious mean sun between successive meridian passages. The solar year is identical in length to the sidereal year, but due to the apparent movement of the sun, it contains 1 day less. The mean solar day is therefore about 3 minutes and 56 seconds longer than the sidereal day. Mean solar time is that used in everyday life. (1) Local Mean Time (LMT) is mean solar time at the local meridian and is the hour angle of the mean sun measured westward from the local meridian. Local Mean Time is 1200 hours at the instant
of upper transit by the mean sun across the local meridian. (2) Standard time is mean solar time at an adopted central meridian for a 15° wide time zone. In the United States, for example, the central meridians of the time zones are 750, 900, 1050, and 1200
West of Greenwich corresponding to Eastern (EST), Central (CST), Mountain (MST), and Pacific (PST) time zones respectively. For any particular time zone the standard time is 1200 hours at the instant of upper transit by the mean sun across the central meridian of the zone. (3) Daylight Saving Time is standard time plus one hour adopted for the general convenience of the public during the months of the year having the longest period of daylight hours, i.e., in the United States from April to October. (4) Universal Time (UT) is mean solar time at the 0 ° meridian, and corresponds essentially to Greenwich civil time (GCT). There are three categories of Universal Time called UTO, UT1 and UT2. (a) UTO is mean solar time determined astronomically by individual observatories and referenced to the Greenwich meridian by application of difference in longitude. UTO is not corrected for polar motion. (b) UT1 is obtained by applying the correction for polar motion to the uncorrected Universal Time (UTO) by the observatory. The correction to the JT2 signal to obtain UT1 and UTO is published in Time Service
publications of the major observatories. UTi is equal to UT2 minus S, where S is the extrapolated seasonal variation in speed of rotation of the earth. (c) UT2 is Universal Time (UTO) corrected' for polar motion and for extrapolated seasonal variation in speed of rotation of the earth. Time service bulletins of the major observatories publish the correction to be applied to the time signal in order to obtain UT2. c. Apparent Solar. Apparent solar time is kept by the actual sun. An apparent solar day is the interval between two successive meridian passages of the sun, and varies in length by about 30 minutes during the year, due to irregular apparent motion of the sun. Apparent time is necessary in computing some observations on the sun. d. Ephemeris. Ephemeris Time (ET) is the independent variable in the gravitational theories of the Sun, Moon, and planets. If it is desired to convert Ephemeris Time to Universal Time, the
following relationship may be used: UT=ETAT. AT is the amount ET is ahead of UT and its value is published in the American Ephemeris and Nautical Almanac.
11. Conversion of Time It is frequently required to convert one kind of time to another. This is done by the following processes: a. To find the sidereal time of a given mean solar time (fig. 2), use the tables in part I of the American Ephemeris or table II in the Apparent Places of Fundamental Stars and DA Form 1900 (Conversion of Mean Time to Sidereal Time). These tables give the apparent sidereal. time corresponding to oh, Universal Time, for each day of the year. This is the mean solar time of the beginning of the day (midnight) at the Greenwich meridian. (1) Find the Universal Time (UT, also called GCT) and date by adding algebraically the longitude in hours, minutes, and
PROJECT. 221 1

LOCATION
CONVERSION OF MEAN TIME TO SIDEREAL TIME
32
(TM
MARYLAND
A
DATE LONGITUDE
1. Recorded
INc.
9 July /963 VV
77
04
20.628
HOURS
MIN.
SECONDS
2
07
52.093
0
0o
03.100
sLCa1std. time
2. .(Wetseh (Chronometer) correction (F, S+) 3
5237)
ORGANIZATION

ted {.Local std. time 5 .
HOURS
MIN.
SECONDS
HOURS
SECONDS
MIN.
HOURS
MIN.
SECONDS
7 48.993
1
4. Longitude or time zone difference (W+, E)
5
00
00.000
5.1 Universal time (UT) (3+4) /0 Jol /963
2
07
48.993
6. Sidereal time of OhUT for Greenwich date
9
0
47.67/
0
00
20.9
1
/6
556/
5
a8
7. Corertionfor sidereal gain + 8. Greenwich sidereal time (GST) (5+6+7)
9. Longitude(hr., min., sec.) (W+, E) +
/7.375
10. Local sidereal time (LST) (89) 16 COMPUTED BY
DATE
WR..am
DADAI
FEB 57 PORM
08,40.186.
m..

AAAS
1®oA
/3 Jul
CHECKED BY
/963
DATE
.

AMS
'4 July /963
:157 0420017 PRINTING OFFICE U.S. GOVRNMENT
Figure 2.
Conversion of Mean Time to Sidereal Time (DA Form 1900).
seconds of the place from Greenwich to the given mean time. If the given time is a standard time, add the number of hours corresponding to the time zone of the place. Longitudes and times west of Greenwich are positive, and east of Greenwich are negative. If the sum is more than 24 hours, subtract 24 hours and add 1 day to the date. (2) Enter the table for the Greenwich date and find the sidereal time of Oh UT from the table in the American Ephemeris, or in table II of the Apparent Places of Fundamental Stars. This term is also known as RAMS+
12h
(right ascension
of mean sun). The sun's right ascension is measured from the upper meridian, while the beginning of the day is referred to the lower meridian. Hence, it is necessary to add 12 hours to the sun's RA. (3) Since the sidereal units are shorter than mean time units, the sidereal time will constantly gain with respect to mean time, and a correction for this must be applied to the interval between Oh UT and the UT of the observation. This is found in table IX, American Ephemeris, or table III, Apparent Places of Fundamental Stars. The tabular differences are minutes of mean time. An auxiliary listing in the right hand column of either table gives the correction for additional seconds in the mean time interval. (4) Add the UT found in (1) above, the sidereal time of Oh from (2) above, and the total correction from (3) above. This gives the Greenwich sidereal time (GST) of the given mean time. (5) Subtract the longitude of the place from the GST to obtain the local sidereal time (LST) of the given mean time. b. To find the local mean time (LMT) of a given sidereal time (fig. 3), use tables as noted and DA Form 1901 (Conversion of Sidereal Time To Mean Time). (1) Add the longitude of the place to the local sidereal time to obtain the GST. (2) Subtract from the GST the sidereal time
of Oh UT for the date to obtain the sidereal interval since Oh UT. (3) Subtract the correction, sidereal to mean solar time for this interval from table VIII, American Ephemeris, or table IV, Apparent Places of Fundamental Stars. This gives the UT. (4) Subtract the longitude of the place from the UT to obtain the local mean time, or subtract the time zone correction to obtain local standard time. c. To find the apparent solar time of a given mean time (fig. 4), use tables in part I of the American Ephemeris, or any other solar ephemeris, and DA Form 1902 (Conversion of Mean Time to Apparent Time). (1) Add the longitude to the given local mean time to obtain UT. (2) Take from the table the equation of time for the date. This value applies to Oh UT. Note proper algebraic sign. (3) Multiply the daily change of the equation of time by the fraction of a day elapsed since 0h UT. (4) Add algebraically the amounts from (2) and (3) above. The sum is the equation of time for the given time. (5) Add algebraically the equation of time to the local mean time to obtain the local apparent time. d. To find the local mean time of a given local apparent time (fig. 5), use the tables. The tables are made for mean time units, and since apparent time is given, a first approximation must be made for obtaining the equation of time. (1) Find the Greenwich apparent time (GAT) by means of the longitude as above. This will seldom differ from the UT by more than 0.01 day. (2) Subtract the equation of time for Oh UT corrected for the elapsed interval in apparent time for the GAT. This gives a close approximation of the UT. (3) Recompute the equationl of time for the elapsed interval of mean time. (4) Subtract this value from the LAT to obtain the LMT. (5) The equation of time is the same at any given instant for all points in the world.
991CONVERSION
PROJECT
OF SIDEREAL TIME TO MEAN TIME
22/32(TM 4A
LOCATI ON
5237)
YZIDORGANIZATION
9
LOCAL DATE
LONGITUDE
w
77
0
HOURS
/7
1. Recorded local sidereal time (LST)
2. Watch correction (F, S+)
4. Longitude (W+, E)
5. Greenwich sidereal time (GST) (3 +4)
04 MIN.
SECONDS
/7
08
43.294
+S
08
/7.37,5
22
/7
o.
7. Sidereal interval since 0° UT (56)
03
SECONDS
HOURS
MIN.
SECONDS
HOURMIN. __
SECONDS ____
9
08
47571/0 51.0/4
/2
Jul
1 963
9 July /963
___
9
08'S
09.655____
30.8 34
o
00 31.jj481_______ 26
74
9. Universal time (UT) (78)
10. Ebeesgitd di e or (time zone)
03
/1
38/74
s
00
00.000
07
42.24
//
38. /74
5
11. Lzzeft Ma. 4 _ (LT T 9 10 Local std.time (L STD T)(  )
22
*If Greenwich date is doubtful use local date for trial computation.
FORM
MIN.
r
0
__
04
8. Correction for mean time lag
DaA
__________ /
_17 _00_66
/9
~~.odsww4.a
HOURS
686___________
03.602
22
______ 0n
20.6~28________
00
6. GST of 0' UT for Greenwich date*
COMPUTED BY
________ 0
0
O

3. Corrected LST
July /963

AIMTE,3
At step (11) determine correct Greenwiich (late and if necessary rew ork comuputation from step (6) to end.
DAECHECKED
July
/963?
BY
G. i.
DATE Teo~,,

/4
M
July 1963
.GVRMNRNIGOFC
90FBD /U
Figure 3.
Conversion of Sidereal Time to Mean Time (DA Form 1901).
970008
PROJECT
CONVERSION OF MEAN TIME TO APPARENT TIME I(TM 5237)
2  2 / 32
LOCATION
ORGANIZATION
/8
DATE
W
LONGITUDE
MA
Y
/963
83
________
I
0
48
24
0
HOURS
MIN.
ii
56
17.8
00
03.1
56
/4.7
35
/3.6
17
31
28.3
0
03
41.0
0
00
02.0
SECONDS
0
__
____
HOURS
HIN.
r
SECONDS
HOURS
MSIN.
i
0
SECONDS
HOURS
IF
/
HIN.
SECONDS
1. Recorded local mean time (LMT) ____ ____ ____
2.
Weatch correction (F,
____
S+)
__
_
_
_
_
________________
3. Corrected local mean time (LMT) _____ ____
____
__
II
_
4. Longitude (TIME)
+5 +
_____________
___
_
__
5. Universal time (UT) (3+4) _____________
6. Equation of time for O1°Greenwich date
7. Variation of equation of time for Greenwich date

8. Fraction of day elapsed (5) _ 24 ___ ____
____
0.73
W
0
00
01.5
0
0
3~
1/
S9
___
9. Correction to equation of time (7X8)

10. Corrected equation of time (6+9) _____________
39.5_____
11. Local apparent time (LAT) (3±10) _______ _______
___
COMPUTED BY
" DAI FORM FEB 57190
54.2_ DATE
u, 
95
A
20 MAY 196.16
_
_
_
_
_
_
_
__
_
CHECKED BY
_
DT
.T.rn
s

q1I
2/ MAY 1.96 3 U.S.GOVERNMENT PRINTING OFFICE: 19570{20810
Figure 4.
Conversion of M1/ean Time to Apparent Time (DA Form 1902).
CONVERSION
OF
APPARENT TIME TO MEAN TIME
Loca2 Date
18 May 63
LocaL Appatent Time (LAT)
12h0d100b
Longitude o6 Station, degnee
Long~ctude of Sttion, houtha 
8 3 48 +
24
W
5h35 13.6
Recotded LAT2hm0 Watch Cotec.tion 00 Colvreated LAT rr
Longitude Vi . enence'
Gkeenwi ch

+
Appaxent Time (GA)  Equation of Timem 18 May (&%om Tabte)+ Covkec/t.Zon Jon In.tehvaL= 2.00 (17.6/24) 
5 35 13.6 13
3 41.0 0 01.5
Co'uleted Equation of Timerr97F
App'ox.Lmacte
tln&,veu1a
Tcmerrrrr 17 31 34.1
CoAuection Jon Inte'wa= 2.00(17,53/24)  F Znae CouLeeted Equation of Time+
0 01.5
3 39.5
Co'uected LAT 12 00 00.0 Equation o Timer+ 3 39.*5 LocaL Mean Time (LMT = LATEq, ob Time)TF520 Figure 5.
Conversion of apparent time to mean time.
Section II. COMPUTATION OF AZIMUTH 12. Method of Computation a. The observation of astronomic azimuth consists of observing the angle between a mark oil the earth's surface and a star or the sun. The computation consists of calculating the azimuth of the celestial body at the time of observation, then subtracting the measured angle from this value to obtain the azimuth of the mark. b. The calculation of the azimuth of the star or sun involves the solution of the spherical triangle whose vertices are the pole, the observer's zenith, and the body observed. This triangle is known as the astronomic triangle or the PZS triangle (fig. 6). Since the body is apparently moving, the time of the observation must be known except in some special cases. c. If the angles of a spherical triangle are designated A, B, and C, and the sides opposite thei as a, b, and c, just as is customary in plazie trigo
nometry, a fundamental formula can be derived for the solution of the triangle when any three of its elements are known. cos acos
b cos c+sin b sin c cos A
All other formulas for the solution of the spherical triangle may be derived from this fundamental equation.
d. In the astronomic triangle, the angle at the zenith, between the pole and the celestial body, is the azimuth of the body, hereafter designated A. The angle at the pole, between the zenith and the body, is the hour angle, designated t. The angle at the star or sun, usually denoted by q, is the parallactic angle. The parallactic angle will seldom be used in astronomic computations. e. The side of the triangle opposite the azimuth angle, A, is the arc of the hour circle between
(900)
ZENITH DISTANCE()
(90d)
OZ
Q ILIW
/
Figure 6.
PZAS triangle.
the pole and the star (or sun) and is known as the polar distance (p) or codeclination (90 °0). It is obtained by subtracting the star's declination from 900. In most cases this subtraction need not be made, since the cofunction of the declination itself can be used instead. The side opposite the hour angle at the pole is the are of the great circle between the zenith and the star. This is known as the star's zenith distance, designated by the Greek letter . The zenith distance is either observed directly, or its complement, the altitude (h), is subtracted from 900. The cofunction of the altitude is frequently used in place of the required function of the zenith distance. The third side, lying opposite the parallactic angle, is the arc of the observer's meridian between the pole and the zenith. It is obtained by subtracting the observer's latitude from 900, and is sometimes known as the colatitude (9000). In nearly all practical formulas, the cofunction of the latitude is used. Jf. The hour angle (t) is obtained from the recorded time of the observation and the right ascension of the body observed. In the case of a star, the local sidereal time (LST) is required. This may be found by observing the altitude, or the time of transit across the meridian of known stars; or from radio time signals, provided the longitude of the station is well known. The hour angle (t) equals the LST minus the right ascension. The hour angle is measured westward from the upper meridian from 0" to 24". For convenience, the t angle is limited to the first 2 quadrants (0h to 12") on the computing forms, and is considered as measured both west and east from the upper meridian. The latter direction is considered negative. Should the hour angle, found by subtracting the right ascension from the LST, fall between 12" and 24 h, it is subtracted from 24 h to obtain the negative t angle. g. The right ascensions and declinations of the stars are given in the Apparent Places of Fundamental Stars. These are given at intervals of Universal Time (UT). Hence, the local observed time must be converted to UT before taking out the right ascensions and declinations, as explained in paragraph 9. The right ascension and declination of the sun are given in the American Ephemeris and many other publications. h. Since only three parts of the astronomic triangle are required in order to compute the azimuth, different combinations may be observed in the field. Thus, we may be given (1) the lati757381 0

65

2
tude and declination, and observe the altitude; (2) the latitude and declination, and observe (indirectly) the hour angle; (3) the declination, and observe the altitude and hour angle. There are also special cases, such as observations at elongation or culmination, when the star's position is found by trial without knowing the time. i. The computer frequently must apply some corrections to observed values. Since the star (or sun) is generally observed at considerable altitude, an error is introduced in projecting its direction downward to the horizon whenever the axis of the telescope is not truly horizontal. For the inclinations involved, the correction, c, isC"=i" tan h
in which c and i are in seconds of arc; i is the inclination of the telescope axis as determined by the readings of the plate level or a striding level; and h is the altitude of the star. This correction is not required in thirdorder computations. (1) Computing the inclination correction. In order to compute the inclination, i, the sensitivity value of the level bubble in seconds of arc per division, and the displacement of the bubble in divisions from its level position must be known. The scale should be read at both the left and right hand ends of the bubble on both the direct and reversed pointings on the star. (a) If the scale reads continuously from one end of the tube to the other, the record appears asDirect
Reversed


Left
Right
07.5 17.7
16.8 08.5
10.2 08.3 +01. 9
The final figure +01.9 is the inclination factor, and is actually four times the mean displacement of the bubble for the two pointings. It is found as follows: The smaller value is subtracted from the larger in each of the columns, indicating readings taken at the left and right ends of the bubble. The difference in the right hand column is then subtracted algebraically from that in the left. That is, if left is greater than right, the inclination is positive; if right is greater than left, 17
it is negative. In case a striding level is used, it should be reversed on the axis during each pointing, D and R. A record identical to the above, and computed in the same manner will be obtained for each of the pointings. The final inclination factor is then the algebraic mean of the inclinations of the two pointings. (b) Occasionally, a record will be made by reading the scale outward in both, directions from its middle, asDirect__________ Reversed 
Left
Right
05.0 05. 2
04.3 04. 0
for varying temperatures and pressures. To use this table, proceed as follows: (a) Enter the table at the apparent zenith distance of the object, and by interpolation, find the mean refraction. This value applies to a standard temperature of 100 C. (500 F.) and a barometric pressure of 760 millimeters (29.9 inches) of mercury. (b) From the table of corrections for temperatures other than 500 F., determine the multiplier (CT) of the mean refraction for the observed temperature. (c) From the table of corrections for barometric pressures other than 29.9 inches, determine the multiplier (CB) of the mean refraction for the observed barometric pressure. (d) Find the refraction correction, r, by multiplying together the mean refraction and the two factors.
10.2 08.3 +01. 9
In this case, the columns are added; then right is subtracted from left, as before. When observing on Polaris, some observers may mark the columns west and east instead of left and right. The inclination factor must then be multiplied by the level factor, d tan h, in which d is the value of a division of the level scale in seconds of are, and h is the altitude of the object: observed. The value d for the instrument used must be furnished the computer. The final figure, iX tan h is the correction which must be added algebraically to the circle reading taken on the star (or sun). (2) Correctionfor refraction. An inclined ray of light is subject to bending in passing through the earth's atmosphere, as a result of which all observed objects appear too high. This bending of the ray is known as refraction, and varies in amount with the angle the light ray makes with the vertical, the temperature of the air, the barometric pressure, and to a lesser degree, the relative humidity. The humidity can be disregarded in all work unless specifically required. Table V, appendix III is used in finding the mean astronomic refraction and the corrections to be applied to the mean
r=rmXCBXCT
(e) The computer should judge whether the class of observation requires the corrections for nonstandard atmosphere.
13. Observation on a Close Circumpolar Star at Elongation a. The stars ordinarily used for observation on a close circumpolar star at elongation are Polaris and 51 H.Cephei in the northern hemisphere, and a Octantis in the southern hemisphere. The reduction formula is: sin AE
cos 0 O
or
sin AE=sin p sec
o
in which p is the polar distance (90 ) of the star. The procedure (fig. 7) is as follows: (1) Obtain 0 for the date from the Apparent Places of Fundamental Stars. (,2) Solve the formula. (3) Apply correction for diurnal aberration, if warranted by precision desired. cos A cos q Diurnal aberration = 0.32 cosA cos h plus in the northern, and minus in the southern hemisphere. (4) Subtract the observed angle, mark to
star, correcting reading of circle on the star for inclination. (5) The azimuth of the star is measured east or west from the meridian, according to whether eastern or western elongation was observed. The field notes should state which, or at least record a time from which it can be determined. (6) The above formula is exact and may be applied to any star at elongation (method 1, fig. 7). b. The approximate formula for close circumpolar stars only is:
and along the top are the azimuths of Polaris from north. In most cases, double interpolation is required to extract the desired value. (2) The correction will always reduce, numerically, the angle between the position of elongation and the meridian. The mean correction for the star is applied to the azimuth of elongation before subtracting the angle, mark to star. e. For the determination of the Local Hour Angle at Elongation, the formulas aretan 4
cos t=tantan
A '=p" sec . This formula is adequate for computing most azimuths observed by this method. AE and p are stated in seconds of arc. 5 is obtained as above and subtracted from 900 to obtain p (method 2, fig. 7). c. In table II, AE&NA, the azimuth of Polaris is obtained by determining the product of the quantity (bo+bl+b 2) and the secant of the latitude. The factor "bo" varies with the local sidereal time, the factor "bi" varies with the latitude and the factor "b 2" varies with the month of the year. All values must be interpolated as accurately as possible (method 3, fig. 7). d. In observations on a close circumpolar star near its point of elongation, it is possible to obtain one direct and one reversed pointing so near to the point of elongation that the observations may be computed as if made at the instant of elongation. In most cases, additional pointings are needed, particularly if a repeating theodolite is being used, since the required pointings cannot be obtained within the time limit. Pointings at some distance from elongation may be easily reduced if accurate time is available. The formulas given in a above are used for the computations. The time of each pointing on the star should be recorded. The formula for reduction to the instant of elongation is: A'=405,000 sin 1" tan AE(TT)
2
in which AE is the azimuth of elongation, T is the observed time, and TE the time of elongation, TTE being expressed in minutes of sidereal time. (1) The correction or reduction to elongation for Polaris can be obtained from table VI in appendix III. Along the left margin are the minutes of sidereal time
o
cot
S.
The body is on the meridian at the instant when the local sidereal time and the right ascension of the body are equal. The body is at western elongation at this instant plus the time interval represented by te. Eastern elongation takes place at culmination minus the time interval represented by te or plus the quantity (24hte). Example: If the observer's latitude is 38°39'33'8 and the declination of Polaris is 89005'15'2, cos to is equal to the product of tan 38°39'33'8 89°05'15'2 = 0.79998800(0.01592651) = X cot 0.01274102.
te= 89016'12"=5h57m04.8S.
Polaris
will be on the meridian when its right ascension and the local sidereal time are equal, or at Western elongation 1 h0 8 m5 8 19 local sidereal time. comes 5 h5 7 m 0 4 .8s later and eastern elongation 5" 5 7 m0 4 .8S earlier. The conversion from sidereal
to civil, or standard, time is explained in paragraph 11b. 14. Observations on a Close Circumpolar Star at Any Hour Angle a. The advantages of this method are that the star may be observed at any time it is visible and an unlimited number of observations may be taken. The two common methods of determining the azimuth are known as the direction method and the hour angle method. Both methods use the same basic formulas which are as follows: tan A=
s t cos ( tan 5sin
sin 0 cos tcos sin t
. cos
0 tan 5
tan A=cot ~ sec 0 sin t (a1a
t
COMPUTATION/ OF AZIMUTH USING STAR AT ELONG AT/ON
Station :
9( 380 39'33."8 A78°44' 373
Tap
Posit&on o1 Polaris : RA :
S/n
52 SI N AE
3
.8)
AE
=
=#0/
o/0'
c
06.7
MarA t'o Star
/58 32 /6.1
Astrono,c Az im u th
202037'50.6
Method O) Solutior by 7'arrrna/a AE =/O$"sc! 6S (89o05'/5S'2) 99o° (89°05'/52)= 0 054'44.
=3284.'8
=/ 2806 /735
AE 12806 /735 (3284.8) Mark to Star Astrono.*nc Azimutht Metfho cl Q LST
o
0.0/S 92449 os$ 0.78087338 0.0203 9318
AE
Sec 0 (38°39133."8)
/963
Jul/y
Time : 0/ h08 "58.~9 L stdT
(89 °05' /52 ) (38°39" 3
9
W
Solut7ion7 by f~ormen4/a
Me t hod O: Co~s 6 Cos 0
CIRCUMPOLAR
Date:
N
01"57"°5515
S6: t 89° 04,
A
=4206.15
=
0/'0'06.6
/58* 32' /6.1 202' 37' 50'1,
Solution~by Table Zi, Arnerican Ephemeris 38° 39' 33.8
Date
.
4,
9
July /963 =~4'
,61= b2=
rot~
( 38 °40 )
Sec 0
.2806 /735
0.0 0.3
= + 54.'8
AE =/2806 /735 (54.'8) = 70.1778 = a 0/*/0"/C7 Mark to Stqr /580 32' /6. Astror'o/',c. Azirmath 2020° 375S4. Figure 7.
Computation of azimuth using a circumpolar star at elongation.
Where: A= Azimuth of star, as reckoned from the observer's meridian in a clockwise direction a=cot 5 tan
4 cos t
t=The local hour angle, reckoned westward from upper culmination. Tables for log (a)
are found in TM 5236.
b. The direction method is named for the type of theodolite used in the observations; this method is the one most commonly used for high order astronomic azimuths. Computation can be made with natural functions, or logarithmic functions (fig. 8). (1) Correct the mean recorded times of each position or set for the chronometer error. When a sidereal chronometer is used, this will give the local sidereal time (LST), the chronometer correction being obtained from observations taken at the station, or from radio time signals and the station longitude. If a meantime chronometer or watch is used, it is customary to obtain the correction to local standard time. This is then converted to UT, thence to GST, and by applying the longitude, to LST. (2) Obtain the right ascension and declination of the star from one of the ephemerides for the date and UT of observation, using the mean epoch of a series of observations which should not extend beyond a period of 4 hours. (3) Subtract the star's right ascension from LST to obtain the hour angle of the star (t) and convert to units of arc. (4) Solve the formula for A, the azimuth of the star. (5) Determine the. correction for curvature (table VII) when applicable and apply to A to give the correct azimuth of star. This correction numerically decreases the value of A. (6) Determine the level correction and correct the circle readings on the star. If the altitude is not observed, it may be computed in the case of Polaris from Table II, AE&NA, or for any star from the formulas: sin h=sin
4 sin 8+ cos 4 cos 5 cos t
cos h=co s S sin t
sin A
cos 6 sin t
tan A cos A
Computation of h to the nearest minute of are is sufficient. (7) Subtract the corrected reading on the star from the circle reading on the mark. (8) Add algebraically the corrected azimuth of the star from North and 1804to (7) to obtain the azimuth of the mark from South. (9) Abstract the results of all positions or sets, apply the rules for rejection, and take the mean of the acceptable observations. Record this information on DA Form 1962 (fig. 9). (10) Determine the probable error of observation. (11) Apply correction for diurnal aberration. (12) Apply correction for elevation of mark by formula: C=+0.000109 h cos2 4 sin 2A where h=elevation of mark. An accurate sea level reduction chart may be used. (13) When the x and y of the instantaneous north pole are known for a given date, the correction to be applied to the astronomic azimuth to reference the azimuth to the mean pole is computed by the formula: Aa=(x sin Xy cos X) sec 4 wherein West longitude is considered positive. c. The hour angle method normally is used for lower order azimuths when the time of observation is less accurately determined. Solution can be made using natural function or logarithmic functions (fig. 10).
15. Observations on EastWest Stars a. Basic Considerations. When high order astronomic azimuths are needed and close circumpolar stars cannot be seen, EastWest stars which reach elongation at approximately 150 altitude may be observed between approximately 7% ° and 22% altitude. At latitudes of less than 20, stars crossing the prime vertical at approximately 300 altitude may be observed. The declination of the observed stars should be four to five times the observer's latitude for the elongating stars and approximately onehalf the latitude for stars crossing the prime vertical. For lower order azimuths, the altitudes of east
PROJECT
LOCATION
BY DIRECTION METHOD
AZIMUTH
A LAD ORGANIZATION
US
MARK
LATITUDE
MAP
MoS
CHRON. NR.
/2460
16.462
T3 63010
o0
Sidereal time
9
659 /88
/
356 450
(star)
8 02 33.8
HA(t) of star (time)
/20 38 270
32.59
89 05
Decl. (a) of star
Sin0
628 65203
Constants for star
.
Sin t
+
29.6
08
/0 06 05.4 /
G. CIVIL DAY
FE..
APR.///
4
/0 33 66.4 08. 29. 7 /0
25 26.7
/
6,6 450
56 45.0
8 0920.4 /22 20 06.0
89
NP(AMS /958)
OBSERVER
3
/4 35.0
/0
295
Chronometer correction
STATION
08 28.86
2
/0 07 48.3
Chronometer reading
t of star (arc)
5 (SIGHAI ,
29.953
/
Date19 63 , position
RA(a) of POLARIS
0267
67S
LEVEL VALUE (d) ECC. (INST.)
INST. (NR.)
(TM 5237) LONGITUDE MX
0~)
/0 40 /6.7 08 297 47Z0
3/
lo
450
/ 56
8 28 41.7 8 35 02.0 /27 /0 255 /28 45 30.0
32.59 89 05 32.59 89 05 32.59 ~ Ten i Cos 0Tan a .777 68672 .63.122 643 49.089 64119 05
Cos
.860 37903 + .844 93526 1.796 80684
.77979344
50o96472 . 534 86859  .604 23412 626o03690
Cos t
Sin m006 t
. 32039547
Cos tanisin ocos t
494/0 03666 4 9425 88 741 4946949419 49483 200"6
sin t Tan A usNtuaaiu*..st
0
A (Az. of star from N.)t
.0/741304
5S9 513
Difi. in time between D. & R.
Curvature
/
. 336 24622
379 85300 . 393 5937
.
.0/61/0703
.0/709499
0
58
458
0 55
22.0
.0/575875
0 54
322
3S/
/0.2
//9
correction
Altitude of star (h)
380 28'
380
2/44~
+
1.2
380
1.283
1.284
d tan h(level factor)
Inclination
69"
+ 2.3
23" 53' /.280
+415
380 22' 45'
1.279 +3.3
+01.5
Circe reading on Mark
+03.0 t+0/.9 t04.2 170 35 09.8 /8 / 36 4 7/ 203 4/ 12.4 215 40 39.2_ /70 35 /1.3 /8/ 36 50.1 203 41 /4.3 2/5 40 43.4 00 00 /1. 3 // 00 44.6 33 0/ 47 0 44 .59 592
Diff. (Mark minus star)
189
Level correction Circle reading on star
Core. reading on star
Corr. Az. of star, from N. t :">;:
.?:;:
UT X
H; as.
var.
MIN.
p
Has. ANGLE
VET ANGLE
__
Logarithms
2
LL ~~
2
14 22
ilL.
4
~
~
P
22 f 11L
o
___
16.
__
SEt.
MIN.
HiS..
ilL
59
~
2L
SEC.
MIN.
HaRs.
SEC.
j4
2
1
per hr
SET NRl 3
VEST. ANGLE
_L4L8
j~J
1
UT
_a
Has. ANGLE
io
Universal time (UT) Oh
SET NR 2
49L4
_
___
Mean time
isat
NR 1 VER. ANGLE
3O.ANGLE
_
it
I~a Sm/f
5gpt52
":.;
COMPARED (Time)
WATCH
DAEOBSERVER .StIp
STANDARD TIME (Meridian) 5
Logarithms
he
P,
Logarithms
Sece
LL
X2
2s.p 0 1 1.
2/
2
4 5q99997
(sum)
a042 15
5j)
2.
47
Sumo2'=log cos A/2
f966
92 Sum.2=lbg coonA/2
J4
A/2
..47LS711_2 2 I WUM q
A, (E or W)
g
Azimuth of S
4
A n l azim ,t auth k tto M ark
06 4Z j5j
Computations:Three sets are computed separately for check. Refraction and parallax from TM 5236. t unversl tme.(s TZC=imezonecorecton touniersl tme.Formulla TZCTim Crretiozoe S(O+h+p)
COMPUTED BY
®A ~E
DAT ~
1
5p
O
,
Su2=gcoAf
Mean true azimuth to :Mark
i
47Grid correction
1522Q 523 Z 22404
A13 9_8True 5159 111 3 5
s
3.~2~Id4 (sum)
m)
Grid azimuth to Mark
M agnetic a n t cdeclination a i u h t E (a ),k_W (+{) 8 Correction = UT X variation per day I 24 If b is (+), p =900  a; if bis (), p 900 + S  p) = arithmetical difference,. always positive. is cos ,s A  /cos s cos (sp) Vcos 0cosh A = astronomic azimuth east or west of north.
n~fHECKED Ca... BY
71908 QLogarithmic functions, DA Form 1908
Figure 13Continued.
/)/
DAsE 4 ,4
(1) The formulas for logarithmic computations are: 1A s2A tan1 A
2
cos s cos (sp) cos 4 cos h sin (sh)sin (sc) cos s cos (sp)
in which h=observed altitude corrected for refraction O=latitude of station p= polar distance of sun (9005) s=32(h++p). This first formula is simpler, while the second is slightly more precise. (2) The formula for machine computation is: cos A=sin
asin h sin 4 cos h cos 0
(3) The procedure of computation is as follows : (a) Compute each position by the direction method, or each set by the repetition method, separately. The observation may consist of a series of positions (D&R observations) or a single pointing on the mark followed by a series of direct pointings on the sun, and an equal number of reversed pointings on the sun, followed by a final pointing on the mark. (b) For each position or set, determine the mean of the recorded times and the horizontal and the vertical circle readings. For each position or set, the angle from the mark to the star is the difference between the mean circle reading on the mark and the mean reading on the star regardless of the number of repetitions. (c) Each mean vertical angle must be corrected for parallax and refraction. Tables of parallax (table VIII) and refraction (table V) are included in appendix III. For lower order observations the parallax may be neglected since it is never greater than 9". (d) Convert the local time of observation to Ephemeris time as explained in paragraph 10d. 757381 0  65  3
(e) Extract, from the AE&NA, the declination of the sun for the Ephemeris time of the observation. (f) The value of p is determined by subtracting the declination of the sun from 90 .. If the sun is south of the equator the declination is considered to be negative. The quantity (sp) is always considered positive. (g) Solve for the value of "A" by use of the formula. (h) Subtract the horizontal angle, mark to sun, from the computed azimuth of the sun to determine the azimuth from station to the mark. The final azimuth is the mean of all the azimuth values. (i) Either DA Form 1.907 or 1908 may be used for this computation. b. Hour Angle Method. The hour angle method (fig. 14) is preferred when the time of the observation is accurately known. It is not greatly affected by errors in the latitude and declination, and should always be used when necessary to observe the sun near the meridian. (1) The formulas are: tan
11
2
(Aq)=
cos ()
1( tan
2
(A
tan A== 
sin (€ ) 1 s ( ) cot ! t cos (++) 2
q)=Sin
sin
()
z(4+)
sin t cos 0 tan sin
1 cot
0
1t 2
cos t
When A exceeds 450 from the meridian,
the last may be stated as follows:  cot A=
cos tan sin t
sin tan t
The algebraic signs of the functions may be maintained, or the following rules followed in the case of the first formula: (a) t is taken less than 1 2h and positive. If over 12 h, subtract from 2 4 h . (b) If t is less than 12 h, the azimuth is to
the west of the meridian. (c)
If greater
h,
it is to the east. 12 (Aq) and a(A+q) are taken out as
than
less than 90 ° . (d) When (46) is positive, add numerically the values of (A q) and (A q) to obtain the azimuth angle between
At
DIAGRAM
PROJECT
V, i in
I
ORGAN IZATION
LATI UDE ()
382'
IMARK 2 _AiS
02
S&nPERVE
77O
I
WATCH F49)
1
054
.
STANDARD TIME (Meridian) WATCH COMPARED (Tion)
SLOW (+)
7.1 DATE
IT HOUR ANGLE METHOD1 (TM 5237) (1))STATION
LONGITUDE
INSTRUMENT (Number and type)
CELESTIAL BODY(S)
:..5
IAZIMUTH I
LOCATION
TRUE NORTH
OBSERVER
Icf
WEATHER r1.1
SET NR 1
egan
MIN.
SET NR 2
TIuNE
HoRn. ANGLZ
Ti E
*Hos.
I
SEC.
a
Hits.
"
Mir.
SET NB 3
I
Horn. ANGLE
SEC.
j
Hag Is.
"
0
TINE
Hon. ANGLE
SEC.
"
I42.81s/A/4343/ SET NR1I
8ET NR3
SET NR2
Has. MIN. SEC. HRs. MIN. SEC. Has. MIN. SEC. 1
_15 _4Z
Mean time of observation
2 Watch correction
f

5
For star
Oh
Greenwich EQT 0jas;:q~
7 (5+6) correct EQT
use factors

#
.
21
a
248
30
t a
_
al
lfhuj
8 (4+7) GATacA fL=2L.Q
in brackets

4 _8
28
6 UT X var. EQT per hour in
[enclosed
.5.
I&L1
f


3
2o. 142
4 UT of observation (1 +2+3)
at. ua
.i"
+.

3 Time Zone Correction (TZC)
SUN OBSERVATION
ISE SL_
& .
9 GHA in time (GAT12h)4a6] 10 GHA in arc
11~Longitude, West
(),
12! LHA (10+11)t t
~
NENR
(
Lat.
.0 )
~aT~s 0001&
I
.)
I4J 7I4Rl~J &21I/AS
Man:zimuth to Mark
Gi
Mgi i
azimuth to Mark
E(), W(+)
.Magnticdeclination
Ssin
t cos* tan asin
~
A 4me W)
1 asJf1Z X2 ~ 
Mark to
8S
AX /4
Tr. Az. to Mark
a
COMPUTED BY
DAI
*cos
t
If LHA is greater than 1800, subtract from 3600 and reverse sign.
Obtaiin a from Ephemeris.
Azimuth of 8 L
.
correction
Sec. (8 Cost
i
Mean true azimuth to Mark
fGrid
105,2410l
be._a_______
5s]3~
oi~.~
I~ _______ & a~. j a~ _________________________________________________
SETNRS
NR
z'.
f
Check signs and quadrants by use of sketch.
vO
DATE
_____________
ss
1g4,&
CHECKED BY
&.CnDT c.I.
1905
7O
Figure 14.
O Natural functions, 1)A Form 1905 Computation of azimuth using the sun at any hour angle
ID
uI
PR~OJECT
i
LONGITUDE
4
L9842'3
7
STANDARD TIME (Meridian)
750 l
CELESTIAL BODY(S)
WATCH
.)
WEATHER
OBSERVER SET NR 2
HOlt
TIME Has. MIN.
SEC.
C, ele
C _. I R~rnnvet
A D .52 0
ANGLE P
TIME SEC,
HRS. MIN.
M
SET NR 3
HoR. ANGLE I
f
0
I
SET NR I
2 Watch correction
For star1

4 UT of observation (1I+2+}3)
W 4
[ ip
56 OUhT.GrX1 ee n wich Er±or Q T [e 8 +dl &jH
ubseratios [oservactons enclosed in brackets
7 (5+6) correct EQT (.~
ph .
.4~
10
GHA in
2. .2247 07 3 2247 10 12, 11
I 76
Log sin 4S
l)
152± l
77 dA
7
.SS.3044i.
2
139 22. 1 75~
O2
a
5714 [39_I913
,2
4 5
________
(Sum) log B
t

Q270
*from
9
tan A) (diff.) W)
Gridgazeim i uth az m to tMatrk o
6383B60
M1_2 .8(j
3600 and reverse sign.
Obtain d from Ephemeris.
9
A83540,1.07%624 013129.i Loi59 0521 24 L& 20 230 2&Q0 22 0 2
slt

If LHA Biso greater a 8sl than 1800, o subtract
Check signs and quadrants by use of sketch.
a12
S
COMPUTED BY,
CHECKD BYDATE
Tr. Az. to Mark 169
ar
Magnetic declination E(), W(l)
Z § 07I 2,5t% 0/40TnA
= 9JZIJ
Log WA) (B
_______
5710
(0Log
IRA
/41 Q4. LQ.
963126UW
5l7925,571

BA7
Azimuth of
6232

E
4 9 81
U 
Mean true azimuth to Mark
5
Log cos. 0
.'=
LQ
Grid correction
(SAM un ) log A
L~o(
4.p
,1
SET NR3
Log cos t
sinl
2L QL
120 .. j~f..* f
SET NR2
SET NR 1
______

~ ()
SEC.
__1.
4 & 14 .2
f.Od 44
LHA (10+11)t (1i.:
MIN.

arc
Longitude, West
i
SETNR 3
SEC. Has.
.2 20 5t
9 G11A in time (GAT12h)4.wo*43J
1. 22°47' O'
ANGLE f
1.~I 4 j 8L2
f
0
SETNR 2

8 (4+7) GAT.~p* JQ£
S
SEC.
SEC. Hal. I MIN.
_
3Time Zone Correction (TZC)
OR.
T~IME. HR.MI
HiRS.MIN.
1 Mean time of observation
SUN OB3SERVATION
WATCH COMPARED (Time)
SLOW (+)
&n,
DATE
SET NR 1
Log
(TM 5337)
OA/ 0846#J5
I NSTRU MENT (Number and type)
CSOlft'D
/hn r
Log tan
ANGLE METHOD
STATION
:A* Mk 1
A
HOUR
(Logarithmic)
LATI'U DE
AOS
MARK fJp ~
Vrain/a
ORGAN IZATI ON
BY
AZIMUTH
LOCATI ON
14w23
TrRUENORTH
functions, DA Form 1906
Figure 14Continued.
e
DATE
the meridian and the sun. If negative, subtract numerically. q is the parallactic angle, which cancels in the solution. (2) The second is the standard azimuth formula. (3) The procedure is as follows: (a) Compute each position or set separately, using the means of the times of pointings with the corresponding means of the horizontal circle readings. (b) Convert the local standard time of the observation to GAT. (c) Subtract 12h from GAT to obtain GHA, and convert GHA to units of arc. (d) Subtract the longitude of the station from GHA to obtain t, the local hour
angle of the sun. West longitudes are plus, east longitudes minus. If using rules to disregard signs, subtract t from 3600 when it is over 1800. (e) Take declination of sun from an ephemeris for date and UT of observation. (f) Apply formula. (g) Subtract angle, mark to sun. (4) All pointings on the sun refer to its center. The method of pointing on the sun should always be recorded by the field party. The computer should inspect this record and apply any corrections for semidiameter or other corrections that may be required. Ordinarily, opposite tangents will be observed so that a mean of the readings will refer to the center.
Section III. DETERMINATION OF LATITUDE 17. Relation of Latitude to Zenith Distance Nearly all observations for latitude, other than those using the astrolabe or zenith camera, consist of measuring the altitude of a celestial body when it is on or near the meridian. The latitude of a place may be defined as the altitude of the pole or the declination of the zenith at the place. Either can be obtained from the meridian altitude or zenith distance of a body of known declination. The equation is: where is the meridian zenith distance and 8 is the body's declination. is positive when the body is toward the equator from the zenith, negative when toward the pole. 0 and 6 are positive north of the equator, and negative south of it. 18. Latitude by the Altitude of a Circumpolar Star at Culmination a. Formula. Latitude, by this method, is detemined by use of the following formula: 4=h±p Where: h=the corrected altitude of the star. p=the polar distance (9006). p is negative if the star is between the zenith and the pole but is positive if beyond the pole. b. Procedure of Computation. The computation is as follows:
(1) Apply the correction for refraction to the observed altitude. (2) Extract, from the Apparent Places of the Fundamental Stars, the declination of the star for the date and time of the observation. (3) If the star was observed at upper culmination, subtract the value of p from the corrected altitude (90°). If the star was observed at lower culmination, the value of p must be added to the corrected altitude (fig. 15). (4) If the position of the star, during observation, was not recorded, the following may help in determining the position: (a) If the local sidereal time and the right ascension of the star are equal, the position is upper culmination. (b)If the local sidereal time is equal to RA+ 12h, the position is lower culmination.
19. Latitude From the Zenith Distance of a
Close Circumpolar Star at Any Hour Angle a. The Formula. Latitude, by this method, is determined by use of the following formula: O=hp cos t+p 2 sin 2 t cot "sin 1" 2 Sp 3 cos tsin2 tsin 1" 3 + p 4 sin 4 t cot 3 "sin 1".
Latidue Ay Altitude JPoo/arn: ato/rrnotion Local Dk le'/~oyS4 Eastern Sfondord ime ofoMar&10:40PM ries ohserved of lower culrrnoa OI'servedo/itudr of str 42°/7' 35
Temperature
45F
Barometric Pressure
30.4 inches
. Neron re,%ction. Tale V AppendiZT
(usezenifi distance ofgA;as arumentJ
2.Correction to refracian for Toperahirsamew Ta/eosl) 3.Correction to refraction for Pressure, (some Tohk as 2) 4 Corrected rerhiorn (if2,3)
S.Observeda/flt/oe ofstar 6. Corrected alitue ofstar (4) 1 Declnatin ofstar (for ate)
8 Poor Dislonce, p, Q Latitude
Figure 15.
(6t8)
0
0/'
t t
04'
0/ OL 0'
0/
06
42
17
35
42
/6
25
8?
02 57
58 02
430
/3'
3/U
ofstar (?o°Deciroti 40
Computation of latitude by altitude of a circumpolar star at culmination.
Where: h= corrected altitude p=polar distance (in seconds of arc) b. Procedureof Computation. The computation (fig. 16) is as follows: (1) Abstract, from the field records, the zenith distance and the sidereal time of observation. Correct the time for chronometer error. Apply the refraction, level, and collimation corrections to the observed zenith distance. (2) Abstract, from the Apparent Places of the Fundamental Stars, the declination and the right ascension of the star at the time of observation. Determine the local hour angle (t) by use of the formula t=LSTRA of star and the polar distance by use of the formula p=90°. Reduce the polar distance to seconds of are. (3) From the appropriate function tables, obtain sin t, cos t, sin 1" and cot .
(4) Apply the formula given above, the terms of which are the star's elevation above or below the pole. DA Form 2839, Latitude From Zenith Distance of Polaris, is designed for a logarithmic solution of this method, and can be used for other close circumpolar stars.
20. Latitude From the Meridian Zenith Distance of Any Star a. The Formula. Latitude is determined from the meridian zenith distance of any star by use of the formula:
Where: b= declination of star; positive if north of equator, negative if south. 1=meridian zenith distance of star corrected for refraction. r is positive when the star is on the opposite side of the zenith from the pole; negative when the star is between the zenith and the pole.
b. Procedure of Computation. The computation (fig. 17) is accomplished as follows: (1) Determine the refraction correction and apply it to the observed zenith distance to determine the corrected zenith distance. (2) Extract, from the Apparent Places of the Fundamental Stars, the declination of the star for the date and time of observation. (3) Should there be any confusion in algebraic signs, a diagram together with a roughly known value of the latitude will indicate the proper procedure. This confusion might occur in the case of a subpolar star observed at a very high latitude. (4) Substitute known values in the formula and solve for the latitude.
21. Latitude by Use of Table II, AE&NA a. The Formula. The following formula is used in determining latitude by use of table II, AE&NA: q ==h+(aofa,±al2). Where: h=observed altitude, corrected for refraction. ao, a, a 2 =factors in table II, AE&NA.
b. Procedure of Computation. The computation (fig. 18) is as follows: (1) Extract, from the field records, the mean observed altitude and the date and time of observation. Reduce the time to local sidereal time if necessary. (2) Determine the correction for refraction and apply it to the observed altitude to determine the corrected altitude. (3) Interpolate in table II, AE&NA, for the values of a0, at, and a2. (4) Substitute known values in the formula and solve for the latitude.
I
PROJECT OBSERVER
LATITUDE
FROM ZENITH DISTANCE OF POLARIS (211 5237)
UNIVERSAL DATE
1.S.12.1e8 Jul '63
I
ASST. OBSERVER
W
FE. IN ST.
LEVEL
WILD T4 No. 3744(

R.A.
"
.28
22
/6.95
LEVEL CORR.
+g
00. 98
/7
32
/7. 9
REFRACTION CORR
+
32. 01
01
S57
SP2/
COLLIMATION
+
.. 70
34 /8.8 2330° 34' 42.0o"
t
NO.
/
to
iS
HOUR ANGLE (t)
/.0239 G
9
/7.
32
2980 "
POSITION
a
0.
LOCAL SID. TIME
FACTOR
04977
CHRON.CORR.
BARD.
314 OF
IdREFRACTION
In
17
CHRON. TIME
TEMP.
R.A. G.
VALUE
h
NORTH
RECORDER
d/2 =
L
STATION
r28
23
01.64
900 001 00 00'"
3.16
LOGp
4957
a
9. 773 584 0  /0 3 290 0797
LOG COS t
LOG 1
9. 698 9700
LOG 0.5
p
89 o5 ° 5
7
2__LOGSINt
9.8112348  0
LOG COS t
9
LOG COTr
0.26733)0
2LOG SIN t
9,811
LOG SIN 1 '4.
46
1.
LOG II
_____
110/ ~
00
/
+
00. 00
sum
+
1 98/ .6(5 330.
0
r)61 62
#A_____
"
h

I
+
58.36
II

III
+
SLOG COT
o28 33

/0
02o
o
4
522
7644/0 10
LOG IV ______________
/0__00.0/
A=
231/0
624
S LOGOSIN i'
.
34
7735s8 10
14.665 98
4 LOG SIN t
IV
sum
10
9. 0%691 10
LOG 1/8
4 LOG p +.
h(90
9
LOG III
I3.34 111
"
2LOSI1"937150
/950 .20
+
_
44.70
/0.549 49
3 LOG p
6855749 10
'
9. 522 88 
LOG 1/3
2 LOG p
032 9914
1530
3284.70
______
10


____
___________
IV
SIGNS: I and III. are PLUS in the first and fourth quadrants. ii and IV are always PLUS.
S
COMPU~~~
DA
FORM
A
2839, 1 OCT 64
Figure 16.
Computation of
M
OTE
JC
Dju
6
CHECKED BY '(Y'63

MS
Juy 6
AM
latitude by altitude of a circumpolar star at any hour angle, DA Form 2839.
LoafA~e from a feridi/n Zenith Dt's/nce
COMPUTATION
OF
LATITUDE
POLARIS
OF
BY
(TABLE
Stfo/n: Iof,' Ce/estial Body: aune
(GCope/o)
DATE:
TIME:
09h
596 /95
POLARIS:
380
28
LOCAL SIDEREAL
Lon9,itde: 75*30'W Terre Zone leridian; 7S W
ALTITUDE
A /ttde fWraction
£cditude Figure 17.
hi)
ALTITUDE
(h)
380
59
28
590
/8"
/6"S
28
//.9
236
As'
/0S +t
06.9
7me
Declination
OF
TABLE Hf AE NA
Local Standard Time Time Zone Correction
Zenith Dstance (,)
AEA NA)
/1 APRIL /63
Date : 20 Feb 55
Universal
11)
ALTITUDE

le
440
/6'
23"
4
1
59
44
15
2
441
36il
57
27
/2'
5/"
5a
45 0f)
az
Computation of latitude from the meridian zenith distance of a star.
22. Latitude by Circummeridian Altitudes of a Star This is a more accurate method for observing latitude by means of the instruments ordinarily available. It is an extension of the method of meridian distance and the accuracy is increased by taking a greater number of pointings. A series of altitudes or zenith distances is observed on a star at recorded times during a period of a few minutes before and after transit. From the hour angle of the star at the time of each observation, the observed altitudes are reduced to the equivalent meridian altitude. The observing period should not exceed from 10 minutes before to 10 minutes after transit. The procedure is as follows: a. Reduce each observed altitude (or zenith distance) to the meridian (fig. 19). (1) Compute the LST of the pointing by correcting the recorded time for chronometer error. The numerical difference
=
00.0
LATITUDE
OF
Figure 18.
Computation of latitude by altitude of Polaris (table II, AE&NA).
STATION
38
57
04.0
between the LST and the star's right ascension is the hour angle (t). 4)scos e5 which 0 is a closely , (2) Let A= cossin  'in scaled map latitude, or is a trial value computed from an altitude taken near the meridian. should be the value nearest
2 sin2 Also, let m~
(t)
s (t) sin ' in which t is in seconds of arc (table IX.). (3) Then hmrh+Am or m,= Am; h or being the observed values, corrected for refraction, for the pointing being reduced. b. After all observations are reduced, take the mean of all consistent values. c. Apply formula: 0= +b. d. If for any reason it is necessary to reduce observations taken more than 10 minutes from the meridian, the formula, the meridian.
sin hm=sin h+ cos 0 cos
2 sin2
(t)
should be used. e. DA Form 2840, Field Computation of LatitudeCircummeridian Zenith Distances, is designed for a logarithmic solution of this method.
23. Precise Latitude by Circummeridian Altitudes of Stars a. Circummeridian altitudes can be used for the determination of precise latitude at latitudes
FIELD COMPUTATION OF LAIUE
ASR
PROJECT
_jSTATION
ASRORGNZT
LOCATIOND
LOAIO
/S.
0/SKO
/SQ//GUA
US'4MS
R. SLVERMO SER APPROX. POSITION
:N
2E
I ASST.
OBSERVER
a: f 03"28
~
SECCENTRICITY
O
/0.0
FROM.
STAR NO.
TO:
1/2
00 35
OQ
R. A. OF STAR
00
./.5
00
34
+o.
36 /3.4
00
36 456
51.0 00
34 51. 0
00
34 S1.0o
S3.1
0/ 22.4
1.5S4
0.17
0.
00
00 35 44.1
341.0
oo 36. 43.7,
0 o
54.6
7/6
3.7/
95862_
O~A
SIN
19
7.6
9.7 72 5056
LOG. Coss LOG.
08.6
3S
NLE()
I
LOG. COS
~
/9
M
(SOUth)
/4
13
00 35 42.2 100 36
.06.7

SIDEREAL TIME
HOR
feNi.0! 17______
POSITION NO.
CHRON. CORRECTION
_
_I J.M.LEW/SIG(0EF
_
6r°46"55.5
CHRON. RAN_
26.81
RECORDER
.,h
DIRECTION
AROM.
 o./ c
/200/
OBSER VER
IELEVATION
TEM
TCHRONOMETER
:4 No/0_37446
(CC.
121 AU6.6WPM
RGNZAINDATE
INSTMMPN
WILD
CIRCUMMERIDIAN ZENITH DISTANCES
(TM 5237)
9.442
JTI

9473
SUM LOG. A[98821 A_
_
_
_
_
_
0..73811
05
/6
ZENITH DITNC RE FRA CT ION
CORRECTED~
A

16
0'
0542.9
/6 ,
.,L/5.5
I
oS 575
/6
05
/6
o 0
I'
o
8
05O 46.9
/S."5 ,
+/5.5 O
""
05 S595 _._6 0 6 02.4
01.1
_5.3 0
02.7
S 05673
/6_
/6
/6 1
0S 44.0 t
05S58.4 
574k
/6
f15.6.
00.
MI
__
DECLINATION
_42. 0
ft'
/6 05
~t~05
1 "t
S392 53.:4o05.2 6 3 40. S5?2S ¢ 66t94 46o ,.5 6?9 46 S6.0 69 4656.3
53 40 S92 LAIUD" REMARKS
COMPUTED BY
6.9
C'
DAECHECKED
BY
JDATE
DA FORM 2840, 1 OCT 64
Figure 19.
Computation of latitude from circummeridian altitudes of a star, DA1 Form ;!840.
between 600 and 900, where other precise methods are impractical. Observations should be made with a broken telescope theodolite. (1) This method requires a number of zenith distances observations at recorded times catalogue fundamental of selected (APFS) stars, during the period extending from 5 minutes before to 5 minutes after transit. The stars are observed in pairs one north and one south of the observer with the zenith distance of each being less than 300 to minimize irregular refraction conditions. For the pair, the difference in their zenith distances should not exceed 30 and preferably should be less than 10. Ten to twelve pairs are required and the time difference between observing the stars of any one pair should not exceed 1 hour. If a close circumpolar star, at any hour angle, is used as the north star of a pair, see paragraph 19 for computation. (2) When a pair of stars are close to the zenith (i.e. their azimuth factors (A) are less than 0.15 as determined from the formula, A=sin see 6), they will move fast with respect to azimuth and the instrument should be clamped in the meridian. The star is bisected in rapid succession with the horizontal crosshair, noting in the recording where the star crosses the vertical hair. A meridian correction can be computed and applied to the hour angle before reducing the zenith distance to the meridian if the telescope was not properly alined. b. The procedures followed in the computation are: (1) Abstract the field data from the field books (figs. 20 and 21). (a) Chronometer times and date of observed zenith distances. (b) Observed zenith distance corrected for the bubble correction. This inclination correction is computed by the formula zenith distance= observed d zenith distance+~ (LR) where d is the vertical circle level value in seconds of are and (LR) is the difference between the left and right level readings.
(c) The radio time signals for computing the chronometer correction. (d) The temperature and barometric pressure readings. (2) When the computations are to be completed using normal observations. (a) The rigorous equation for the reduction of the observed circummeridian zenith distances of a star to the meridian is:
'
(cos
4 cos
) (2 sin 2
sin
t
_\ sin 1 '
sin 4 it) ±(cos 4 cos b)2 (2 cot sin 1" sin Substituting: 2 sin2 t sin 1" cos 4 cos sin j
2 sin 4 It sin 1" S
B=A2 cot
We have: rl=rAm+Bn, and 1I=f+Am+Bn, for stars. In the above equation: '=
Observed
zenith
(1 subpolar
distance
cor
rected for inclination, refraction, and for index error of the vertical circle (collimation and zenith point error).
'1=
Zenith distance of star on the meridian. 4= astronomic latitude of the station. t= hour angle of the star at the instant of observation. S= declination of the star at transit. In the above equation, the third term was neglected. This third term is: +4/3 (1+3 cot 2
sin 1"
1)
Asin 1t
2
(b) After finding the mean zenith distance of the two stars of the pair, the latitude for the pair can be computed as follows. s+0n=
2
the latitude for the pair,
where '~,=6+ i, for stars south of the zenith, and ~,= n=  , for stars north of the zenith.
STATION]S//V
ft'
CHRON TIME
POS. H'
RE. OBJECT OBSERVED
Souz
S
N
STOP WATCH
'
p/jn4
__3
3/
42(7
67
MICRO. / VERN.
O4ffee 'e
WEATHER 7G'4 1V /
~iDR/N
MEN
06

_

#
6
Zz Y19.610
44,
o~
2.
*/
__
5
Os
Figure 20.
,f
'
/,
107./ 17/
Field observationsStar 17.
o~
__
_
' EVL
REMARKS
/'
20B'
IST/A(')
22s
/
RECORDER
6./
3s2~
36
6/
CIRCLE
TEL D/ R
y,
___
3 6.'ee
I9
DATE~./A9dU
ECC
_____
Ii STRUNTAN STATJON POS. I_
I
S/
G
E CC. TIME
IIRON 
OBJECT OBSERVED
H

F T
WATCH
1=
71,c W4 _
____
 >
V ,/er/7' 55
/V
DI/R

RECORDER
IST/A('
Ofr E___ WEATHER,:
G
7l  MMEAN
MICR0. /VERN
CIRCLE
TL
*
S
96/I
'
. DATE''
!P 0_ NSRUhEi7T
MEAN DIR /JAN
37A i/Z
tTG.~I
EVELS
REMARKS
D/R
"T)/B!
T'
,
W
___

__
__z
V__
7Jse .3.5
Figure 21.
S_;,5 /
Field observationsStar Na.
"p7
_,
_


y~a
3
1t,~
(

E
(c) Southern latitude and declinations are to be considered negative, the sign of the cosine being plus (+), and the sign of the tangent, cotangent, and sine being minus (). (d) The factors m and n are functions of the hour angle t. Values of m may be found in table IX, appendix III. Values of n are tabulated in table X, appendix III. (e) Constants A and B are computed for each star. Since closely approximated values of
0 and
1 are
required, it will be necessary to recompute the meridian reductions if the values used vary by more than 5" from the true value. (f) The approximate values for 4 and (1 may be determined as follows: Select the observation nearest to the transit (minimum zenith distance) for the north and south star of a pair. Correct these two zenith distances for refraction and level error. Compute the latitude for the north and the south star. The difference between the two results is equal to the double collimation error (index error) of the vertical circle. The mean of the two results is equal to the approximate latitude. If the computed latitude for the north star is higher than the one for the south star, then half of the double collimation error will be added to the zenith distance of the north star and will be subtracted from the zenith distance of the south star to obtain the approximate corrected meridian zenith distances. (g) Having found the arithmetical means of the zenith distances of the two stars comprising one pair, we have the latitude:
n=8(1,for the north star, and ,=8 +[, for the south star Then the latitude derived from the pair is: 22
,
(h) The arithmetic mean of the latitude results of all acceptable pairs determines the observed astronomic latitude of the station. (i) No further adjustment is necessary if the hour angles are distributed symmetrically before and after transit; if the observer has bisected the star always at the center of the cross wires; and if the recorded time in reality corresponds with the exact local sidereal time of the bisections. An approximate method of adjustment is employed in order to determine the error in t; to correct all individual meridian distances; and to obtain a more accurate final zenith distance of the star. (3) The computations of observations, when the instrument is clamped in the meridian, are as follows(a) The reduction of the observations to the meridian are made by the following formula: m
,
sint
t
sin sin 1" Xsin 26, where
t=the true hour angle of the star at observation. m'= the correction to reduce the measured zenith distance to what it would have been if observed at transit. Table IX may be utilized for this computation. Using t as the argument, the m values from the tables sin 28 must be multiplied by 2 'which is a constant for all observations of the star. (b)Since the horizontal center wire is seldom exactly perpendicular to the observer's meridian, it becomes necessary to determine the inclination of the cross wire in order to reduce all obseivations as if observed along a perfectly adjusted cross wire, that is, in a plane perpendicular to the observer's. The adjustment of the reduced zenith distances due to inclination of the cross wire is shown in figure 22. (1) Explanation of the terms:
PROJCT LOCATION
HE HYPO TCA ITABULATION a.'
o
OF GEODETIC DATA
(TM 5237) ORGANIZATION
Oc0 tZDo 0500o  300 04 /2 252 03 25  205 02
/01/30
0/
30
00
/0
00.53 o/
47
025S/
Itt
VO
=SSTATION

9.76
90

/0
26.06
20.00
+.o3
______10.25
20.0o3
.0o
_
____6.50
20.40
_
____4.50
_
____o50
6.38 24$g 3.78 20.09 +o.43
+107
/7.65 +2.87 /4.70 4.5S82
, 17/
11.41
+9.11
07.92
+ 12.6a
04.85
+IS.67
+
53
04 04 + 244 01503 # 303
~
__ _
_
03
20.523
#535 ¢ +8.55 +"
,5'991.
#07
20.12
20. oo
. 09 +.03
20.027
+.03
+1/515 +
0.03
10.9
+0.00237
______________+
________
415 473.0000

080.0908 1____
. 2 976  20 724.6276
+414,392,9092 __________~~~~~ __
_
_
_
20 724.3300 
0.0500/202
_0
AZD
+
ZD DATE /
TABULATED BY
D DAI FORM 16
GPO
5716
22.
2
92196f
_
_
_
_
.00237
.49557
_________
Figure
20.30
CN
___________49.90909
ES
 .37
.. 27 20.05S . 02
+ 12.20
+'
/lI,
_
. 62
19.80+.23 /.959 +.44 +____ 2.5
y,
109
V
/2.60
____
2.6o 12.08J o.2
I
Z
/500
1453
35.05
26.90

[~O]
04)
20:S230
+
.909)
____
DATE
CHECKED BY
U. S. GOVERNMENT PRINTING OFFICE : 1957 0  421182
Adjustment of zenith distance for inclination of cross wire.
_
t=the hour angle of the star ZDo= the observed zenith distance corrected for curvature and refraction. v=the residual from the mean AZD= correction applied to ZDo a=inclination of the cross wire, or AZD per unit of t. The normal equations are: n(AZD)  [t]a+ [v] = 0
 [t] (AZD) + [tt]a [tv]=O (2) After solving for a and AZD the individual AZD's may be computed as follows: at=AZD; then the final ZD is ZD=ZDo+AZD (c) There remains another source of error of the observed zenith distance if the line of sight (collimation axis of the telescope) does not lie in the plane of the meridian. The equation for the correction is: A01 = Am,
ment expressed as an hour angle (local sidereal time at transit minus right ascension of the star). The correction is applied so that it will numerically decrease the zenith distance. c. Following is a completed example of the field notes and computations for one set of stars using this method. This pair of stars was observed at station Test on 21 August 1961, P.M. date. The longitude of the station is 03h 28" 10.08 W.
Star 17
00 h Local sidereal Time 03 Longitude Sid. Time at Oh  22 Universal Time
where
cos 0 cos sin 1
a
2 sin2 't sin 1"
Greenw. Civil Day Greenw. Civil Day
A ==the correction to be applied to the reduced mean zenith distance of the star, and t= the meridian error of the instru
34m 28 00
Star N. h
518 01 03 10 14 22
03m 28 00
21 06 31 (22 Aug 61) 391.35 1444 22.271 Aug 61
02 47 06 (22 Aug 61) 362.7833m 1444 22.251 Aug 61 d¢= 0.042 dE= 0.077
Star 17
not applicable
Star Na
n
0.145
0.8145 (0.1855) B"  0.0378
4
OOh + 090) (0. 077) +
Right ascension
00
34m 50'. 745 . 2395 .0036 .0048 34
RA for 22 Aug UC 0. 145 (0. 24)
50. 99
01h
03m
25. 328 0.03
01
03
25. 35
860
02'
49. 47"
Note. The FK4FK3 corrections are to be applied for 1962 and 1963 only.
Declination 14 Aug UC
530
40'
56. 85"
Declination 22 Aug UC
(0. 8145) (20. 95) 0. 0378 (0.29) (0. 39) (0. 042)+(0. 15) Declination
+
2. 403 .011 .028

(0.077)
530
40'
258 10 14
(2) The right Ascension and Declination for each star are then completed.
Longitude 0. 145 day west of Greenwich n 0. 8145
RA for 14 Aug UC 0. 8145 (+0. 294) 0. 0378 (0. 099) (0. 067) (0. 042) + (0.
The
chronometer correction for both stars is 1.9". The constant for the vertical circle level is 2.845" per div. The stars observed are APFS stars 17 and Na (1) The Greenwich Civil Day and the short period terms are computed for each star as follows:
59. 21"
Note. The FK4FK3 corrections are to be applied for 1962 and 1963 only.
(0. 145)
+ 0. 048
(0. 33)
860
02'
49. 52"
(3) Approximate Meridian Zenith Distances and preliminary Latitude are then determined as follows: Star 17 (South star)
Star Na (North star)
Observed Zenith Distance (position 11) 16005'42/' Refraction + 16"
(pos.
9) +
16015'40" 16"
Corrected Zenith Distance Declination
+ 16005'58"
 16°15'56"
+53°40'59"
f86O02'50'
Latitude (g) Mean Latitude (preliminary)
+69046'57"
+ 69°46'54" 69046'55"'5
Zenith Distance (p') of star 17: 16°05'58"1.5"= 16°05'56.5"' Zenith Distance ('1) of star Na: 16015'56"1.5" = 16015"54.5/ ' (4) For the computation of the Corrected Local Sidereal Time, refraction, and corrected observed zenith distance (figs. 23 and 24), Roman numerals were assigned to the columns for clarity in the explana.tion. (a) Column I. Number of position. (b) Column II. The corrected local sidereal time is equal to the observed (chronometer) time plus or minus the chronometer correction. (c) Column III. The observed zenith distances as taken from the field records. (d) Column IV. Level correction.
=(e)
R) (2.845").
Column V. The refraction correction is always plus, increasing the zenith distances numerically. The formula for the correction is: (rm) (CB) (CT) (table V). (f) Column VI. The corrected observed zenith distance is equal to the observed zenith distance+level correction+refraction correction. (5) In the computation of the final zenith distances (figs. 25 and 26), the Roman numerals are continued. (a) Column VII. The hour angle t is equal to the corrected local sidereal time of the observation minus the right ascension of the star (IIR.A.). (b) Column VIII. Factor m to be found in table IX.
(c) Column IX. table X.
Factor n, to be found in
(d) Column X.
Am, where A= cos € cos 6 sin
1
2
(e) Column XI. Bn, where B=A cot f'. (f) Column XII. ,, the meridian zenith distance (VI+X+XI). (6) The adjustment of the reduced zenith distance is based on the assumption that there is a constant error in the corrected hour angles at the instant of each bisection. There are a number of sources producing this error, some of which are listed below: (a) The star is not observed at the intersection of the cross wires. (b) Delay in time from the instant of bisection to the instant of reading the chronometer. (c) Personal error of the recorder in reading the chronometer. (d) Horizontal collimation error of the instrument. (7) The adjustment of the zenith distances (values listed in column XII of the sample computation) is carried out by solving two simultaneous equations: Equation I: n (ZD)+[t] a[j]=0 Equation II: n (ZD)+[t] a[ l]=0 (a) In equation I, n denotes the number of observations before transit; ZD is the final adjusted mean zenith distance of the star reduced to the meridian; [t] is the algebraic sum of the hour angles before transit (in seconds of time); a is the correction for the zenith distance per second of the hour angle t; and [1] is the sum of the zenith distanc'es before transit. (b) In equation II, the explanations for equation I holds, except that "after transit" should be substituted for "before transit." (c) After the unknown a is computed, column XIII may be computed by the formula: XIII=(a) (t), in which t is to be taken from Column VII (in seconds of time). Apply Column XIII to Column XII to obtain the adjusted zenith distances.
G.C.D. =22.25/ AUG. /96/
STAR #/7(SouA)
BARO. =:26.8/
S92/1
Cos
o0/734"
509
Cog
If,
/6<
05"
5"65
51I
0
69 °
46'
S 555
COT f,
3
S.
RA
40"
0
,
0.34S 59/7
Chrotn. Corr :
+1S9
0.592 2507
C =CB 4
0.928
0.277 2984
/6
06 23.55
Level

.28
Refr.
Corr Obs. Z.D.
+
/5 53
/6
2/73
06 38.80
_____2
30
01J.7
14.45
14
f
_____3
30
3559
06.35
 .14
+
1S2298 1/.62
3/
/1.5
00.20
.00
4
/552
/5.72
31
48.4
/6, o5 54.55
,oo
t
/552
/0.07
6
.32
23.7
49.6o
00 4 /552
0S.12
7
.32
53.5
46.10
oo
33
21.0
43.50
00o
4 _____
_____8
53.o
42.00
*oo
24.7
42.05
+.28
*f
08.6 44.1
41.90
+. l.4
/2
3S 35
42.80
+14
1.3
36
13.4
43.90
14
36
456
46.80
4.14
+ 15.51 4 15.51 5/59.55
1S
37
/,54
4 9.40
.
6
37
48.8
53. 95
/7
289
24.9
18
.39
04.9 /6
1___ 9
39
38.8 /4.7
1/
20 00
40
TABULATED BY
1 FW
/S651
34
/0
_____
1 /552 f
33
_____9
DA
0,6Os. Z.D.
29 24.8
___________
3464 8020
Po_____Fs. Corr Loc. S31W.
/ 0
TEMP:_0./C
DATE/2
GPO 921961
571962
Figure 23.
06
,.4
+
01.62 16
05
/5.5/ /6.5/
+.1/552
59.0/ S________
5784
57.55 58.45 /6
06
o2.46
14
+
/552
04.78
+
16.52
09.33
.59. 75
/4 14
07/0
1.4
.
t 15.52
i.S.
1.3
/552
22.48
/4.85
.o0o+/.52.
30..37
23.20
.00
38.73
CHECKED BY
f
+ 1S.53
RQ U. S. GOVERNMENT PRITIG
Correction to observed zenith distanceStar 17.
DATEjl
OFFICE : 1957 0  421162
GREELAN
A rR
TABULATION OF GEODETIC DATA (TM 5237)
bSiA4et
Sta. /SL/NG(UA
STAR
ORGAN IZATI ON
Ec c.
USA M11S ,
7/
6. C.tD:22.2
COS COS
16*15'S4'5
05 = 46'5..
0 (6
0. =
.6
S/Ny
0i=. 280 0827
COT f,
3. 427 4733
____
/
____
2
00
3 4 5
o/
.57 32.3 58 /2.8 68 53.5 59 35.6 00 /6.8
TEP +02 C
Chron. Corrc+15 C =CB CT
0.927
Obs. Z.D. 16
15
45.75
"
Level
Refr
0.00 + 15.68
+o.43
+
42.40
1668 +0.43 + /5.68
40.95
4#0.85
f
44.10
/6
01.43 00.2/
/5 58.51 57.48
/568
28.2
03
25.3
38.80
t 1.85
+1568
6'6.33
04
04.3
38.65
+ 1.99
4 /6568
56.32
+2.28
+15.68
66.46 56.59
01.3
7 8 9
0/
50./
4.5.9
38.50
/2
0
24.9
39.36 +1.56
13
06
03.2
39.30
+ /.56
 1668
14
07
/1.7
42.80
1.28
t 15.68
1___ 5
07
54.3
44.50 1/.0 oo
/6
08
22.7
45.00
1___ 7
09
/1.6
45 90
09
.. f6.o
48.00
1104
0/
/6
02
0/
/8
Corr. Obs. Z.D.
41.00 +0.85 + 1568 39.00 i.42 + /5.68 39.35 # 1.42 +'15.68 38.85 ,j56 +#1568
6
/0
S.
BARO.2 ~8 /g
.ZifF1
Cor. Loc. Sicl T
____Pos.
5917
0. 068 9366~
II
____
R9.
AUG./1961
Nq~ (orth) 6:802'4952 RA 0/ h03'5. 3 5
0,hs.
57563 6 .10
56.45 66.09
+16568
66.5S4 6720 59.1/8
/5.6o8
1.00 15.68 0.43 +/568 0.281+ /568
59.68
/6
16
01./6 03.40
48.00
TABULATED BY
16
D®AEI,FORM FE
GPO
Figure 24. 757381
0

65

4
921961
U. S. GOVERNMENT PRINTING OFFICE: 1957
Correction to observed zenith distanceStar Na.
 421182
PROJECT GARENLAND
SIR
TABULATION OF GEODETIC DATA
I(TM
GREELAND,,~Jw
5237)
TORGANIZATION
60GAT O,
1/7
STAR Fs.
Ecc.
/SL'AGUA
Sta. sTTe'COS60COS
A=
USAMS
S~
si
M
t
2 S= AcotL
0.73811 Am
8rf7
1.88765 AdSj.
atv
.
.00
. 5
2652
58.03
.0082
42.84
+ .02
55.98
458
56.56
2
4
49.3
45.65
.005/
33.69
,,.0/
56.15
f. 5/
56.66
.3
4
151
35.49
.00.3/
 26.20
+.0/
5554,
# 45
55'99
+.5S7
4
3
395
26.28
.00/7
/1940
.56..32
4.39
56.71
.
1
51
3
'026
/8./P

13.43  8.74
56.64 56.38
f.32
S6. 96

.q0
+ .26
56.64
. 08
/
6
2
273
/18.4
7
/
575
753
8
/ 0
30.0 6'8.0
442
'o0
0
26.3
0.36
/4
0 i
176
o.17
/2
0
S3.1
1.5S4

1.14
/3
/
22.4
3.7/

2.74
/4
/
6S4.6
7/6
,'S
2
24.4
/137
9
/6,
2
S6. o6
3.26
65.75 56.16
135
___
__
0.28
___57.66
0.13
_
47.8
201
S
23.7
.03
57.39

.83
6722

.66
56.66

.10

.42
56.81

.09 .165
57. 18

.20
56.98
. 26
S6.13
__56.39
_
+1.43
.56.28
7*.28
6.33
'. 23
.0/
5703

.5/
56.52
,.o4
57/14
.00 79
4218
*"0/
66.56
. 58
55S.98
'. s8
/0ZD 
1711.8a
56.60
56.7/

%68.
a__
/OZD: 568.5
55 6.0/
a+
____3381.0
6
 562.54
/669.2 a.
.t32
a71
1.05
33.35
TABUATE
FEe
5676/
 30
00oS0
=
:0  o.
BYCorr
56.558
__
_
_
__
_
GPO 921961
figure
2?5.
+#05
_
00j,77
3
Z'D =56. 6.002 in~It, DATE
CHECKED BY
962
f .47
0
4For error V/62
56.0,9
O
 2.?67 =56$5
TAUAEYDATE
DA
OS
f . /0
4518
/OZ0
/P . . = f .
*.66
56.54
+2.~2
IPf9 .
655.9/ 56.26
.oi/
23'16
A= +.0241 ___
/6

56.554
t=
+
 25.95
/8.42
4
t
.0030
24.96
/'9
56.27
 .32 . 38  .45
12.73
4
2/
____573/
 8.3 9
.29
#.

__5742
28 6__
/7.24
18

5.56
___
678 33.9 /3.9
17 3
1.83
.00
. 10

_______/______
U. S.
GOVERNMENT PRINTING OFFICE: 1957 0  421182
Computation of final zenith distanceStar~ 17.
4
STi? Pos
'Na
l
tfm
5
3
0.08506
0osC_
A=C
As
f,
55.65'
5574
. 05
.02 f'. o7
5. 76
55.15
. 07 t . 54
5 03
+.06
55.09
9 60
5588 5. 4
t . 5
55.93
. 24
.0o4
55/8
t. 51
+ .02
56o.05
36
+0/
55.95
. 26
.00
56.33
. 64.
.0/
,56.24
55
.o2 . 03  .04
56.14
. 45
55.90
. 2/
55.34
+.35S
54.82
 .06
.
55.83
. 07
54.76 576
56 38 S_ 59 .
. 08
56.32

68.00

/2.6
53.29

3
4
31.9
40.32
3.43
55S08
4
3
49~8
28.80
2.45
s
08.6
6
3 2
24.1
/2940 /133
/66 __ o. 96
7
/
353
4.95
8
0
1.78
9
0
572 00.!
0.00
38.9
0.83
/0 f0 (I
/
2o.5
3.54
/2
/
59.5
778
/3
2
378
/3.S8
/4
3
46.3
15
4
/6 /7 /8
_
_
55.94
___0.00
___56.33
o.0o7 _
0.30
'i.
0.6 _ 6
C
___
//6538
27.3
____
2.38
28.9
3943
___3.35
S
04.3
50.5 So
S
46.2
6o537
6
36.6
83.2/1
4.3o
_
___656
. 7.08
__
f=
:2
5
/6352.7at
92D
1933.Oc'.

__
i6
_5_0.76

a
+ .:39
. 09
55 30 555so
.1/0
56.22
. 53
RZD = +
/>
7.0
=0___
.90
+1
93
. 07
5568
_
5/.66 =
6.7
S7

56.
__55.69
92D
_358
56.25 5.93
2
_____
_
_56.03
___0.15
___
__
8
___5561
_0.42
v
,
.0 9
53.1
5
5.78 4.53
0.02480
a
2
/
cot f,
Adj .
Bn
Am
=
__
_______
0
251
0.0
____02
55.686
00.?76Z
__
*
Olt
P~s /. =± .3/
+
4/5
____
4.00
H'
. 07
Rej. L. =
__
_
50/ /7
5
/.'Q
TABULATED BY
pDATE
1 FesR11
96V2
GPO 921961
Figure 26.
_
__
__
*
Cor.
__
CHECKED BY
'/62
DA
17S____
86,VOIK
_
4ror e rro
in
Rq&DATE COM.PU/TER
_
"it
7
/62
U. S. GOVERNMENT PRINTING OFFICE : 1957 0  421182
Computation of final zenith distanceStar Na.
_
(d) ZD, the mean of the final adjusted zenith distances, may be computed by substituting a in one of the two equations. (e) In the event that observations before and after transit are not balanced in number of observations, it is advisable to employ a least squares adjustment of the reduced zenith distances (values in column XII of the sample computations). Figure 27 shows the least squares adjustment of star 17. The two normal equations are: n(as)[t]a+[v]=O [t(A 1) +[tt]a[tv]=O (8) In the above adjustment, the zenith distances will be reduced to a meridian which is, by the amount of dt, off the astronomical meridian. Normally, not a large error in t is to be expected if all precautions in the determination of the local sidereal time of the bisections were taken. The correction to be applied to the mean of the adjusted meridian zenith distance of the star is: CorrectionA sin2 Idt CorrectionA sin 1 Am, where dt is the average error in the hour angles. (a) The average error in the hour angles (dt) may be found in the following manner: Differentiating the formulal'1 = Am+Bn, and neglecting the second term, we can say that 2
sins sin2
d (A)
(c) From this, it follows that the average error in t may be computed by the formula:
dt
a is
0.0019(A ) 0.0010 9(A)
plus, then the chronometer time of the bisections is less than the correct local sidereal time, and the final adjusted mean zenith distance will be lower (numerically) than the zenith distances from column XII of the sample computation. If a is minus, the opposite would apply. (9) The latitude from one pair of star (fig. 28)
2,
is equal to
2
where
s=
+1
for the star south of the zenith and 4= i, for the star north of the zenith. The mean of the summation of the results of all pairs will give the latitude of the station. Eccentric reduction, reduction to sea level, and the correction for the variation of the pole must be applied (fig. 29). (10) The probable errors (PE) are computed as follows: (a) PE of a single pointing on a star: eo= ±0.6745
(nv) (n1i
(b) PE of the arithmetic mean of the reduced zenith distances: ro=
e
t t
,, sin t". dt(radians) =
sin 1"
A •sin t".dt" (b) Assuming that dt (error in hour angles) and t (hour angle) are both in error by 1 second of time, then: d ( 11)=A (0.000,072,722) (15)= 0.00109"A, which means that the change in the zenith distance due to an error of 1 second in t is equal to 0.00109" (A) . (dt). This relation holds true since the sine and arc of t are linear, when t is small.
(c) PE of a single latitude pair:
e=±0.6745V([v
1
(d) PE of the arithmetic mean pairs (latitude result);
of all
r=n In the above equations, v=the residual, and n=number of observations. should be understood that the It (e) probable errors thus derived represent the probable errors of the observations only, and do not include constant
LEFIST SQU/ARES PROJECT
GREENLAND
LOGAT4ON,1
Sta.
ADJUSTMENT
I
/SUNGUA
STAR~ /7 6 fl
5237)
ORGAN IZATION
Ecc. V
TABULATION OF GEODETIC DATA
I(TM
A S7RO
U'SA lviS
[te[ W11
Ad~j.j
a6
/6° 05,
/6x05" /

3216.2
2

2893
V2
568 S6. /S
0.3S7
fo. 45
56.43
7o 1
0.40
40.40.
S6.55
#
5
0.0/
/10/
V0.35
89
*067
4

2/95
56.32 4 0.23
A0.30
56.62
. 06
5

/82.6
56.64 0O.09
*0.25
56.9
0o.33

/47.3
56.3.8 90.1/7
__
i0.240
56.56
 o.o 2

1/75
56.06
_
"0./6
56.22
f
.~0.69
255/1
3
7
55*54
_
1.49
0.34
8

90.0
557
*0.80
#0. ,_ 2
55.87
9

58.0
S6. /6
+0.39
f+0.32

26.3
5756
/10/
+0O.08 +10.04
56.24
/0
57,60

0.02
.5740
o.84
 0.07
5724.6
__o.//
56.70
/1+
76
/2
+
.53.1
57.3/
0.76~~
/3
#
82.4
56.8,'
0.26
/4 /S
+146
~
/6 /7
^
+0.16
177.8
56.60
0.06
56.7/
0/16
56.54
A0.0/
0.35
S6/2 S6,11___
___o__4
56.156 .00
_____
.
"'
0.20
56.1,9
o.25
,56.35
3703
0.48
56.56
0.0/
04o  o. 04S  o.
42.6
56.55
0.09
+0O04
+~2878 f
AJ,
CHfECK

.3
0.07

4
0.07
56.56
A42.6
+2o.ooo

0.o8
757
4
+757
704. 62
0045
fo. f.
+
052. /7/0 .19/7 1/0523627
6/3.882 A9
JDATE
0.090
2.13 90. 738

TABULATED BY
.1
0.
____
_____
0.2/
f
56.63
[N2 ]nrz
+ .0/
56.56
____o._37
56.4/
__0.3,0
323.7
/9
t
04
56.39
20
FE:
___.__4
50
144.4
253.9
__

5.S~.6
42/3.9
/3f
5742  0.87
/.o4

_____
._00/7______
CHECKED BY
DATE
7/62
DA
I RED
71962
GPO 900e47
Fig~ure 27.
11. S. GOWZEBnm.6 PRDITIN
Adjustment of zenith distancestar 17.
OflCE. 1"1? 0 
21 is
GPRNOAJEACT?
TABULATION OF GEODETI 5237)
GREELAN A STRO(TM
LCATION,
Stu.
/SIAG A
DATA
ORGANIZATION
MS
Ccc."USA
SCIO4N
STAR
DECINI/A TION
ZEN/ITH DIST.  /6 °/5.569
/7 (S)
4/16o
#86°oO2 49s2
065'.66

TABULATED BY
DA
07PAI
DATE
1FED 71962
GPO 921961
Figure 28.
6~9
592/
40
t S3
______
t
(142)
(2)
(1)
Noa (N')
LA TI/TUDE
'Nr2s
46'53.83
_____
5S.7
69 46 54.80
oo>
o=o
DATE
CHECKED BY
U. S. GOVERNMENT PRINTING OFFICE :1957 0  4Z1182
Observed latitudesingle pair' of stars (17 & Na).
SUMtMARY OF LATITUDEREUT RESULTS PROJECT
GREENLAND
I
ASTRO
LOCATION,
TABULATION OF GEODETIC DATA (TM 5237)
ORGAN IZATION
Sta.
ISUNAGUIA
Ecc.
UISA MS
STATI ON
0V
PAI/R NW'. /
S4.8 ,
69 46
2
53.0
3
654.70
(o
1
V2
.1
00
0 67)
R
4 0.23
0/15
/0.06
6
54.87 55.02 54.76
7
54.67
8 9
Ao. 26 0.20
54.83
Mn. or Su
S.5493
4 S
55/3
_0.0o9
0.02 0.17
0./7
'o A0./8  2 +4.D 2 t0.0
0/
e2
r
MEAN
x'10745
±
.V
08 ERVED
'
.8
R~j. Limit =.Zo x 2.84
.20
Limnit
O.Rej.
7
=.t
LA T/T DE
ASTRONOMIC ___
____
DATE/6
.1/ x2.76.
___
__
_
____
690
Figure
GPO 921961
29.
±0.30
___t
46' S485 o o4
DATE
CHECKED BY
7/6
'42~
D,FORM 16
=+0.17
0.04______

____
TABULATED BY
(0.6769) 0. /736
0o.02
54.85
6,~±1745
v v]
U. S. GOVERNMENT PRITING OFFICE : 1957 0  4ZI182
Summary of latitude resultscircummeridian altitude method.
errors of the instrument and the errors introduced by inconsistent refraction conditions. (11) Chauvenet's rejection rule should be applied for the rejection of single pointings on the star (Column XIV of the sample computations), and for the rejection of latitude pairs. The rejection factors are found in table XI.
24. Latitude By the HorrebowTalcott Method a. The HorrebowTalcott method of firstorder latitude determination is used in most of the world today for latitudes up to approximately 600. This method utilizes pairs of stars on or near the meridian and of approximately equal zenith distances observed north and south of the observer. Excellent results are obtained because the method depends entirely on differential measurements which can be accurately determined. b. The basic formula used in computing latitude from observations made with a Wild T4 broken telescope theodolite is:
O=2 (8+
')+1 R(MwME)
subsequent to July 1st must be computed for the epoch of the following calendar year, since the Besselian and Independent Day Numbers in the AE&NA are tabulated on that basis. The best available star catalogue is the "General Catalogue of 33342 Stars for the ]poch 1950," by Benjamin Boss. In the HorrebowTalcott method the right ascensions need be accurate only to the nearest second. (1) This accuracy will usually be attained by adding algebraically to the catalogue right ascension the product of the annual variation and the number of years between the epoch of observation and the epoch of the catalogue. If the epoch of observation is earlier than the epoch of the catalogue, then the difference in years must be considered negative. (2) If we denote by ao, the mean right ascension for the beginning of the year nearest the time of observation (to) and by am the catalogue right ascension, then the complete formula for use with the Boss General Catalogue, Epoch 1950, is: o=am,+ (to1
9
50) An. Var.
+±1 (t1950) 2 +l (d+di) [(n+nil+s)
1 0)
w(n+nl+s+s8)E
1 1 +I (rr')+ (m+m').
Sec. Var.
(to1950\o) 100
Where: 8 & 8' are the apparent declination of the stars of a pair. are the micrometer readings with Mw & ME ocular west and east respectively. n, nl, s, s, are the readings of the north and south ends of the two levels. are the values of one division of d & dl each of the levels. rr' is the difference in refraction of the two stars. m+m' is the sum of the meridian distance corrections. R is the value of one turn of the latitude micrometer.
In the case of circumpolar stars, it may be necessary to use more than two terms of the above formula. (3) If rm represents the catalogue declination and ro the mean declination for the beginning of the year (to), then the complete formula for mean declination is:
Note. If the Wild T4 is not used, the micrometer and level sections of the formula (i.e. m, n, s, and d) must conform to the instrument used.
All the terms of the above formula may be needed to determine the mean declinations with the accuracy required by the latitude. Usually the mean declinations are computed accurately to the nearest hundredth of a second. (4) The above formulas for reducing the mean right ascension and declination of a star
c. The first step in the computation of the apparent places is to obtain the mean place of each star for the epoch of observation from a standard star catalogue. It should be noted that the apparent places of stars for observations made
o= 0 8,+(to1950)
+~
An. Var.
(to1950)2
(to1950
1
Sec. Var.
3dt.
trom the epoch or the catalogue to tile epoch of observation contain all the elements necessary for the reduction. These elements consist of first and secondorder terms of the precession and proper motion. The terms of the formulas are arranged according to ascending powers of the time interval from the epoch of the catalogue to the epoch of observation in the manner indicated in these formulas. d. After the mean places of the stars for the epoch of observation have been obtained, the apparent declination of each star is computed by means of Besselian and Independent Day Numbers. Using the trigonometrical expressions for Bessel's star constants (Formulas for the Reduction of Stars, AE&NA, any year) the formula for apparent declination customarily expressed in terms of Besselian Star Constants and Besselian Day Numbers has been transformed into:
6=6,+rI+t
cos 6o+X sin ao+Y cos ,+AbaIFaa,.
Where: X=(B+C sin 5,) Y=D sin o+A (1) DA Form 2865 (Reduction, Mean to Apparent Declination) has been arranged for this computation (fig. 30). (2) ao and 6, are computed for the "appropriate year" and entered on the form along with their natural sines and cosines, each in its designated column. Four place tables provide the required accuracy of the trigonometric functions. Be sure to use the correct signs of these functions. (3) The proper motion in declination, u', is taken from the star catalogue (G.C.). (If the change in proper motion per 100 years, namely, 100 A ', is large enough it must be taken into account in taking the value of ' from the catalogue.) (4) Usually when the period of the observations does not exceed 4 hours, the values of A, B, C, D,
T,
and c can be obtained
with sufficient accuracy by using the mean Universal time, commonly abbreviated UT, of the observations. If the mean right ascensions of the set of stars are spaced with approximate regularity, the mean of the first and last right ascensions will provide a sufficiently accurate mean local sidereal time of the observa
tions. Kometmes
when there is a pronounced break in the regularity, the set should be divided and the mean epoch computed for each part. (5) To the mean local sidereal time, add the longitude if west of Greenwich or subtract if east, thus obtaining the corresponding Greenwich sidereal time. _From this Greenwich sidereal time or as abbreviated, GST, subtract the Greenwich sidereal time of the nearest preceding Oh UT. Divide this interval expressed in minutes by 1444 and the result will be the fractional part of a civil day from Oh, UT chosen.
(6) After quantities A, B, C, D, r, and c have been determined, compute the X and Y values for all the stars. When these have been entered in the proper places on the form, all the necessary information for computing the apparent declination of a star is on the same horizontal line with the exception of T and I which are at the bottom of the form. By machine, all the steps of the computation from mean to apparent declination may be carried through in one continuous operation. The signs of the products must be carefully adhered to. (7) The corrections Ass and A,5a must be applied to the apparent declination. These corrections reduce the values obtained from the system of the General Catalogue to the system of the FK4. The requisite tables may be requested from the Americas Division, Department of Geodesy, Army Map Service, Washington, D.C. Distribution of the tables will be made as soon as they are available. e. The information which should be entered by the observing party on DA Form 2842 (Latitude Computation) (fig. 31) consists of the following: (1) At the top of the form, the names of the station, chief of party and observer, the date of observation, the kind and number of the instrument used, the number of the chronometer, and the elevation of the station if available. (2) In the designated columns for each star, the star catalogue number; the position of the star with respect to the zenith N or S; the position, E or W, of the ocular during the observation on the
AONR
AP
2
LCIN
LAND
ATOMARY? CATALOG
STRN.
MEAN
Mag.
RIGHT ASCENSION
/47  4/
4
23
m4
/22i02 34 A
/ 9(87j /97424
MEAN
aI
DCIAONsin
2 _0/_ '578
B
4387 3973
A.
55 6.4
37
05
54
/o
5709
40
/312/
/6
62.25
2907/ 6._ 20/5/. 7/
_5
/9800
22l_5 20 02 74 2 66 ~ 7~
6
S6
20489<
6.2
2058o6
6.8 r
7
1/
21
20696: 2074h
20 298
0S
2o4211
:
2
690 3 46
2/4 S51/ 4132
59
216841 6.6_ /6 _ 27/ .51 2/863L 54 i 1
22 0
22094 22216
6. 6.9 6.0
'
2234 232 64
13
5O82 157
204.1
43
54
17
33
S7
S9f5
GREENWICHSIDEREAL TIME
SIDEREALINTERVAL GREWC CVLDY!MAY
__
)
I
08.5
2/
30.5
/4
40.7
6
49.8 3.284
I1.6/7' 27
j.
7919 4584
'
x.03
6936
03.88i
.5585
.438
m .
.3790
_.9254
2/ .8922
_i
~
+.L
D
+
'
8935
 .4
...
)
1
['
t
f . 04 4.2.495 '. . 9004 . 04 + 7389~ 9053 L786/K 9126 ) 004 +1.6724,9233
!.358
0./08
0A
.3.607
13+D34 6.041
0. 3346
o.9/ NAMBERS
.4,System
V
+8.820
.9324 .
9344
7.665'1
/113021 5200
 3.445 12,248
43.1_
"/8: .7 .181
.9
56 20 57
46 02 4
/4.44
29
22
03.68
3/
S5
2746
47
45
9!1830 .2/' /8' .8.19
5 1/ 28 38 4
47 40
.191.37
l09_2 .26. 20 /S 191 2 .2029 47 22' 1' .20; 9 43 .2/ 19 33
1 .19 i
4350
 6,/2
/0.8311 . 4247
)
. 4088 3842 3614 . 3562
/9 "/1
.2/
.26 ./
. 04.07
243/8
.2/: .181 45
.204
. 4490
/732 490
33.38
/1.57 /.1 22.65' /8.95 03.04
453
57
20
44.64
36
_0795, 04.89
S54
51.8/
66
58. 95
.19 1.2 5 .19 52 .19' 55 8 22 .17 14 /14 63
59
03 4
07 25 /16 32 08
28.23 00.43
2039 51.53 42/16V
The mean epoch should be for a period not exceeding .4 hours Use oil sines and cosines to 4 decimal places; compote X and Y to to 2 decimal places.
3 decimal
places.
*Reduction
from Boss General Catalogue System to nR
_______
Fomls
1
.=
5____
OE
a° F
BCs,,
A
dl
snd
T
j'..
AECECDNSRT

AIVS
July 63
DA FORM2865,1 OCT 64
Figure 30.
9
E
/4.&f
.20' .18:
0
+ 4.172,
96 7? .4517
13; /3 29
4
L o~o8o;
{086
BESSELIANAND INDEPENDENTDAY
A
1/42
43.348 +66801 4 054
.8989 .639 .5676
/ .7894 .8233
08.09 33.65 25 06,1~ /6 25.14 2656 08 48.55 h
8295
A8,772!
 .0/5 .309
.7204 '
S434
,o6I
I.22'
4,672
. 8441 1 3,63 5362 . 8660 _6.852, . 500/ .4776 "878(0_/0/381 8854 49.52/1 4448
_0_5_
S6.81~
+3.44
23
.7
22; ./4
 .65289 . 77 94 632 /0.070 6 9434  . 62 95 5/~.36'.76'.3~#.,77 ./32890._5860 8.89/0 .5770 83 . 834/ /2.200 51
.51#44
70
2647 .9643 .4923L.84 0958~_ 7385 i'86
07
22
5
69 .6/ 06 .8887
S9
m
b
//
CORRECTIONFOR LONGITUDE
3
23 06.61' 48,21
'.195
71,27'
/2
38 12 6.6 /6"_ 36 32 63
*i41 .16
0/
23
. 73/8  .7/82 . 6940
9.833'
47 02 4 3 37 06+.62
3
7509
/.483'. 4.702!
. 7576
8623
./6
.2
+6.370,
L
_2
7(S
+.004
13.23~ .50o63
52 i9) .,6.i,26 1..9lIt./5 /
47,
.6977
25 1/ 6 3 38 42
APPARENT
7716
7163
13.80~
L
74
= 6.73077 7.258 .
30
29
S45'
. 7448
30.17
5237) DECLINATION
 /.132
+~3.7S?9
40 56
21 2 5 28) 52
6 _
8462
8092 8009
027
45
/9
~
/0.820,00.
.848.7
45
_6743~
09
.3427
645
*1
*1 *
.5288__.
oo0
9
/997'
01+44
1 Y
.490
55
0534
MEAN EPOCHOBSERVATION
SIDEREALTIME Oh O.C T.
938
.6/.1636 480 005S+80451 . 6604 .9395 . 070 #1.166'IS . 68/5 Io .529.025 4 8.1.80! . 95 . 756.949, 3224746 .02 032 1.22 . 7353
MEAN TO APPARENT DECLINATION
PM
3 2220.9187/5
3/
5/2
'
"7
3629
95
3Si
213
2/8
I
20 !S
Stu,
I
4
MAY1963
1404 "6/54473724 58 75 o2.52s4 . "987 .88 9/77 _4.015 +1927' 624 .585 2 ;.024 4.68 . 633
.7882
",
/08
2j3_4.73
C~a
SAREDUCTION, , UA2
Reduction, mean to apparent declination (DA4 Form 2865)..
f.lR. n. do

AMIS
DEC. X63
STATION~~~ LCTOPRJ
75 m~ PAIR
RO N BOS OR STAR NO.
E ON N
/9467
N
E
LEVEL
RE
READINU N t
d t
l/
d
45.9
i 9528 .5
N
S
WEU
d
d
d
30.1
13.248 8
21.1
CHRONOMETER TIME OF OBSERVATION h
32.0
S.t
N
IM
4 24 /14 1__ .66___o_+6.6
51~7
/28.
W
3
R. SAL VERMOSER
MICROMETER
an
14
27
07.2
2 MAay 196

/2R
RMENHLESMO RDISDECLINATION
(TM 5237)
76.5
=
7ZII
LATITUDE COMPUTATIONPAEN.
No.)
r4 No. 56095
LAOW/LD
CADDESS
H/A'.
INSTR.IOENT(Type
T
MPATO24A
d +d
.//168
/ 16
LEVEL
MURRO
REP
LATITUDE
/9687
5
W
/2 93.y
/9742 {N
E
S6700
49.8 21.o.1
+Z239 1470/24.+ 258 494
14 .94
27.4
REMARKS
EI
NS
/6
52 0/
/6.69
.03
39  l
26 0/
0/
/038
04
38
S6
08.47 +0,71 . 08
TIPPED 62.63
0407
/52.4 /300
2
OPGS
/246o
CORRECTIONS
DECLFNATIONS DELNTOSTANCE
N
2
23 24 35,18
Lo .97 074
3
54 /0 .5762
TEMP.
ME4ANJ 38
BAR.
M6EAN= 03 Ojn
3
46.40
09
60,.47
/3.78 +0/2 +.17
__124.214
198001s 3
1/EI83.0
32556.1 /30.0/52.5 +4.3
/4
40 /4.4
2
21/16
8 28.3
573 33.
/4 45 275
8
56 46 /4.44
___3.547
1_9907
N1 W
4986
20041 4
S
20151
N1
20308
8 876
E
W 10 41.8
2
1496/126.9 20.7 445
20 o2 _
/1504 /3.2
5932 297
5 02.1
E
SI 5.3
,4 S6 23.3  2 /46,9241______________________
+5460/ 20421 .5
/ 4/ 0.4
114,1146.8 48.7 25,0
N W /0 62.2
6
0/' 32./S
04
31.35
'0,5
. 08
61..23 0/
______
TIPPiD
LATE
_______
252
.. 42
_____
39
/535S /3t0
 2
/7.32____________________ a/
41.26___
5796 o o5 +,o4
47 49.30___________________ _____
48 17 33.68
+333
_____________
38
___5.
03.31
55
____38
57
09
670 375 29 22 03.68
/3
05.0
49
____
_______
48.68
07
08.48 3,89 +.12
__61/33
______
16_______TPPED
____
lArE
/190 141.9 20489
215 451
9 40.4
5S E
6
/'.
20586 Nd W 1465/1
43.9
~/5
20.0
/43,01/20.0
_
__ _
_ _
_
_
3o
3/ 55
27.46
3,3
+S,247 /20.6/43.0
_
_
_
./
38 /70O.S
24
44
/9.52
50
+06 41.40
0.39
TIPPED
+,12
LATE
60.76
/157
____
_
_
_
.SESP
SAP4
______________
____________________

A
S
DATE
July
DA FORM2842, 1 OCT 64
Figure 31.
OF
2
(,.o24)(/.o0.1= 1.034
47
38
14
LATE
_______
Latitude Computation (DA Form 2842).
CHNEDBYNATF
X63
0,RNk

AMS
D0EC. (03
f.
star; the micrometer reading in turns of the screw and in divisions and tenths of division of the micrometer head; the respective readings of the north and south ends of the two latitude levels; and the chronometer time of observation. In the office, the computer will complete the computation. (1) If the observing party has not furnished the approximate elevation of the station, the value of one half turn of the micrometer, and the level value, then these data should be entered at the top of the page. (2) The apparent declinations computed on DA Form 2865 (Reduction, Mean to Apparent Declination) should be entered in the column designated "Declination", and the halfsum of these declinations, for each pair, computed and entered in the next column. (3) The algebraic difference of the micrometer readings for each pair (in the sense ocular west minus ocular east is positive) is then placed in the "Diff. Z.D." column, usually in decimal form. This difference is then converted to seconds of arc by multiplying by the value of one half turn of the micrometer and the result placed in the micrometer correction column. (4) Next, the algebraic difference of the sum of the level readings for the star with ocular west minus the sum of the level readings with ocular east is set down in the designated column. This difference multiplied by the level value, i.e., by
(d+d ) , constitutes the level
correction.
(5) The approximate meridian distance is computed by the formula ao(t+At), where ao is the mean right ascension, t the chronometer time of observation and At the correction to the chronometer time obtained from a radio time signal. This distance is entered in the proper column on DA Form 2842 (Latitude Computation). If, for any reason, the observer has not observed the star on the meridian, it should be noted under "Remarks", giving an estimation of the time of observation before or after transit. (6) If a star is observed off the meridian while the line of collimation of the telescope
remains in the meridian, the measured zenith distance is in error on account of the curvature of the apparent path of the star. Let m be the correction to reduce the measured zenith distance to what it would have been if the star had been observed on the meridian. Then, sin2 r sin 2 m=in sin 1" in which r is the hourangle of the star. The signs are such that the correction to the latitude (2) is always plus for the stars of positive declination and minus for star of negative declination (south of the equator), regardless of whether the star is to the northward or to the southward of the zenith. 2 or m is then al2 2 ways applied as a correction to the latitude with the sign of the righthand member of the above equation. For a subpolar, 18008 must be substituted for 6, making the correction negative for a northern subpolar, and positive for a southern subpolar. Table XII gives the corrections to the latitude computed from the above formula. If both stars of a pair are observed off the meridian, two such corrections must be applied to the computed latitude. (7) Although the difference in refraction of a pair of stars used in the HorrebowTalcott method is small, it must be applied as a correction to the latitude. The refraction for each star of a pair is very nearly proportional to the tangent of the zenith distance, so that the differential refraction will be very nearly proportional to the square of the secant of the mean zenith distance. In addition, the differential refraction depends upon the pressure and temperature of the atmosphere at the time of the observation. For a mean state of the atmossphere (pressure 29.9 inches or 76 cm. and temperature 500 F. or 100 C.), the
correction to be applied to the latitude for differential refraction will be given by the formula: rr' 57''9 s 22 2 sm ( 3') sec2
are the refraction and where r and zenith distance, respectively, of the star observed with ocular East, the primed letters referring to the star observed with ocular West. Differential refraction will, therefore, have the same sign as half the difference of the zenith distances as measured by the micrometer. The two zenith distances of a pair of stars used in the HorrebowTalcott method are so nearly equal that either may be used to determine the sec 2 in the formula. (a) Table XIII has been computed by means of the above formula with half the difference of the zenith distances as measured by the micrometer for one argument and the mean zenith distance for the other. (b) In as much as the refraction obtained from the above table is only valid for the assumed mean state of the atmosphere, it will be necessary to apply to this differential refraction, factors obtained from the regular refraction (table V) in order to reduce it to the differential refraction for the pressure and temperature at the time of observation. (c) When the micrometer, level, refraction, and meridiandistance corrections have been combined algebraically with the mean of the declinations of a pair of stars, the latitude as determined from observations on that pair of stars is obtained. (8) The correction to the mean latitude and the correction to the value of onehalf turn of the micrometer are computed by the method of least squares. Separate computation of each night's observation is not necessary unless there has been a distinct change in the value of onehalf turn of the micrometer. Such a change should have been recorded in the field records.
(a) DA Form 2843 (Astronomic Latitude Summary) should be used for the adjustment (fig. 32). The data in the first four columns are obtained directly from DA Form 2842, Latitude Computation, the micrometer differences being taken to the nearest tenth.
(b) Before proceeding further with the adjustment, it is necessary to find out which, if any, of the results are to be rejected. An absolute rejection limit of 3" from the mean of all the latitudes in column 4, each considered to have unit weight, is first used. Then a mean of the remaining latitudes is taken, and the probable error of a single observation, e p, is computed. Any latitude with' a residual, 0O, equal
to or greater than 3Y2 e is automatically rejected. In addition, other values may be rejected if the residual is excessive when compared with all others. Before final rejection of any value, the records should be reviewed to determine whether a star has been misidentified, level values follow pattern, or the turns of the micrometer may be a full turn in error. These are the most common causes of error. Notes by the observer may indicate doubtful observations. Another criterion for rejection is table XI. For a small number of observations, its use may increase the number of rejections. (c) After all rejections have been made, the accepted observations remain to be adjusted in order to determine from the observations themselves the most probable value of a turn of the micrometer and the most probable latitude of the station. The information at the foot of the third column is now entered.
Then a mean
4m is taken
of the unrejected latitudes in column entered 4 and the difference A¢= in the next column and summed algebraically at the foot of the column. (d) If p is the number of accepted latitudes, c the amount in seconds by which the mean latitude deviates from the probable latitude, and r the amount by which the preliminary value of one halfturn of the micrometer is to be corrected, then there will be p equations of the form. c Mr +A = v. Inserting the condition that [v2] or [vv] must be a minimum, the normal equations to be solved for c and r are
I
PROJECT
ASTRONOMIC LATITUDE SUMMARY
I OBSERVER
(TM5237)
DATE
A'.A/.CADDESS CHIEF OFPARTY
19467
/9528
3.2
/9687 /_9800
1/9742
+72
20041
20/51
20308
2042/
+
,S.6
20586(o +2
20696
20744
2/032 21246 21534
2/086 2/32/ 21684 2/86 3
+&6.9
4.8 3.7 +8.2 +0.8 4.9 +4.9 +4.9
22003
21937 22094
!7~
IMARYA Mr
2
/44S .0s
01.23
0
01.26 0/.33
0.0.8
_
4.o61.32
 oS
4. ?$. +. 2/ +__ 1. 28 4_
+o.42
00. 76 01.54 00.67
___
0.36 +i05S1
. 20
__
+0o.50
00.68
00.40 + o. 78 00.32 1+0.86 02.23 i/. oS
 .
/S
___.20
24003 24 /SS 24279 24433
#3.1
0o.9
+ o. 19
.13 +.o2
24538
24413
2.7
00.86
+0.32

24699
24816
+2.9
01.41
24 92o'
0. 5
4.8
25.040
26290
26358
2.3
26475
26 542
1.2
26996
26749 27047
1.9 6.9
26,632
50.3
ALGEBRAIC SUM
+8.0 0.32
. 14  .0/
_

0.62
40.67
. 13 1.0o/
 0.30
0.6o9
01.0/
+0.19
00.73
+0.45
0o.08
.1 ./12
0/.53
.20
00.66
00.40
. 08
00.39 01.93
0.33
+ 0.54 f 0.80
_____ +.7/ 1.03
___.28
6.48
/2.12 /7.47 29.59 01.18
(o.57 __
_
_
0.73
1__ 4.15
6.54
5.71
6.65
__
+.08
9.2468
__
.09/0.0279
+0.81
___29.810
_01.20
_
UNADJUSTED MICROMETER (1)
MEAN
0
ONEHALF TURN
121 MEAN 0
13
DIFFERENCE
121
01.0/
01FF.
0.'. 34
PAIRS WITH MINUS MICRO. 0IFF.
(1)
+
2/
_01.
.33
+
03
NORMAL EQUATIONS
MASERMI
COMPUTED
N
LATITUDEFGDEISTIN
BY
DATE AMS
IJU l
CHECKED
'(p3
0.
38n
57
Astronomic
Latitude
DATE
BY
R.fli.oka~  A MS
Summlary
.0
(03.240
DA FORM2843, 1 OCT 64
Figure 32.
(DA. Form
_
ADJUSTED
76.54/26 0.1/8
76.5
VALUJE
PAIRS WITH PLUS MICRO.
12
__
+0.73
01.89 01.50 01.28
.09

0,36 +0.23
00.89 +0,31
.
0.23 00.86 + 0.32 00.49 + 0.69 02.21
42.3 l
 o. / / #0.421
004
 SUM
MEAN
+~ .14
o.46
b00.764
01. 33 o2.2/
.o0
0.57
R
+±SUM
+20
__o5
/.o3
01.82
0o.,2__
+0.46 00,35 +0.85 02.03  0.83
+__ 03
. 1
+.0 o
00.97
130 +0.43 + 0. /1
00.74
R___ 01.13 0 2.2/ R 0 1.7$ 01.64 0. 29
01.66
00.47 00.53
+__ "34
22785 22980 23225 23574 2.39/9
f
02.50
1. 3
+__~.30 00.77 . 1/4 0/. 09
oo.47 +0.7/
223,42
0.2
___2
Q,
22646 22866 23/32 23433 23 770
#9.6 3.3
ND
ADJUSTED
22216 22382
3.9
2
i
63
Io2
 3.5 i.S
/8907
_
AS7RO
LOCATION.
r4 (No. 56o95
WILD Mic. Diff. j M 381
STAR NUMBER BOSS GEN. CATALOG
2/761
MAP
PM
}INSTRUMENT
R?.SALVfiRAVSER
_20489
STATION
2 MAY /963
2843).
DEC. '(03
pc [M]r+[A4,]=0  [M]c+ [MM]r
[MA4] =0
where [ ] is the standard symbol indicating summation in operations with least squares. In the solution by the Doolittle method (fig. 33), two columns are added to obtain the proper coefficients necessary to determine the probable errors of the micrometer and the latitude. The equations take the form: pc [M]r+ [A+ 1.0+0=0
 [M]c+[MM]r [MAC]+0+ 1.0=0 (e) Each micrometer difference, M, of the accepted latitudes is now multiplied by the value of r and the results placed in the proper column. Then to each preliminary latitude there is added algebraically the corresponding Mr to produce the corrected latitude which is entered in the designated column of DA Form 2843 (Astronomic Latitude Summary). A mean of these latitudes is now taken and entered at the foot of the column. In the next column the difference, mean latitude minus individual latitude, is set down for each pair of stars. The sum of the squares of these new A¢'s is entered in the next column. The mean of the corrected latitude is the mean observed astronomical latitude of the latitude station, uncorrected, however, for elevation of station above sea level or variation of the pole. (.f) The probable errors have been computed by means of the following formulas with data used in the adjustment. =
±
/0.455[A 2] p2
V=
[M
P
2
[M]2
(p2) ([MM]
==
eP
[MM]
[M2
(9) The correction to the latitude to reduce it to sea level is given by the following formula: A=0"000171 h sin 20 where 4 is the correction in seconds of are, h the elevation of the station in meters, and 0 is the latitude. This correction can be obtained directly from table XIV. (10) When the x and y of the instantaneous north pole are known for a given date, the reduction to be applied to an astronomical latitude observed in west longitude (X) to reduce it to the mean pole is as follows: A0= (x cos X+y sin X), x and y are in seconds of are. (11) If the observations have been made from an eccentric station, the reduction to the geodetic station is computed by cosine of the azimuth x distance in meters divided by difference per second of are in meters. The sign of the correction should be checked on an oriented sketch of the eccentricity.
25. Latitude By Meridian Zenith Distance of the Sun This is a rough method used for convenience in observing and it will not yield precise results. The computations are similar to those in paragraph 20, except for the necessary determination of the sun's coordinates. a. The field observations are usually made by following the sun in altitude near noon and accepting the highest obtained altitude as the meridian altitude. This ignores the slight difference in the meridian and maximum altitudes due to the changing declination. In finding the maximum altitude by trial, it is seldom possible to secure a reversed pointing. Hence, the observed altitude must be corrected for index error of the vertical circle, refraction, semidiameter, and parallax. b. A more accurate method consists in knowing the meridian from a previous azimuth observation, or in computing the exact time of the sun's transit from a known watch time and the station longitude. The vertical circle is then read a few seconds before transit, the telescope reversed and the other limb observed. The index and semidiameter corrections are thus eliminated.
MAP ASTRO 2
OF
SOWT/oON
c 25 ooooo C:=
ADJUSTMVENT
LATITU E
r
aR
 8.ooooo
 0.0 9000
+ 4. 00000
0.00000
+0.32000
+0.00360
0.04000
0.00000
/8.99200
0.00000
______+463.52000
2.56000

+460.96000
s.
0.02880
+0o.32000
/902080
+0.32000
r=+ oao4126
0I0000O 0.00000o +
~00069
1.00000o
0.00217
r= +. 04126__ c: +. 0/(680
____
_____
+./0.02790_
_
0.00032
0.04000_____
0. 78480

9.24278,(0. 45495)
 0.002/7
.04022
. 00 217
e e
=
Al
&. Ifly
61
7
_
___
_
_
_
±0.08S5.
2
.0021~7
TABULATED BY*
DA
±0.42758
23
______
_
0.00022
9.2 4 278
e1
_
0.01/P92
=±
DATE
.
11MS
CECKED BY
Jul363
b
GPO 380647
Figure 33.
_____
DATE
0.1?.
aU.
AM s
S. GOVUBUmWa
Solution of latitude adjustment.
DETC.
PR99Th63 OFMZ : 1957
6
 421132
3
c. The computations for a above, are(1) Correct the observed altitude (or zenith distance) for index error (if known), refraction, semidiameter from the American Ephemeris or equivalent, and parallax from table VIII. Parallax may be neglected in rough work and when the index error is unknown. (2) Find the sun's declination as follows: (a) Accepting the observation as having been made on the meridian, the local apparent time is 12". (b) Add the longitude to obtain Greenwich apparent time (GAT). (c) Subtract the equation of time from GAT to obtain UT. (d) Correct the apparent declination for the date for the elapsed UT from Oh. (e) In case the local standard time of the observation is recorded, the UT is found at once by adding the time zone difference. (3) Apply the formula: 4= +(90
or
0
h)
26. Latitude By Circummeridian Zenith Distances of the Sun This is an extension of the previous method for greater accuracy. It is similar to the method in
paragraph 22. The observations are made starting about 10 minutes before local apparent noon and continued for about the same interval after noon. Pointings are made in direct and reversed positions alternately upon the different limbs. The computation procedure is as follows: a. Take the means of each pair of D and R pointings and the means of their vertical readings. b. Determine the hour angles (t) of the mean time of pointings on each pair. These are the difference between the observed time and the time of transit. c. Scale the approximate latitude from a map, or compute a trial latitude using the highest observed altitude. d. Apply the formula, finding the corrections Am for each value of t, and find the equivalent meridian altitude or zenith distance by the equation: hm=h+Am or ,'m=  Am e. Mean all the consistent values of hm (or tm), and apply corrections for refraction and parallax. f. Obtain the sun's meridian declination by finding the UT of transit and using tables of the ephemeris. g. Apply the following formula:
4=
(+fm
4= 8+(90°hm)
Section IV. DETERMINATION OF LONGITUDE 27. Basic Method The longitude (X) of a place is the arc of the equator between the meridian of the place and the primary meridian of Greenwich. Since there is a direct relationship between longitude and time, determination of the time at the place with respect to the time at the meridian of Greenwich will establish the longitude of the place. Present day radio time signals broadcast by WWV, WWVIH, GBR, JJY, and several other major observatories have been synchronized and provide an excellent means of obtaining time at the meridian of Greenwich. Time at the place is determined by observations on various stars using several different methods and procedures. 28. Determination of Longitude By Star Transits The most direct method of determining longitude is by observing the instant of transit of known stars over the observer's meridian. At that instant, the observer's hour angle is 0h and the 757381 0

65

5
local sidereal time is equal to the right ascension of the star. This method is applicable to any class of observation but is seldom used except for first or second order work since the preparatory work of placing some types of instruments in the meridian (para. 29) will provide a longitude having the required accuracy for lower order work. a. The instruments used in this method are usually large meridian transits or universal type theodolites with very sensitive levels and impersonal type, automatic recording eyepiece micrometers. b. The following formulas and identities are applicable: AXAa+ (a+AatAt)=v A= sink secb= sin tanb cos4 B= cos sec 3= cos O+tanb sine C=secb k= 0.0213 cos
4 sec6
l=2 (m+s) C
b=i(d)/15 (n) The symbols in the above equations are: AX= correction to an assumed longitude A= azimuth factor of the star a= azimuth of the line of collimation (amount. by which the instrument is off the meridian assuming the collimation error of the instrument to be negligible) a=right ascension Da=the short period terms of right ascension t=mean time of transit corrected for levels, diurnal aberration, width of contact strips and lost motion At = chronograph correction v=residual of a star in the solution of a star set (usually six stars) = zenith distance
= declination 0= astronomic latitude of observer
B = level factor C = collimation factor k=diurnal aberration. The sign is minus for stars observed at upper culmination and plus for subpolar stars (i.e., observed at lower culmination) 1=correction for width of contact strips and lost motion R= equatorial value of one turn of the micrometer in seconds of time m=lost motion in terms of divisions of one turn of the micrometer s= average width of contact strips in terms of divisions of one turn of the micrometer b=inclination error in seconds of time i=mean level value in seconds of are per division of bubble d=difference of bubble readings (refer to instrument manual to determine sign) n= number of level bubble readings c. The following data should be furnished by the field party: (1) DA Form 2844 (Longitude Record) containing the following information (fig. 34): (a) date and headings (b) star names and/or numbers (c) level records (d) time of radio time signal comparisons (e) remarks as applicable (2) Chronograph sheets and/or tapes which contain the record of the star trackings and radio time signals (3) Instrument and level constants including
information as to when and how determined (4) All data abstracted on the proper forms including the scalings from the chronograph sheets and/or tapes (5) Field computations d. Scaling of the Favog Chronograph record is fully covered in TM 5667521015. In scaling other types of chronograph records a suitable glass scaler or a variable scale may be utilized. Figure 35 is an example, using a glass scaler. (1) In scaling radio time signals on DA Form 2845 (Radio Time Signals) (fig. 350), 20 breaks should be adequate if good reception was obtained. The mean epoch of the radio time signals is reduced to the nearest second and the mean chronograph time is corrected to that epoch. (2) In scaling the star transits (fig. 36), at least 10 matching pairs of breaks are required. If the residuals of the least square solution for the star set appear erratic, it may be desirable to scale all recorded matching pairs of breaks, make the obvious rejections, reject others on the basis of pattern, and then obtain a new mean value for use in the computations. With an experienced observer, it is seldom necessary to scale additional pairs except under conditions where it was difficult to track the star. e. DA Form 2847 (Comparison of Chronometer and Radio Signals) is used for radio time signal comparison computations. The procedure is as follows (fig. 37): (1) Fill in all headings. The latitude and longitude should be the closest approximation which is available. (2) Enter the year of observation and the meridian of the local time which is being recorded. (3) For each column, enter the local date, recorded local standard time, the chronograph time of signal, transmitting station, and frequency on which received. (4) To the local standard time, add if west (and subtract if east) the meridian of the local standard time expressed in time and fill in the appropriate date and Universal Time (UT). (5) From the American Ephemeris and Nautical Almanac published for the year of
LONGITUDE RECORD (Original Trait Level Readings)
ROJ, CT
(TMt 5237) SAION
LOCATION
MAR~YLAND
MAP ASTRO
ORGANIZATION
CHIEF OF PARTY
IUSA MS
INSTRUMENT
R. SAL VERAMO1SER __ _
531 (E)
(ta __
_ _ _
_
_
A .3(6 .36
1A
72.0
34.1
572 (E)
33.3 38.7
71.1
oC
37.0
+./6 33.1
71.0
/430 14 IA
72.2
34.3 36.7
540(E)
14
. 20
/6
A
14 54 24 Pike4 u~ ate 555(w) A /5 00 44
EA
di
/50410 fvt z 1
563 (w)
A
/S /368 IA
70.2
30.8
IA
+.20
33.0 372
72.6 41.8
A
+,23
.33.0
72.5
+.43
65!.4 29.8 36.4 42.7
/5
578 (w)
/5 33 04 IA
70.3
71.1
.34 392
32.S 378
15 44 25 Picked a late ZA
+.42
71.0
32.1
32.o
71.0
+.08 39.0 . o4 32.1 71.3
392
39.8
+.i0
32.0
71.3
73.0
33.7
41.0
376o
. /0
REMARKS
W WV
/S
s Er 2
N
/4.18
(2) /434
s ETr 3
(1)
I523
(2)
1541
IA + .5(o 36.7
42.3
/5 56654 Pkke4 ~Ite IA
38.7
A 48
33.8
73.4
691
29.3
+,08 36.3 __
71.8 _______
_______
MC
(V) /440 (4.) (;3) /6.59
144.5
(.s) /5/6
sint proper sequence with stae to show time of receptiom, include following data for each tine signal received: Time (Local, standard, etc.), radiostation identification.frequency. DA FORM 2844, 1 OCT 64
Figure 34.
73.0
73.1
32.2
.20
43.5
33.6
.08
/S S/ 2/ 595(w)
32.0
31.3
73.3
70.3
A
71.1 38.9
71.1 4.o4 37.9
29..8
41.7 30.8
/416(EW
'38.9
713 32.6
24
A
/5 26
E.
+,20
A
.36.3
+.02 31.9
FA +.04 13 95(E)
LEVELS W.
40.1
72.0~ 33.8 .3/ 9 70.1
Z A .36
1386 (w) A /4 4739 Di thy&4haze EA
________
/42 (E) A . 20 65'9.3 /S537 03 33.8 Pcked fte la FA .+.23 35.5S 583 (W) A +.41 33.2
A .. 3728
S51 (E)
39.1
3
STARS (Or Signals)
E.
A
534 (W)
Ver
W.
_
/4k23'.565
cc
H. N. CADOESS NO:
LEVELS _
RECORDER
_________SET
Sidn"i*) __
0.
( oca
/8 JUNvE 063 pm
WILD T4 No. S4095
SET NO.__2
STARS
P9
DATE
1247i4
R.SALVERMOSER
OSERV[R
2
CHRONOMETR
Longitude Record, DA Form 2844.
4441
NOTE:
As shown here, it is not uncommon to scale the chronometer time of the ending of the JJY signal rather than the resumption, 0s02 is then added to the scaled chronometer time.
4% i
Il
IL
(
IIcL
I
p
ryO
F,
U,
iI 11111111 _
H
p Figure 35.
I
i I I/
' * '
01
1
// l
Graphic sample
Time signal scalings.
the observations, obtain both the sidereal time of OhUT (apparent sidereal time, HA of first point of Aries) for the UT date and the change in nutation. "Change in Nutation" is the proportional part of the UT day multiplied by the tabular difference of the Equation of the Equinoxes for the date. The sign of the tabular difference is determined by reversing the sign of the equation of the Equinox at Oh and adding algebraically to the equation of the Equinox at 2 4 h . (6) From table IX in the Ephemeris (AE&NA) determine the correction mean solar to sidereal time for the UT or multiply the UT expressed in minutes by 0.1642746 which is the rate of change per minute. (7) The transmission time between the transmitter and the astronomic station may be determined by first using the formula; Cos D= sin 41 sin 4,+cos 01 cos 42 cos AX where
1
and 42 are the latitudes
of the respective stations, AX is the difference in longitude, and D is expressed in degrees of are and decimals thereof. Then the correction for transmission time becomes AT=0.000401 D, the constant being based upon a speed of 278,000 km/second for short wave reception. (8) The "correction to signal" is obtained
f.
from "Time Service Bulletins" published periodically by the Observatories monitoring the time signal. If the monitoring observatory is not close to the transmitting station, it will be necessary to correct for the time of transmission between the two points to obtain the correction to signal at the transmitter. The UTO corrections are currently being used. If a common pole is adopted in the future, it may be desirable to convert to UT1. The correction used should be identified so that future conversions may be made if warranted. (9) To obtain the Greenwich sidereal time, add (3) through (8) above. (10) From (9) above, subtract the approximate longitude to obtain the local sidereal time (LST). (11) The chronometer correction is the difference between the LST and the chronometer time of signal. The correction is positive if the chronometer is slow and negative if fast. (12) The rate per minute of the chronometer is determined by dividing the difference between two chronometer corrections in seconds by the difference in their chronometer times in minutes. The rate is positive if the chronometer is losing and negative if gaining. Computation of factors A, B, C, k and 1 are
IRADIO
PROJECT
TIME SIGNALS (TM 5237)
OBSERVER
.
DATE
RECORDER
STATION
MAP AS TR O 2
&1'vE. /1963 PMq
/8
SAL Vi/?MOS'E
TIME ZONE OF SIGNAL
H. CADDESS
FREQ OF SIG NAL
WW' 1/'
R
Local Chrono Sending Correction t Local ChronoLclCrnTime of meter SednReduce Timeto SednIieo meter Time Time of of Sga Reduce to SinlSignal Mean Epoch Mean Epoch Sinal Sina ___h. ~ . __ /h. 4_ 12 m. 2/ h.
m.
34
S.
m.
22 24 26
/2
S.
S.
.668
.6o2 .6o7
.033
.673
25
.601
.677
27
/(6.56 /8.56
.04/
.027
20.57
.030
.600
3/ 33
22.57 24.58
.025 .0/9
.595 .599
26.59
.014
.604
28.65 31.58 33.59 3655
.008 .000
.598 .580
OO
.85
. 0//
.579
26. 66' .684
.675
23.66
0/4
.008
.669 .664
39 41 43 45
26.66
.003
0/9
.
14..56
29 35
.036
.596

.008 . 0/4
.662
.666
33.68
= .019
.b6/
48
396/
. 022.
.588.
35.69 3769
025 . 030
.665 .660
.50
. 027
.583 .S77
 .036
.654
S2 54
41.6/ 43.61
. 0o41
.659' .663
56 .58
.67/.
60
32969
.
41.70
S57
43.7/
. o47
S8
44.72
. o49
20 34
EPOCH
MENSCLD EDIG
ADOPTED
42
2967 3/.68
.003
55
0 .,2
/9
37 40 42 44
2767 .
CMN TO SID (
o4
.668 .663 .667
669
____MN
MEAN
42 m. s.
s.,
31.597
.025
S/ 53
/4 h.
S.
.052 .047
1765S
9
h.
.057
/6.65
47
14 m.
LcaChnLclChnMean Epoch
/255
30 31 33 35
37
S.
meter Timo meter Time of
/0.54
.044 .038
/9.65 21.65
m.
MCS oretno Crecint ofLoaCrnMean Epoch
2/ 23
.049
08.62 /0.64 /2.63 /4.64
28
__
S.
/
.40.2
.675
)
MEAN EPOCH
CORCHO TM/4 CMUEYDAECHECKED
OOOS
20
.
.
033
45.62
'.038
47.63
. 044.. 049 . 055
49.64 51.64
.582
.1586 S9 .S85 '5/
___M
___________
2( 04
MEAN EPOCH
3,92 .5895
MEAN SCALED READINGS CORR, MN TO SID
(.8).0022
34 40ADOPTED MEAN EPOCH /2 26.667 CORR CHRON TIME BY
DA FORM 2845, 1 OCT 64
® DA Form 2845 (Radio Time Signals)
Figure 35Continued.
2/ 04 40
/4 42
31..59/7 DATE
STATION
MAP
PROJECT
2
ASTRO
LOCATION
531 14
TR IN
h
17.5
STAR
22
SUMS
37.7 36.o 34.2
55.2 55.4 55.2
5 24.5 67 26.0 7 < 277
30.9 293 27.5
55.4 55.3 552
8 9 D'
292 3/.2 32.6
26.1 24.4 22.6
553
2
36.3 37 7 39.3
/9.3 /7.5 /6.0
556
S.52
5' 41.0 6 42.9
/4.4 12.7
71 8
44.4 46.1
9 c
I 2 3
4
WILD T4
534 h
TRIN
2
INSTRUMENT'
R. SALVERMOSER
/4 h
TRIN
LOCAL DATE
56095
/8Ju.63 540
STAR SUMS
9
/4
Ii
SUMS
68.7
91.2
2
/8.5
396
58/
5 6 7 8 9
67.5 66.4 65.1 63.9 62.8
90.9 R 91.2 91.0 91.1 91.2
3 4
20.1 21.4 23.1 24.3 25.8
38.0 36.7 35. 33.7 32.3
58.1 58.1 58.2 .5.0 58
C / 2
29.6 30.8 31.8
61.5 60.4 59.2
91.1 91.2 9/.0
8 9
27.3 28.8 30.1
30.8 29.5 28.0
58.1 58.3
58.1
33.2 34.3 35.6
579 56.7 55.7
91.1 91.0 2 91.3 R 3
31.6 33.1
26.6 25.1
563
3 4 5
34.5
55.4 556
6 7
36.8 379
54.4 53.1
91.2 91.0
4 5
35.9 37.2
11.2 09.4
556 555
8 9
39.1 40.3
52.1 50.9
91.2 91.2
6 7
47.7 49)4
077 06.2
55.4 55.6
b 1
41.5 42.5
49.6 48.3
9/.1 9 0.8R
/
51.1
04.6
2
52.8
55.7 R 55.6 EAh 55.40
3 4
OUT
21.0
/4
TRANSIT TIME
02.8 h
25 /4
m
23
.56 56 2
57 700
Scaled By
R..±2V
TRANSIT
30
TIME
/4 h30 DATE
t*1?6s L  AS
m*+MEAN
JUNe
14
h
I 39.1
46
43. 2
m
SUMS
82.3
40.4
41.9?
82.3
41.7
40.6
4 5 6
3.o
392
44.3 454
38.o 36.7
82.3 82.2 82.3 82.3
7 8 9
46.8 48.2
49.4
355 34.1 32.8
82.3 82.3 82.2
58.2 58.2
C
50.8 52.0
31.6 30.2
82.4 82.2
23.8
58.3 58.3 58.0
2 3 4
53.4
22.4 20.8
28.9 27.6 26.3
82. 3 82.4
38.9 40.3
/9.4 /79
58.3 58.2
57.2 58.6
8 9
41.6 43.2
/6.5
58.1 S8.3
8
44.5
/3.8 /2.3
58. 3 58.3
25.0 23.7 22.4 21.2 /99 /8.6
82.2 82.3 82.2
/5./
5 6 7 8 9
S
6 7
C
91.12
OUT
.5 560
TRANSIT TIME
/963
/386
IN
2
46.0
/4
TR
.3

OUT
4
STAR
36
22.5 23.4 24.8 25.9 272 28.4
/9.4
STAR TRANSIT SCALINGS (TM 5237) No individual sum shall exceed the mean by more than ± 0.2
SET NO.
USAMS
OBSERVER
MARYLAND , USA STAR
ORGANIZATION
TEST:
h
458.18
14
im *S
/
37 29.090 .
55.9'
59.8 6/.1 62.4
8
8OT4
OUT
Checked By{
54.8
63.7 m
TRANSITTIME
AMS
h
48
14
82.2
82.3 82.3 82.3 MEAN
82.28 4m411
DATE
July
1963
DA FORM2846.1 OCT 64
Figure 36.
Star Transit Scalings, (DA Form 2846).
made directly on DA Form 2848 (Astronomic Longitude Data) (fig. 38). (1) Extract tan 5 and sec S from the fundamental catalogue at the bottom of the page for each star. If greater accuracy is required, the should be determined to the nearest second and tan 8 to five decimal places. (2) Apply appropriate formulas as listed in b above, and as listed on the form. g. The level correction to the time of transit is computed as follows: (1) On DA Form 2844 (Longitude Record) (fig. 34), subtract the west bubble readings, the difference is considered positive; then subtract the east bubble readings with the difference considered negative. The inclination is the algebraic sum of these two results, which is entered on DA Form 2848 (fig. 38). (2) Multiply the result of (1) above, by i/60, where i is the mean level value in seconds
a
per division of the bubble. This result is the inclination error in seconds of time (b). (3) The total level correction in seconds of time is the product Bb which is entered on the appropriate line. h. The uncorrected transit time is the mean of the 10 or more matching pairs of breaks from d(2) above. i. The mean corrected t then becomes the sum of the scaled transit time + (l) + (k) + (Bb) which is entered on the t line. j. In computing a and Aa, it is necessary to proceed as follows (fig. 39): (1) Determine the mean epoch of each star set (usually six stars). The mean of t for the first and last stars of 'the set will be sufficient if the star transits are evenly spaced. (2) If the chronometer correction is greater than 10 minutes, it will be necessary to
OF CHRONOMETER AND RADIO SIGNALS
PROJECTCOMPARISON LOCATION
(TM 5237)
HRNMT
STATION
M1ARY[IA N0 D_______
/24 74
ORGANIZATION
LAIUE
USA MS
/8 JUiNE
TIME OF SIGNAL MERDIA
SIG~NLTE
H
M
20 34 40
26.667
14 12
TIEO
wr w vW
FREQUENCY
/S Mc
OF SIGNAL
UNIVERSAL TIME (U. T.) OF SIGNAL
DATE
JuN. /7
T..
H


ACHRQNOMETER
COMPUTED my
AM s.
Mc
WV /SMC
M
S
M
3.0032M3.0
.00/
}
.00/
M
T
26.210
S
.00/
33.7S8
00
0
.372

UTO
08 2? 6 08 29. /7 29.9S5 16 0334.503 /7 30.3/6 /6 0334.858
31.3S
.367
f
.0. 006
8
correct ion
25 58.955 2/ /2 03.503
*0.
006
74m
7
si9 nal).
IDATE ICHECKED BY July33 6'(oca
DATE
AM 5
JulyI63
DA FORM 2847, 1 OCT 64
Figure 87.
34. 858
03
10
M0
SPre/itninary
/5S
39
.480
#0005
CORRECTION
cf
2
/9 20 5.2 95 /9 5/ 00.225 20 5 0 8 29 S 08 29 5 1_ 4 /2 26o.29.5 /4 42 .3/.22.5 /S 14 12 26.6167 /4 42 31.5S9216'
CORRECTION
30. 316 /(p
4~5 .59.845 /7 45 .59845 /7 45 59845 /7 45S9.845

LOCAL SIDEREAL TIME
____3_22_25_J
17/7
/9
0
CRNMTRTOFSIGNAL
0
I6
WW
0M4,00
,55120
STATION
CHRONOMETER
2______
WIVIw
S
TO SIDEREAL. TIME
LOMGITUDE OF
04 40 42 3/592
. 000
SIGNAL
/8 JUNE
1.19
N
NUTATION
CORRECTION TO SIGNAL
/8 JUNE
MS
/M
TRANSMISSION TIME
G.S.T. OF
/4
34 40,.000 2
CORRECTION MEAN SOLAR
21
H
/9 /
TIME
TIME OF OhU.
CHANGE IN
/8 JUNE
S
TRANSMI TTING STATION
SIDEREAL
T RO 2
T__m____ST m 08 t_29__00_A__
Jv)NE /963
LOCAL DATE STANDARD
MAP A
LONGI TUDE
Comparison of Chronometer and Radio Signals (DA Form
2847).
PROJECT
OBSERVER
TEST
LATITUDE (40)
1425
8ec
+
INCLINATION
b(1ncl.xrdt)
~
Transit Time
1.7
+.0317(a
2.3
I
H Bb t
/4h 23
At
a
Im
s
0
+.07098
s /5.560
mo
s
.120
. 023
052
+..050
+.
57.862 30
15.692 37
009'
. 369 /S.3/5 37 . 0//
. 137 l =Y2. ' l
(WE)d/60
(4)
(5)
(6)
(7)
 .0/7 +.002
54
. 0/0 .
027
.022 (In+
s) sec S
.
a
m
10
04
34.907

04
34.525 04  .0/0 . 028 . 0340
1/03
A=
da (tA)
g01
~
a
/4
02.300
./O2 .
.030
+.
14.093 /4
13.808
AT LOWER
/4
0o2.o63
. 009

. 043
. 244
+.//s .
d qi+:da (E)".df
.~L.
BY
6 6
GATE
BY
CHEKE
.0/0
+. 069
______jmA________MS___
CULMINATION K IS POSITIVE.
20 076
02.458 . 3(02
. 3(o3
8
0seSCMPUTED
OBSERVED
in
s
14.020 .128 .. 025
. 344
+. 4260
1/ 93
5h/
. 022 + .007
. 012
+.0635/
1.483
.112
.088
.3.4
July_____&3_
M
DEc
64
Figure 38.
(3)
40.877
34.
1.1/99 +
IS ALWAYS POSITIVE.
DFOR STARS
DA FORM 2848, 1 OCT
m
s
/4
+0. 66 /
. 02055 h
.20.06500
.108  .021 +.033
0/7
+.
/70
C
m.
54
.663
1.1
1.315
00
+.
 tan 8 CoG I/ + tan 8 sin 9S +seS
0.941 /4
OF
+.33 27
/. 503
+. 00.660
29.682
. 348 0
S
41.140
. 0,0
A
GiD
m. 47
28.945 47
019
COG
.+.00/87
39 17 46
+. /22
+0.3
00
 .
sec s
+
29.28(4 47 41.260 54' 3o. 138 . 368 . 36(a . 365
.F. 041
B =COs
/.31/6
+.099
a+IAa(t+At)
A = sinsec=
.033 0./
14
.099
_
1.24~8
PAGES
2
+48
18
+0.8.55
1.268
29.090
. 0/0
_05
I
555 + 40 32
+ 0.260
+.026/5
 .019
141 23 575 42 30
551
(+
s)=
+0.780
1.397 14 3
to
Y2
+ 14 35 45
+1.4
.139
. 370
Da
0. 98S 1. 404 + 3.8
de . 207
PAGE
8"'295
/38(c
. 0o27
+)
t
540
1/ 49 14
.57.700
5%
+ 44 33.59 +37 .57 55
+.04483
15S83 14
8
K
534 32 06 + 0. 590 1.16o/ + 2.4
+.520/28+.30
52/
LONGITUDE()
*.7776o9
COIN
2
dTP
l_1/9.09/J114.(,
i8 JuN (3
6 8
6286
INE
+ 1.281
Scld
18
MAP ASTRO 77
GREENWICH DATE
93IE0COIE1
N 38 .57 0/ .531
s
CC
s0
d/60 =
(TM105237) STATION
56095
LOCALDATE
0
DECLINATION(s) TAN
WILD T4
LEVEL VALUE (dt)
u5 AM S
ASTRONOMIC LONGITUDE DATA
2
INSTRUMENTTy'pe a.dNo.
Hf. CADDESS
ORGANIZATION
STAR
12474
RECORDER
MARYLAND
SET NO
CHRONOMETER
R. SAL VERMOSER~
LOCATION
Astronomic Longitude Data (DA Form
apply the chronometer correction to the mean epoch. Add albebraically the X (+ if west,  if east) to the Mean Epoch. Determine the time interval in sidereal units by subtracting the value of the sidereal time of ohUT for the nearest preceding date. Convert the time intervals to minutes and divide by 1444 to obtain the decimal part of a day. This is the Greenwich civil date or the UT date. Use the UT date to interpolate for d1, and d& in table 1 of the Fundamental Catalogue. Interpolate for a in the Fundamental Catalogue including the second difference interpolation as explained in Chapter 2. Since the interpolation factor 77is equal to the X expressed as a decimal of a day for one day stars, then for 10 day stars it is equal to '10A plus 1/10 the number
2848).
of days between the date of tabulation and the date of observation. As a check, subtract the UT date which is listed to one decimal for the date of tabulation from the UT date of observation computed in (5) above and divide by 10. The first two decimals of this approximate 77should agree with the precise
77.
(8) Compute Aa by formula [da N,) X d4, +fda (e) X dE] and enter on form. (9) If a and LAa are not in the FK,4 system and corrections to that system are available, it will be necessary to apply these corrections to either a or Aa. k. The chronometer correction may be determined from. the following formula: zAT=Co+ (tt 0 )r Where Co is the chronometer correction of the nearest preceding chronometer/radio comparison, t is the chronometer time of the star observation,
TABULATION OF GEODETIC DATA
PROJECT
(TM 5237)
MAYADORGANIZATION
LOCATION,
USA MS
ARYLAP4U
STATION
INTERPOLAT/ON
MAP ASTRO 2
t t+
+f
At
/2 43 2S
LAST)
/3
/6 54
/4 2 38 /S14 02
/2
59 40
/4
49 00
/,526 20 /S 657 /6 41 38
08
S
08
S
APPROX.
S
G. S. T. S.Tooh U.T SID. INTERVAL G.C. D. d
2 00 43
/965729 /746 00 21/l
.02/ .062
29
.52/
.02
17 2647
29
08
2200o43 /7 46 00 4 14 43 3 04 07 JUNE /9.1/28 JUNE /9.176 . 205 .0/1 .203 . o/3 ,52/ . 062 .52/ 0162. 2060o07 /7 46 00
u
Figure 39.
_
_
_
____
_
_
_
0. 2/4
HCE
YDT
_
______
24
4EOF'
/65S2 /4 S 08 29
_
.521
°. TIM BY A,4M sI DA 11w51962 FOR
/6/68 40
/9.1 /3.9 . 52
25.1 24.9
17.021
C'oMPurATON
29
MY26.084 JUNE /9.0 9/ o082 .2o7 . olo +o7
G.C. D TABULATION DATE APPROX.A
TABULATE
29
/8 08 09 /6 07 26
2L.005
ACTUAL
SET 4
Ss T3
S6T'2
(14) MEAN
d
se r
FACTORS
_
FOR
_
_
_
_
/0 DAY
YDT
c QR.T~. BYGO984 AMS GPO *G6647 '6o3
U.
I. GOVERN
Computation of interpolation factors.
I Oc. 6o3
,1 ParrTah OFWWZ: 1957
 4211!2
to is the chronometer time of the signal from which Co was obtained, and r is the rate per minute by which the chronometer fails to keep sidereal time within the interval containing the star observation. The algebraic signs of Co and r will determine the sign of the correction. 1. Compute (a+AatAt) for each star. m. Abstract on DA Form 2849 (Least Squares Adjustment of Longitude) (fig. 40) A and (a+AatAt) for each star, set up and solve simultaneously the two normal equations: nAX
[A]a+ [(a+AatAt)] =0 At)]= 0
 [A]AX + [AA]a [A (a+ at
Then use a to determine each stars residual by the following formula: AXAa+ (a+AatAt)=v n. In the above least squares solution of the star sets, it is not anticipated that good observations with present day equipment should result in any v being greater than 0.080 second. If any v is greater than this amount, an examination of that star should be made to determine whether any errors in scaling, level reading, or other factors may be the cause. If the value is not consistent with the remainder of the star set, it may be rejected. Any star with a v of more than 0.2 second of time is always rejected. o. The summary of results is computed on DA Form 1962 (fig. 41). (1) In the "Summary of Results" observations by a competent observer using good equipment should be such that the difference between the mean observed AX and any individual AX should not exceed 0.04 second. Any star set having such a large residual should be reviewed to determine whether any errors have been made or whether one of the stars of the set might have an "A" factor which was distorting the AX of the set. In a final examination, any star set having a large residual not balanced by one, equal to or only slightly smaller, with an opposite sign is to be rejected. (2) The probable error of the longitude is computed from the standard probable error formula: PE=0.6745/(_
1
In(n1)
where v=the deviation from the mean and n=the number of values
(3) The correction to the geodetic station, if applicable, is computed by the formula C=S sin A.H. C = correction in seconds of arc. S=distance in meters. H= Reciprocal of the meters per second for longitude using the 4 of the station as the argument in the extraction from the appropriate tables of meridianal are from the Spheroid used. A= azimuth to geodetic station. An oriented sketch of the eccentricity will provide a means of verifying the sign of the reduction. (4) When the x and y of the instantaneous north pole are known for a given date, the correction to be applied to an observed astronomic longitude to reference it to the mean pole is: AX"(arc)= (x sin Xy cos X) tan
4
West longitude is considered positive.
29. Longitude Using Time By Transits of Pairs of Stars Over a Great Circle Approximating the Meridian a. This method (fig. 42) is suitable for use with a small theodolite or transit, and is the method used for placing an instrument accurately in the meridian. It consists of selecting two stars, one north and one south of the zenith, which will transit at a convenient time and interval, say 2 to 5 minutes apart. These stars must have high azimuth factors, that is, at least 0.75 and preferably higher. The azimuth factor (A) is computed by the formula: A=sin
r/cos 6
A value of the latitude is necessary, preferably within a few minutes. The instrument is carefully leveled and pointed as nearly in the meridian as possible by a ground azimuth, corrected pointing on Polaris, or even by magnetic compass. The zenith distance of the first star is computed by r=46, and set on the vertical circle. When the star appears in the field, it is placed on the horizontal wire, and the time recorded when it crosses the vertical wire. Without unclamping the horizon motion, the telescope is pointed to the zenith distance of the second star, whose transit is recorded in the same manner. If the telescope
SET NO.
LEAST SQUARES ADJSTME NT OF LONGITUDE2 PROJECT
LOCATION
STATION
MAP ASTRO 2
MARYLAND OBSERVER
INSTRUMENT
R. .
__
T4
__WILD
STAR
(T p
CHRONOMETER
/24 74
No. s6oOPS
a+Aczttit
A ____ ________t ____
andNo.)
Adat+Aa
AA
/5'.09'
A.
eta
_
DATE
JUNVE
A _____
63 ,_____
____
S31
._ 368
1..o41
+073
4.o32
. 0/9
534
+./70
 0/5'
. 034
 oi5
j..o028
540

.1/37
+.0/7
+.o27
+. 0/0
t,.o03
1386
+.o22
.027
.
+. o23
.
S5/
+.426
. 103
. 084
#.019
. 006
6S55
. 036
. 028
+~. oo7
0,.3S
. 022
1395
. 244
+.06'
___
6S63
+./5S
. 043
_
+.048
__
__
004
+.o20
___.o23
.093 +*4391/ . 0858 9
____.05S2
+ 8.00000 o
+.05200
09300
. 006.50
+. 01162
+.43911
+.08589'
ooo34
..ooo60
+.438 77
408649

,
:
. 02/
.013
/0
fo34 .
007
+00/
.197/2
NORMAL EQUATIONS
nmx [A] a+[ (a+eateAt) 1.=0 [A]
ea+[MA] a
COMPUTEDOBY
[A (a.&atset) 1.=0
.DATE

A MS
DAjITE
CHECKED BY
IJu Iy' 63 D.&R.
n oka
 A MS
DA FORM 2849, 1 OCT 64
Figure 40.
Least squares solution of star set.
Iy16
G. C.D.
_____
5 ET
2 4.70o
9.128 9o l
J
V
adaX
+o./86
25.084 MAY w3
1
SLIMMAARY
LONGITUDE
ASTRONOMIC
MAP ASTRO 2
"N
S5
/9.248
"
6
.28.5

1'3 + 0 .14 .1 30o 0 74
+0.027 0 033 o
0222 0.,297 N.A~
______.
. 0.0/2
+0. 024
.0 / 6
+0.018
 0,006
0.004
+0.0/6
+0.012
Zv2 =. 000918 e =± 50037 (or) ± + '0(o
ASSUMED
MEAN MEAN
LO GITLJDE
A
0.5" 08'' 29.000 +______I
A A LoNGiTUDE
0SERY ED
ASTRO. LoNG I UDE
(rimrI)
(ARtC) MAP AST 0 2)
GEODETIC STArbON ASTRO. LoNGI UOE NP(AMS
RenucTioAI To
DA
~ 958))
DATE
TABULATED BY
o° 17962
00.012
+
AM'S
July
63
GPO 921961
Figure 41.
CECED
1
?.0 2 t S 0 0 4 07 /5/ 8 + 7'06 f 0 07' 1/.! 02.54
08
+ W W E
05 77 0
W
77° 07 /2. (4
770
.06
+
DATE
BY
D .R. ,ika.
2

AMS
July
(03
U. S. GOVERNMENT PRINTING OFFICE : 1957 0  4z11SZ
Summary of results.
LONG/T(/iUD BY TIME OF TRANSIT OF PAIRS OF STARS GREAT CIRCLE APPROXIMATING THEF MERIDIAN STATION: CASS
DATE : /5 FEB. 56
STARS
T/E: 20h40FS.T.
R/G7ASCENSION (og)
22 H CAML SCANIS MAJORIS
6h 6
14 18
05.5 39.5
30
A
9
369 0
314 68
03
A
LATITUD(): 38 08,
S

OVER
/47 +/.07
OBSER VAT/ON S TA S S 22 N CAML. $ CAMS MAJORIS
SOLUTION
t
6 6
/3 47.1 /8 44.6
(at ) + /8.4 51
SID. T. AT
OF 4T
0 0
CIIROA'. T OF SiGNAL CORR. (4T) L. S. T OF SIGNAL G.S. T OF RADIO 51G. L.S. T.OF RADIO SIG. (ARC)
00
00.0
2 00 00.0 9 39 53.6 + o /9.7 ii 40 /3.3
01U.T.
LONGITUDE (rjME)
Figure 42.
2/ + 5
CORR. FOR SID. GAIN G.S.T.
dT 1.47Q /8.4 4T + /.07a + 05.1 / 074lT  1.51290 19.688 47AT +. 512 90 + 7497 ,2.54 4T /2.191 4T =4 80
RADIO SIGNAL /5 FEB. 1956 TIME ZONE CORR. U. T, /6 FEB.
Ch fh S 6h 3)154.2
04.8
+
6 3/
59.0
1
40 3/
/3.3 5920
5
0 8
6
77 03
1 43
34.5
Computation of longitude using time by transits of pairsof stars over a great circle approximating the meridian.
axis is not accurately leveled, it will be necessary to obtain bubble readings and apply corrections. b. The computation procedure is as follows: (1) Determine the right ascension and declination of the stars from the ephemeris. To accomplish this, the approximate local standard time must be converted to UT by the difference for the time zone. (2) Determine the azimuth factors of the stars. (3) Apply level corrections to observed time of transit. The correction to the time of transit for the level error is: BXiXd/60, where B, a factor of the star=cosg/cos, d is the value of 1 division of the bubble tube in seconds of arc, and i is the inclination factor as explained under computation of azimuth.
(4) Write an equation for each star: AT+Aa (at)=0 Where T is the desired chronometer correction; A is the azimuth factor; a is the angle between the meridian and the line of collimation in seconds of time; a is the star's right ascension; and t is the recorded chronometer time. The values of A and (a t)
are known.
(5) Solve the two equations for AT. (6) AT+t=the required local sidereal time. There remains an error due to the collimation error of the telescope. This should be made a minimum by careful adjustment. The effect can be reduced by observing a second pair of stars with the telescope in the reversed position, and meaning the values of AT.
(7) Obtain the chronometer time of a radio time signal by comparisons. Add the chronometer correction (mean AT) to obtain the LST of signal. (8) Compute GST of signal. (9) Subtract LST from GST to obtain longitude.
30. Longitude Using Time By the Altitudes of Stars Near the Prime Vertical a. This method (fig. 43) is most commonly used by surveyors. It is frequently used to determine the LST for azimuth observations in the absence of radio equipment or of an accurate longitude. An approximate value of the latitude is required. b. The observations are taken on a star near the prime vertical and having an altitude of from 300 to 500. A number of sets, each consisting of one pointing in the direct and one in the reversed positions, are ordinarily taken. The accuracy is greatly increased by observing pairs of stars at nearly the same altitudes and relationship to the prime vertical to the east and west. When this latter system is used, the reversal of the telescope between consecutive pointings can be dispensed with and all pointings can be made with the telescope in the same position. DA Form 1909, Longitude by the Altitude of Stars Near the Prime Vertical is used for the computation. c. The computation procedure is as follows: (1) Correct the mean altitude (or zenith distance) of each star, for refraction. (2) Determine the mean time of observation corresponding to the above. (3) Obtain the declination and right ascension of the star for the date and UT of observation from the ephemeris. An approximate local standard time converted to UT is sufficient for this purpose. (4) Apply the following formula: 1
1 t sin 22 [1+( 8)]
sin' 2
1 sin 2 2 ['(8)]
cos € cos 8
where t is the required hour angle,
8 the
declination, and 0 the latitude. The 1 value of 2 t will be positive for a west star, negative for an east star. (5) Add the hour angle t (in time) to the right ascension to obtain the LST. (6) Subtract the chronometer reading from LST to obtain the chronometer correction. (7) After comparing the chronometer and the radio time signal, that is, obtaining the chronometer reading for the time signal, add to this reading the chronometer correction found in (6) above, to obtain the corrected chronometer time of the signal. This is the LST of the signal. (8) Compute the GST of the signal by the rules for conversion of time. (9) The longitude is the difference between the GST and the LST.
31. Longitude Using Time By the Altitude of the Sun This is, in principle, the same as the method of paragraph 30. The differences lie only in the calculation of the sun's coordinates and time conversion. The computation procedure is as follows: a. Extract the mean vertical circle readings and times of each pair of pointings, D and R. b. Correct the vertical angles for refraction and parallax. c. Obtain the sun's declination and right ascension for the date and time of observation. d. Apply formula, preferably using DA Form 1909. In this case, t is the local apparent time (LAT). e. Convert LAT to local mean time (LMT) by subtracting the equation of time. J. Subtract the chronometer time of observation from the LMT to obtain chronometer correction. g. After comparing chronometer and time signal, add the chronometer correction to the chronometer time of the signal to obtain the LMT of the signal. h. Subtract LMT from UT of signal to obtain longitude.
BY THE ALTITUDE
LONGITUDE
OF STARS NEAR THE PRIME VERTICAL STATION
PROJECT ORGANIZATION
LOCATION
Ohi INSTRUMENT (Type and number)
DATE
Aisfi.,
APPROXIMATE ANGLE BETWEENSTARAND POLARIS 0 CHRONOMETER TIME OF ANGLEREADING
COMPUTATION OF TIME
Chron. Reading
Refraction
log cos
(335
a
I9+(8
log sin i it(manJ +}[r(ma)1
4 0
27.0
8
+__________
44 1
________
23
24
115
Sum two log sines=.log N
._____0__
92291L4.
59.2472&2
log Nlog D=log sin' j t
h.
m.
a.
h.
74 4/ 04.3 3l 2 4 0.
t(arc)
t (time)
01
I
o
a.
9. 6413230 IL46,4L .Z14330. L 46 4L 1L6842.3 3L 23 9. 91/L 30 02. 2J.iZ2Q.Q7a.Z L 37~b 9 IS 871 13....2 ....
coB 8=1og D, #a
log sin [f+()J,
l1
an.
4AL 1 .234
S8Z932
8
#+log
2.20
45f0JL.
*
a
h.
+
_________
Corrected Z. D.=
cos~#
1
0
582 45 o
18 f6
Zenith Dist.
_________
}
a.
in.
____
h~fSTAR { i
STAR{Eas
h.
log cos

Dc.43
CHRONOMETER
OBSERVER
log
(TM 5237)
t
Right ascension of star
an.
a.
0,2 0 4J .3/k3LL.2 22ZO
Sidereal time
_______________3L
Chronometer reading
_____________
Chronometer correction
_________________
.9 0L..2Z0Q.
______
QdR
The chronometer correction is plus if the chronometer is slow, and minus if fast. Carry all angles to seconds only, all time to tenths of seconds, and all logarithms to seven decimal places. COMPUTATION OF LONGITUDE TIME
Chronometer reading
F
RAIO
(Sid.T.)
Chronometer correction
9
SGNALTRANSMITTING
S
STATION
Std.
+
8I
LST_________f
1.5
UT
23
52
479
4
43
47.804
3
£37
0
37
Sid. T. at Oh UT
Longitude (X)= GST LST
Corr. (table III) GST L.ST
COMPUTED BY
DA,
FEB 571
Figure 43.
17""14 DATE
47
TZC
TZC= time zone correction
Longitude (a) (arc)
2
timel71 mer.18
2
23 13
45?oo
Longitude (X) CHECKED BY
DATE
909
Computation of Longitude By the Altitudes of Stars Near the Prime Vertical (DA Form 1909).
Section V. COMPUTATION OF LATITUDE AND LONGITUDE FROM OBSERVATIONS MADE WITH THE ASTROLABE 32. Basic Procedures
sin H,=sin 0&sin +cos
a. The astrolabe is an instrument used to obtain latitude and longitude by the observation of the times at which stars cross a circle of fixed altitude. These stars are distributed in the observing procedure, to each quadrant of azimuth, to minimize the effect of refraction. b. The solution of the observation set requires an approximate or assumed geographic position of the observing station. This position should be near the true position in order that the intercepts in the graphic or analytic solution will be small. c. Two procedures for reduction of an astrolabe observation set to a latitude and longitude are discussed. The graphic solution is the established method for computing third and fourth order astronomic position and the analytical solution is used for the reduction of first and second order observations using impersonal timing equipment. The number of significant figures used in the computation is dependent upon the order of the work.
d. The Field Record, DA Form 1910 (Observations, AstroFix) (fig. 44), contains the names of the stars, approximate azimuths of the stars observed, observed chronometer times, and chronometer corrections or stop watch readings. When an electronic method of timing is used (as in the example shown), the scaled results of the time comparisons and corrected chronometer times of star transits are recorded in column (d) of DA Form 1910. This form is used both for an observing program and for a record of the field observations. While sidereal time equipment is preferred, mean time equipment may be used. If more than one transit wire is used, it is customary for low orders of work to mean all transit readings on the several wires, but for higher orders of precision a single wire transit is required. e. Computations for observations up to 700 latitude may be made by the socalled sinecosine formulas using DA Form 1911, Altitude and Azimuth (SinCos) (fig. 45 or DA Form 1912, Altitude and Azimuth (SinCos) (Logarithmic), (fig. 45 ®). Astrolabe observations at higher latitudes are of little value. The sinecosine formulas are as follows:
,()),
sin Z=cos
a,cos
cos t
sin t
cos H
where H, refers to the computed altitude of the star distinct from the observed altitude; 0 is the assumed latitude; and t is the hour angle of the star in the shortest direction from the meridian. f. The values of H, are computed for each star. Each value is then subtracted from an assumed approximate altitude of the fixed circle to obtain differences called intercepts, which are then adjusted to reduce the assumed position to the computed position. The azimuths of the stars are used in this adjustment. 33. Computation Procedure a. From the recorded time, compute the UT for the purpose of finding the declinations and right ascensions of the stars. An average time may be taken by inspecting the record, or if the observations are at fairly well distributed intervals, the mean of the times of the first and last stars is sufficient. b. Enter all the stars and their declinations and right ascensions on the form. Declinations should be to the nearest second of are and Right Ascensions (RA) to the nearest OSl of time, for third order work. If impersonal timing equipment is used, and higher orders of precisions are desired, the declination should be determined to the nearest 0.01 second of are. c. Correct the recorded time for the chronometer error found by comparison with radio time signal, and covert fromn GST to LST by subtracting the assumed longitude. d. Compute the numerical difference between LST and RA of each star in such a manner as to make the difference always less than
1 2h
.
One
method is to apply the equation, t=LSTRA, and if t exceeds
12
h,
subtract t from
2 4h
.
Dis
regard algebraic signs. e. Solve equation for H,. Complete the computation for all stars up to this point, and inspect the values of He for uniformity. Reject any outstanding values after checking to eliminate errors.
PRJCT
SA
OBSERVATIONS,
',(A(TM
LOCATION
R YUK Y!/ WATCHFAST()
ASSUMEDLAT. (LA)
300 sTR~LAE/AToMATc TIMER DESCRIPTION
27 03.2
se
1ONES
R.L.
STATION
4o"52,r
08'
As
OF STATION; REFERENCES,
22.,43/% O . 69
ASSUMEDLONG. (XA)
(Number and type) OBSERVER
INSTRUMENT
DATE
4C,,4MS
45*
SLOW(+)
ASTROFIX
5237)
ORGANIZATION
Fgqa
/ 0.00
RECORDER"
#0.
=
W. A:
CROSS BEARINGS. SKETCHES,
AIMS
ETC.
For an astrolabe, put L. Sid. T. in column (e); for a transit, put vertical angle in column (e) NR
STAR (a)
rSAGITTAE
752
1602
WATCH TIME (b) ____ S H M
_
892 C PISCwM~
THESE
8
TWO
33 A(ANDROMEDAE
NO TI
765 2'cY6NI
CH ON
ANDROMEDAE
17 9CASSIOPEiAE~
COL
MAIS
A PLIC SL E
CAL
IF FROM
D
GfPA N.
RECORDS. ____
777 a( CYGNI (DENEB) 45 I
2 I CASSIOPE/AE
__
52 SI ANDROMEDAE 64
a rRIANGL/I
So
'? P/,Sclum
73 4ANDROMEDAE P RIET/S
1 7r2 CYtN/
74 a
_
878 a' P/SCW
COMPUTED
33 15.323
65
04 45
17
56
39 3.699
/38
37
/_8
.57 L70O
03.356 298
29
19
02 18.254
45 45.316
7/
0/_______
19
06 07.308
49 34.997
3/
44
_______
1__ 9
/0
54 11.305 308
43
_______
0/
______
BY
53
19
54 20.7/5
_
20 02 39885
__U_
to
20
23 0
,
6
192
CHECKED
24 /9149 256
/0._
28 57258
44
/2
______
56.733
83
02
______
36 45.745 /14 56.327
_37
42 59667
46
16.863
/0/
49 49829 3/7 53 49984
_
25'
______
28_______ 39_______
24
______
30_______
24 06 48&533 206 33
BY
____
23_______
287 43 57
_______
______
DATE
M
Nov.65
.
U. S. GOVERNMENT PRINTING OFFICE:19570420636
Figure 44.  65 
11.769
______
24
31
8.227
20
88
_____
______
14
14.8.831
06 12.249
__20
7DATE
59
00 48.089
07_______
06 03.226 285 S/L
10.327
19
DAIFEB571.1
0
18.562 23
1_9
NvOV.59
757381
41

2.862
__
__
/9
49 50308
8~ 22. 104
____
21957 235
/8
45 23.119
ARIET's_______
46
/8
___9
__
150
27 43.724 /43
____19
iVcya~
05 12.635
44 /26/5
1_ 9 " 40 45.769
______
37_______
/8
28 17500 _
255
/a
00.692
22 32.838
_
0
00 130.051
27
__/9
804 1 PEGAS /
66 '
2.215
REMARKS (&)
AZIMUTH (a) (f)
/8
___/9
8
8
/0.403 22
1_ 9,/7
P/SC/A
78 CYGNI
5
17 2/
______
902 w. P/SC/elM
42
S
Mf
CORRECTED LOCAL SID. 7 TIME (d) (e) ____ M 5 H
_________/8
768 6 DEL PHI NJ
4
___
___(C)
______
__
4~ P/ScluM
STOPWATCH
Observation schedule and field data.
SPAN
PROJECT
ALTITUDE AND AZIMUTH (SINCOS) (TM
5237) DATE
ORGANIZATION
LOCATION
0151A MS
R YUIK YU /IS.
2.4Nov 59
ERA SU30
1089 40
27 #o3.2
ASTROLABE/AUTomATic TIMER 7 2
Star
R.IL. JONES 12
S4
f ~0. /0
SF
=0.00
3
748
/602
36'
SLOW(+)
52.'0701
OBSERVER
INSTRUMENT (Number and type)
WATCHFAST()
ASSUMEDLONG. (lXA)
ASSUMEDLAT. (LA)
STATION
4
&9'2 /2.077 05 024' 38.074
9°2
/a 3.267
03 °
0917
/0.403
09I 2 522/5 09 27~ 00.692 09 44 196/5
Declination
21.079
//1
/0'
Watch Corr. slow, fast
.9
UT G. Sid. T
2243
1
___
j4g0
doh UT
Mean time itra
to sid. time
(NA) E+, W
35990 0/ 33.146 01 32.30/31.529 /9 37981 /3 24 20.565 /3 29 29887 /3 46 5/.654
089
40 52.070__
0/ f
__0/
00 3005/ 22 05 /2635 22 /0 21.957 22 27 /9 56~ 57615 23 0/ 50.370 20 3/ /72/9 23 37 +02 03 32.436 00 56 37735 +0/ 39 04.740 01 /0. 300 I306..54 345 50 33.98~ 240 46 //V34 2 27 22
Local Sid. T.
R. A. M.
/3
(corr.)1
G. Sid.'T.
Long.
6.049
A.
M. A.(are) t
506 799'63
Sin LA
0
Sin a
0. 33/ 94 742
A(product)
0
CoSa
/6823083 0.86206388 0.94329789
COS t
+0O858/9798
Cos LA
4.724
53.454 0930 3.0o5
_____
0.0628925/ 0.03187390
0.19372/07 0.09817777
0.0 9429208 0.047787/9
0.9980203/
v.98/06665
0.99554458
+o.94962821
_____
+0.9079988/
'+0.95349955
B(product)
0.69787204
0. 83422667.
0.7679250/
0.8/83/.527
A
o.16823083
0. 03/873 90
0.09'8 /7777
Sin Hc*
0. 8(o0/0287
0047787/9
HO
0.866/005'7 0.866/0278 0.86o6/0246 700j319256 60j 00 31.j7934 60 00 31.9628 .607 joi 0124760 60 00 33.5800 60 00 33.5800 604 00 3.5800 60 00 L3.5800
intercet ne pt "To") H°+ Cos a
0.94329789
0.9980203/
0.98/0.5665
v.99554458
Sin t
0.5/33/884
0.4189775
0.30/39443
C(product)
0.4842/258
0.24458361/ 0.2440 994/
0.411036 00
0.30005/5 9
COS HC
0.49986580 0.968 68516
0.49986 972 0.48832599
0.49986.59/ 0.8222 9252
0.499866.51
H0
Sin Z (C.Cos Hc) Z
01.6/72
75
37
255
Azimuth ZN
0.64
+ 0.576
2
37
/SO
1
46
235
COMPUTED BY
DATE
JAN. DA.
FORM
,
60
/93
CHECKED BY
0.60026344 6
05kl
*When L, and a have same sign: Sin Hc=A±B if M. A.)H0 =
Log cos a Log sin t LogOC (sum)
Log
cos.Hc
Log sin Z: (diff.)

997464886 97/038720
.68503606
95988.5342 ~9986/8264 75°37"
Z Azimnuth ZN
4#
0/.6172
255" 37'
02.5676
0/65544
#
9999/3938 9.38842736 93876674 9 69885688
9.99/6 9409' 9622/8578 9 6/387987 9' 698853.55
9568870986
99/502632
2901/4
ISO D46' 0
/9
9.998 06072 969885403 977834191 3653
,43 °07'
"
0
When L and a have opposite sign: Sin H 0 =AB if M. A.90° .DATE CHECKED BY S9 DATE COMPUTED BY _P
D
0/ 7866
9479/3522 9477/9594
55a/
235 0
*When L and a have same sign: Sin Hc=A+B if M. A.90
FXoV. '5
*
k
~/5
J
:1957 0420844 PRINTING OFFICE U. S. GOVERNMENT
1 FORMD7191
® Altitude and Azimuth (SinCos) (Logarithmic) (DA Form 1912)
Figure 465Continued.
f. Select an arbitrary value for Ho to an even minute or second of are slightly greater than the highest retained value of He. The quantity HoH,, the intercept, will then always be positive. g. Solve formula for the azimuth, A, of each star using a single value of He. h. On DA Forms 1911 and 1912, the first quadrant angle is represented by the letter Z. To determine the quadrant the azimuth angle falls in when measured from north (ZN) one of the following methods can be used. (1) There is no satisfactory way to show the quadrant of the azimuth angle by using signs and therefore they are omitted on the computation forms DA 1911 and DA 1912. (a) A North star with a negative meridian angle (MA) would be in the first quadrant. (b)A South star with a negative meridian angle would be in the second quadrant. (c) A South star with a positive meridian angle would be in the third quadrant. (d) A North star with a positive meridian angle would be in the fourth quadrant. (2) When the assumed latitude and a star's declination are nearly the same, determination as to the star being a north or south star can be determined from DA Form 1910, Observation, Astrofix, where the approximate azimuths of the observations have been computed.
to errors in the observations, the circle can never be drawn truly tangent to all lines. The discrepancies should be equalized in amount and direction. d. Mark the center of this circle. This represents the true position of the station. e. Measure the seconds and fraction of a second by the scale of the graph in the direction parallel to the NS axis of the plot between the assumed and true positions. This amount applied to the assumed latitude gives the observed latitude. f. Measure similarly the EW difference between the assumed and true positions. Divide this value in seconds of time by the cosine of the latitude, and apply the result to the assumed longitude to obtain the observed longitude. g. The algebraic signs of the differences are apparent from the plot. h. Should the assumed position be in error as much as 10 minutes of arc, the scale of the plot will be too small for the required accuracy. In this case, plot four stars in different quadrants to obtain a close approximation of the position. Recompute with this as the assumed position.
34. Graphical Solution DA Form 1913 may be used for plotting the intercepts derived above (fig. 46). This form does not require a protractor. The solution is as follows : a. Plot the intercept of each star to a suitable scale from the central point of the plotting circle in the star's azimuth direction if the intercept is positive, or in the opposite direction if the intercept is negative. Small variations are displayed best by adding a constant to all intercepts before plotting. In the example given a constant of 50 seconds (50") has been added to each intercept. b. Draw lines through the points so laid off, at right angles to their azimuth lines. These are the line of position. c. By trial, draw a circle as nearly tangent as possible to all the lines of position. A set of concentric circles drawn on a piece of transparent plastic will facilitate this operation (fig. 47). Due
Where:
35. Analytic Solution a. When the circummeridian intercepts and azimuths from North are known, the observation equation can be written in the following form: v=cos0o(AX+AT){sin AN}+ A¢(cos AN})
{1} AH (HoHC)
{},
braced quantities indicate determined coefficients; AX, is the change in longitude from assumed longitude; A¢, is the change in latitude from assumed latitude; AH, is the adjusted intercept from adjusted position, AT, is the correction to the radio time signal; AN, is the azimuth of the star from North; (HoH,), is the circummeridian intercept based on assumed position of station; v, is the residual. The international sign convention is used. This convention specifies that longitudes east of Greenwich, hour angles west of the meridian, and latitudes and declinations north of the equator are positive. Corrections are to be added al
N
SP
PROJECT
ASTROLABE PLOTTING (TM 5237)
LOCATION
RYUKYLS
/SLAND5
ASSUMED POSITION LATITUDE (0) 30027I LONGITUDE ()
3 0
032 Al
0/'
E
ORGANIZATION
DATE
USA MS
S
CORRECTIONS
om
O52
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PLOTTED POSITION LATITUDE (~ )
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IDATE CHECKED
COMPUTED BY
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LONGITUDE BY
0
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DATE
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U.
71913
Figure 46.
Graphical solution.
LI
.GOVERNMENT PRINTINGOFFICE :1957 0420713
Figure 47.
Concentric circle overlay.
gebraically. See sample formation of observation equations (fig. 48). b. The normal equations are obtained by requiring that the sum of the squares of the residuals be a minimum. This requirement is obtained by finding the sum of the partial derivatives with respect to each variable in the squared observation equations. The symmetric normal equations are namely: I. [(sin AN) (sin AN)] cos 0 (AX+ aT) ± [(sin AN) (cos AN)1AO + [(sin AN) (1)]H[(sin AN) (HoH,)] =0 II.
[(sin AN) (cos AN)] cos 40 (AX+
AT)
+ [(cos AN) (cos AN)1A4 + [(cos AN) (1)AHH[(cos AN) (HoH)I =0 III. [(1)(sin AN)] cos Ok(aX±AT) ± [(1) (cos AN)IA41AH+ [(HoHo)]=0 The brackets indicate a finite summation of the set. These equations are solved by the Doolittle Method. See sample formation of normal equations in figure 49. c. Substitution of solved values back into the observation equations determines the residuals (v's) for the star intercept observations. See figure 50 for sample forward solution and back solution and figure 51 for residuals and probable
P
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PROJET Al
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LOCATION,
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TABULATION OF 5237) GEODETIC DATA
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STATION
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Formation of observation equations.
PROJECT
TABUTION OF GEODETIC DATA
SPAN
LOCATION,
RYUKYC/
(TM 5239)
ORGAN IZATION
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+Zcois 3x3 + hCoLes 3x4 + ScoLs .3x5 + Icotss 3X6 + FcosAmecosAN + EcosAN * aH + Icos AN(c +ZcosAN.(i4a + (75232) + 1.3059) + (/./969') +(+4.8/00) +ZGCOL's
4 x4
+ ZCOL's +
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+
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23.oooa) +
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COEfFFICIENTS I
(±
3.3324

1.3059 + 23.0000 
DATE

'4MS
JAN .00 SPO
:?:'e 51962
Figure 49.
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4 x5
H
L'ATIONS
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(+
(+49.06290)+

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+ 4.8/00 + 674307
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4 x6
Fcol's
ajoO+
/4.2536

+
DATE
G.T.T&.)nA

AMS
U. S. GOVERNMENT
Formation of normal equations.
PEUTD=
JAN.600 OFFCE : 1951
 431162
PROJET CO
NI
II(TM
.JU~I'
LOCATION,
SAINRUY
I
FORWARD SNDUASsSOLUTION
BO
+ S. 4776 4L =
 14.2536
+ 0.0/36
+0.2/53.
+
o.9209

1. / 96 9
1.3059

0.029 ____ 75203
1.35/ 2

58 4=2.07.58 (+0,797)+ 0./1849
+o.1849
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0.4992 0. 8587
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+ 4.8/00  0.0315
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7900
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=2.07
 0.1881 + 0.9209 =+0. 47/4
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( 4 &+6)
k 4(411 X
ARC  AL/cos o
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TABULATION OF 5237) GEODETIC DATA
ORGANIZATION
M
±01 DATE
A.10
GPO 921961
Figure 50.
ELEVATION
of ERAU~
.T7Ywu.
CHECKED BY
M
=6(00.06
A.6
DATE
U. S. GOVERNMENT PRINTING OFFICE : 1957 0  421182
Forward solution and back solution.
SPAN
PROJECT
TABULATION OF GEODETIC DATA '.JI~I1(TM
5237)
LOCATION
ORGANIZATION
RY/KYU~S
/SLAND
USAMS
STATION
$
NO. AV si(AN
4
ERROR
(HV
" cos AN.
4
+0.2302
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AND
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9,+0.4457
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:c'9_ __I~~~i~~iU~~iv =±=068 FV =t
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.19
TABULATED BY
'§.K
DA
FORM
16
DATE

A M5
/44
0
I
A. 60
GPO 921961
Figure 51.
!o
e
70r
t±./
9.1929
.
rM2
e
c~ =+0.13
7.
_±O64 , 5
CHECKED BY
G. T
4 5
oe
am o.
095
DATE
rc,
AM$S
JA.
60
U. S. GOVERNMENT PRINTING OFFICE : 1957 0  421182
Residuals and probable error.
error determinations. After the residuals are obtained, the probable errors are obtained with the following equations: [vv]
Sn3
P=
x (arc)= ax (Time)=
± =
P
 [sin 2 AN] a
1 cos 4o
x (arc)
[cost AN]
o'H e= ± 0.6745a
The statistical results are given in figure 51. Further rejection may be considered by using Chauvenet's Criteria, table Xl in appendix III. d. Considerable savings of computation time and effort will be realized by rejecting observations before the observation equations are formed. Plot azimuths and intercepts on DA Form 1913 as is done in paragraph 34. A transparent overlay is placed over the plot of azimuths and intercepts. The overlay is inscribed with a number of concentric circles whose diameters differ by two times the rejection limit. The rejection liinit is generally five times the probable error that will be tolerated for the class of work. Thus, for firstorder astronomic work, the probable error should be less than two tenths of a second (0'2) ; and therefore, the rejection limit would be set at 1 second (1'.'0) and the diameters of the concentric circles would differ by 2 seconds. The overlay is moved until a ring is found into which most plotted intercepts will fall. Those intercepts falling without the limiting rings are rejected. The formation of observation equations proceeds as in paragraph 34a. e. The graphic and analytic determinations of astrolabe astronomic position are concluded with corrections for elevation, mean position of the pole, correction to signal, transmission time, and diurnal aberration. (1) Corrections to latitude. (a) When the observing station is at some elevation other than sealevel, a correction must be applied. This is a
correction for the curvature of the vertical or, otherwise stated, for the fact that two level surfaces at different elevations are not parallel but converge slightly as the poles are approached. This correction can be determined by one of the following equations: AE=0.000171 h sin240 AOE=0.000052 h'sin24o in which h is the altitude of the observing station in meters, h' is the altitude of the observing station in feet, and ,o is the assumed latitude of the station. The correction is always subtracted when the observing station is above sea level. (b) Where the greatest accuracy in the astronomic latitude is required, as in first and second order astronomic observations, it is necessary to reduce the observed latitude to the mean position of the pole as is outlined in paragraph 24. (2) Corrections to longitude. There are four corrections to the longitude, two caused by small errors in the time, one caused by a small error in the observed altitudes, and the fourth caused by the variation of the pole from its mean position. (a) The transmission time correction is generally computed at the same time as the chronometer error is determined. This computation is outlined in paragraph 28e. (b) The correction to signal is the correction to the time service clock as transmitted. The radio time service bulletins contain this correction generally for OhUT and 1200 h UT for the date and year of observation. See example in figure 37 and explanation in paragraph 28e. (c) Because of the rapid rotation of the earth, a star when observed is apparently displaced. The displacement is in the direction of rotation of the earth so a star toward the east will appear to be at a slightly lower altitude than the true altitude of the star. In the same way, a star toward the west will appear to be at a slightly higher altitude than the true altitude of the star. This effect is known as
diurnal aberration. The correction to the longitude of the observing station for this effect is found by the equation: AXD=0"'2771 cos 40 (ARC)
in which AXD is the diurnal correction to longitude and €0 is the assumed latitude. The correction is added to the east longitude and subtracted from the west longitude.
(d) Where the greatest accuracy in the astronomic longitude is required, as in first and secondorder astronomic observations, it is necessary to reduce the observed longitude to the mean position of the pole as is outline in paragraph 28o.
f. The corrected astronomic positions are recorded on DA Form 2850, (Astronomic Results) (fig. 52). This report card is described in paragraph 36.
Section VI. ASTRONOMIC RESULTS 36. Use of Form The results of the computations of astronomic data should be summarily tabulated on DA Form 2850 (Astronomic Results). This form was designed to provide a final summary of the astronomic position, azimuth, and LaPlace azimuth computations for a station, and to further provide a convenient reference source for this information. An explanation of its preparation is outlined.
37. Tabulation of Results a. All entries on the astronomic results form must be complete in all respects to avoid any ambiguity. This involves an accurate station designation, appropriate hemispheric reference in the latitude and longitude entries, reference pole for the azimuth entry, unit of measurement for the elevation entries, and careful identification of personnel, equipment, and dates. The form should not be considered complete until all entries have been verified by someone other than the person who made them. b. The latitude, longitude, and azimuth entries are extracted from the respective computations. If cardinal directions are used, as on the example form in figure 53, there is no chance of error in the proper application of reduction data in the latitude and longitude entries. Care must be taken in indicating the proper sign of the corrections applied to the azimuth entries.
c. The present design of the astronomic results form does not include provision for certain corrections which are not habitually computed. Effects of polar motion on latitude, longitude, and azimuth are not included for this reason. Occasionally an astronomic azimuth is observed from an eccentric station and the necessary eccentric reduction may be entered as shown on the example. d. The computation of the deflection components and the LaPlace correction for azimuth can only be made if the geodetic position of the station is known. It should be noted that the sign of the prime vertical deflection will be influenced by the consideration of East longitudes as positive, and west longitudes as negative values. Also, this consideration will affect the LaPlace azimuth and the note concerning the application of the correction must be carefully observed, i.e., aG=aN
LaPlace Correction (sub
tract algebraically). The deflection in meridian is computed considering the sign of north latitudes as positive and south latitudes as negative. e. A sketch of the geodetic connection should be prepared from data available from field records and computations. Reasonable care should be taken in depicting the relationship of the points involved. Sufficient data should be provided in order that eccentric reduction computations can be made.
ASTRONOMIC RESULTS
STATION
(TM 5237) SPAN
PROJECT
ERABU
RYUKYU ISLAND
LOCATION
LATITUDE OBSERVER
R.L.
AND ORGANIZATION

Jones
INSTRUMENT
USAMS
I.
ELEVATION 600.06
OF STATION OBSERVATIONS ACCEPTED
m
496
OBSERVATIONS REJECTD
22 Nov. 1959 ur,
30
N
LEVEL REDUCTION
23
DATE OBSERVED
0
MEAN OBSERVED LATITUDE
2SEA
I
CHRONOMETER
Astrolabe/ Auto Timer
03.01
27
S
±
0.17
00.09
ECCENTRIC REDUCTION
REJECTED
1
00.00

PROBABLE
+
ERROR (PAIR)
ASTRONOMIC LATITUDE
0.17
3
IN
30
27
0.17
02.92
REMARKS
Haddox
COMPUTER

DATE
AMS
Jan.
OBSERVER AND ORGANIZATION
R.L. Jones

INSTRUMENT

AMS
STAR SETS REJECTED
MEAN OBSERVED TIME MEAN OBSERVED ARC
1
Jan. 60
DATE OBSERVED
496
22 Nov. 1959
H
23
DATE
CHRONOMETER
Astolaber
USAMS
STAR SETS ACCEPTED
Tennis
60 CHECKER LONGITUDE
M
S
S
+
08
40
52.116
±
0.009
E
130
13
01.74
±
0.13
REMARKS
ECCENTRIC REDUCTION 00.00
ASTRONOMIC LONGITUDE Haddox
COMPUTER

AMS
Jan. 60
DATE
E ICHECKER
130
13
01.74
Tennis  AlIS
DATE
0,13
Jan. 60
AZIMUTH OBSERVER
AND ORGANIZATION
INSTRUMENT
CHRONOMETER
MARK
DATE OBSERVED
 p
MEAN OBSERVED AZIMUTH ELEVATION
OF MARK
DIURNAL ABERRATION
OBSERVATIONS
ELEVATION CORRECTION
ACCEPTED
ASTRONOMIC AZIMUTH
OBSERVATIONS
FROM
(MEASURED
REJECTED
)
+
REMARKS
COMPU TER GEODETIC
DATE LATITUDE
GEODETIC
SKETCH
DATUM.
DATE
)CHECKER LONGITUDE OF GEODETIC
CONNECTION
NOTE: NORTH LATITUDES AND EAST LONGITUDES POSITIVE
DEFLECTION
IN MERIDIAN
DIFFEPENCE
IN LONGITUDE
(OA4iG
"I
If
(XAXG)
PRIME VERTICAL DEFLECTION LAPLACE CORRECTION
(XAXG) SIN4G
LAPLACE AZIMUTH (aG)
NOTE: aG=ALAPLACE
I
DA FORM 2850, 1 OCT 64
Figure 52.
Astronomic results (astrolabe).
CORR (SUBTRACT ALGEBRAICALLY)
ASTRONOMIC RESULTS
NP (AMS, 1958)
jSTATION
(TM 5237) PROECT
MARYLAND
CATION
LATITUDE OBSERVER AND ORGANIZATION
INSTRUMENT
H.N.C.  AMS
Wild T4
ELEVATION
OBSERVATIONS
OBSERVATIONS REJECTED
ECCENTRIC
3
PROBABLE
PROBABLE ERROR (PAIR)
+
N
SEA LEVEL REDUCTION
25 pair
ACCEPTED
56095
MEAN OBSERVED LATITUDE
75 m
OF STATION
CHRONOMETER
pair
ASTRONOMIC LATITUDE
r
38
57
2 May 1963 01.20
S
00.01
N
02.45
0.09
REDUCTION
__
Q?.43
12460
DATE OBSERVED
N
38
57
03.640.09
+
009
REMARKS
Horrebow LES (AMS)
COMPUTER
Talcott method. DATE 23 July 1963 CHECKER LONGITUDE INSTRUMENT
OBSERVER AND ORGANIZATION

R.S.
Wild T4
AMS
STAR SETS STAR ACCEPTED
SETS
ORN (AMS)
MEAN
6
CHRONOMETER
12474
56095
H
TIMEH
OBSERVED
DATE
05
M M
SRETSD
SETS1
,
18 Jun 63
24 May, SS
±
0.004
±
0.06
±
0.06
it
MEAN OBSERVED ARC
0
REJECTED
DATE OBSERVED
29.012
08 o
STAR
11 Dec 1963
W
077
07
15.18
REMARKS
ECCENTRIC REDUCTION
Reduced to UT 0
E
_______
ASTRONOMIC LONGITUDE
LES (AMS)
COMPUTER
W
077
1963CHECKER
22 July
DATE
02.54
ORN
07 (AMS)
12.64 DATE
9 Dec 1963
AZIMUTH OBSERVER AND ORGANIZATION
CHRONOMETER
INSTRUMENT
F.E.W.  AMS
Wild T3
12460
53010
DATE OBSERVED
10 Apr, 23 May 63
o
MARK
MEAN OBSERVED AZIMUTH
MAP (AMS,
1958)
ELEVATION
008
348.78
OF MARK
Ft
OBSERVATIONS
DIURNAL ABERRATION
+
Eccentric Reduction
+
ELEVATION CORRECTION
32 p0$
ACCEPTED
25
06.43
±
0.19
00.32 03
02.93
00.00
ASTRONOMIC AZIMUTH
OBSERVATIONS
i
REJECTED
0
(MEASURED FROM
South
008
28
09.68
0.19
REMARKS LES
COMPUTER GEODETIC
N 380
(AMS)
57'
, W 0070
05'.'159
DATUM
2
DATE GEODETIC
LATITUDE
Oct
LONGITUDE
07'
16'.'394
NOTE: NORTH LATITUDES AND
1927 NAD
EAST LONGITUDES POSITIVE
DEFLECTION
IN MERIDIAN
DIFFEPENCE
IN LONGITUDE
PRIME VERTICAL DEFLECTION
LAPLACE CORRECTION
(4AG)
(XAXG)
(XA)O) COS 
i
01.519
'
+03.754
+02.919
0
(XAXG)jSING
+02.360 LAPLACEAZIMUTH
(aG)
008
28
07.32
NOT E:aG=CAiL4PLACE

DA FORM 2850, 1 OCT 64
Figure 53.
Astronomic results.
CORR (SUBTRACT ALGEBRAICALLY)
CHAPTER 4 DISTANCE MEASUREMENTS Section I. TAPE MEASUREMENTS 38. Data Required a. In the establishment and extension of horizontal control, the distance between control points is a primary requirement. In triangulation, selected sides designated as baselines are obtained by precise measurement, and these are used with observed angles of connecting polygons to obtain all other lengths of a triangulation net. In trilateration, the sides are measured by electronic methods, and taped distances are seldom involved. In traverse, the distances between stations are generally measured directly. One method of determining these distances is by measurement with standardized metallic tapes, using special tapes, with a small coefficient of thermal expansion, for baselines, and steel tapes for other measurements. b. Base line computations should be made on DA Form 1914, Computation of Base Line (fig. 54). The first five columns and the temperature column are filled in from the data in the field records. These six columns are used to record the section measured, the date of measurement, the direction of measurement (forward or backward), the number of the tape used for each measurement in the section, the number of supports used for the tape for each measurement, and the average temperature of the forward and rear thermometers (a mean of the temperature can be used for all the fulltape lengths). The column "Uncorrected Length" is used to record the total number of tape lengths and the corresponding total length of each section. Odd tape lengths, such as half or quartertapes, or measurements with a steel tape are also recorded in this column. Setup or setback measurements of less than a meter are entered as corrections in the column provided. c. Computation of other tape measurements should be made on DA Form 1939 (Reduction of
Taped Distances) (fig. 55). All field data is entered on the form in the space provided.
39. Corrections to Tape Measurements a. The field measurement of any line must be corrected for temperature, tape and catenary, inclination, and elevation above or below sealevel. b. The information for the temperature correction and the tape and catenary corrections is found on either a tape standardization certificate furnished by the National Bureau of Standards (fig. 56) or from results of field comparison of tapes with previously standardized tapes (para. 41). The NBS certificate contains the coefficient of thermal expansion of the tape and the standardized length of the tape under several support conditions. Every tape used in a base measurement should be standardized by the National Bureau of Standards, Washington, D.C., both before and after the base is measured. The average lengths of the two standardizations are used for the computation of the tape and catenary correction. (1) The temperature correction is found by multiplying the coefficient of thermal expansion (shown on tape certificate) by the number of degrees difference between the temperature at the time of measurement and the temperature of standardization, usually 250 C. for invar tapes and 20
°
C. (680 F.) for steel tapes,
times the measured length. The thermal expansion as given on the tape certificate usually states a certain expansion per tape length per degree Celsius (centigrade). For example, a tape may have a thermal expansion of +0.020 millimeters per 50 meters per degree Celsius. (2) As an illustration of a temperature correction computation, Tape No. 5123
PROJIECT
COMPUTATION OF BASE LINE
NAMEOF BAE
1442 LOCATION GAE
zER.
__
.
NR
OROER OFBASE iMh;4
SPPOT TATN
r
_
TAPE i[R
_r
EMP.REDUCD APE .G i N.
Tr
n0
zu
I __
PAGE NR
euttro
S. Iv
LENGTH (Meter. c
PAGE NR
NROFPnES
ADPTED LNGTH
(V) m)
__
HECKE B
COMPUTEDDBYDATE DA A1..
2 NR FIELD BOOK
EN"
DIR. SECION
'Pis,,kTi5.37
ORGANIZTION

P.E. (mm)
_
DATEV
1914
Figure 54.
Computation of a Base Line (DA Form 1914).
(fig. 56) has a thermal expansion of 0.020 millimeters per 50 meters per degree Celsius standardized at 250 C. Section A of a base line was measured in
the forward direction with this tape. There were 12 full tape lengths and 1 half length. The average temperature of the front and rear thermometers for the 12 full tape lengths was 300 C., and the temperature for the half tape length was 290 C. The correction for the 12 full tape lengths is: 12 X 0.020 X (3025)
x±
1.2 miliimeters
This correction is plus because the measurement temperature was higher than the standardization temperature; therefore, the tape was actually longer than recorded. When the measurement temperature is lower than the standardization temperature, the correction is minus. The correction for the halftape length isX X 0.020 X (2925) = + 0.04 millimeter Corrections are ordinarily entered only
to Yo millimeter on the base line computation form. c. When the tape is supported in the same manner as when standardized, the tape and catenary corrections are combined. If the tape is supported at different points than it was at standardization, it is necessary to compute the tape and catenary corrections separately. (1) Using Tape No. 5123 as an example and the same section of the base which has 12 full tape lengths and 1 half tape length, the tape and catenary corrections are computed using the standardization data. For the 6 tape lengths with 3 supports, the correction is: (6) (+ 0.00067)= + 0.0040 meter For the 6 tape lengths with 4 supports, the correction is: (6) (+0.00021)=+0.0013 meter The half tape length is supported at 2 points, and the correction is: (1) (+0.00028)=+0.0003 meter The example illustrates a very simple case where standardization was available for all the tape lengths used. A more
PROJECT
30(TM
Q/,
LOCATION
TRVREN.FROM
STA.
TAPE
A IS
6602 TEMP.
MEASURED DISTANCE
SUPPORT
SECTION
TO STA.
2
TEMP.
TENSION
SUPPORT T
CAL. LENGTH
TEMP.
TENSION
SUPPORT 2
EVTON
__
_
__
___
_
___
003% 0L48.
_
0040

4_
A0.
.2
0.
_
t~a /
_
Q4
Z&
___
2____ ow _
DISTANCE

150.865
__
CORRECTED
4.ol
L1o
200.000
SLOPE
CORRECTION
DIFFERENCE
CALIBRATION CORRECTION
TEMP. CORRECTION
200.000
_____
I94
CAL. LENGTH
200.000L
L
TAPED DISTANCES
5237)
L
ORGANIZATION DATE
OF
REDUCTION
BOOK NR
1O/,
_
2
70A4
05B
to
1Q46
0
A689
*3
2 0 0 .0 0 0
~
L
t _o _ a
____~2Oo~oo
____
____
_
0.065
_
_
_
_
0.25
. _
___
DAIFR
C.A,.
0

65

7
7L~8

__________
DATE
vjeS4a9
.2
. 048

QQ8625
CHECKED BY
1939 Figure 55.
757381
t
_2
42.aw0046  0Q30
4j).
_A
0  000
20
COMPUTED BY
_
_
Computation of Taped Distances (DA Form 1939).
to
OATEJ
orm 1 UNITED STATES DEPARTMENT
OF COMMERCE
WASHINGTON
Alationat ureu of 'tatbirbg (Certficate Fr
50METER IRONNICKEL ALLOY TAPE (Low Expansion Coefficient) NBS No. 7871
Maker: Keuffel & Esser Co. No. 5123 Submitted by
InterAmerican Geodetic Survey Liaison.Office c/o Army Map Service 6500 Brooks Lane Washington 25, D. C. This tape has been compared with the standards of the United States under a horizontal tension of 15 kilograms. The intervals indicated have the following lengths at 25° centigrade under the conditions given below: Interval (meters)
Points of Support (meters)
Length
(meters
0 to 50
0, 25, and 50
0 to 50
0, 12 1/2, 37 1/2, and 50 with the 12 1/2 and 37 1/2meter points 6 inches above the plane of the 0 and 50meter points. 50.00021
0 to 12 1/2
0 and 12 1/2
12.50088
0 to 25
0 and 25
25.00028
0 to 37 1/2
0 and 37 1/2
37.49701
0 to 37 1/2
0, 25, and 37 1/2
37.50134
50.00067
For the interval 0 to 50 meters, thermometers weighing 45 grams each were attached at the 1meter and 49meter points. One thermometer weighing 45 grams was attached at the 1meter point for all other intervals. Test No. 2.4/G15354 Figure 56.
Tape standardizationcertificate.
NBS Certificate continued
 2 
NBS No. 7871
These comparisons were made on the section of the lines near the end on the edge of the tape marked with small dots near the
gra duation.
The weight per meter of this
tape, previously
25.8 grams.
determined,
is
The values for the lengths of the intervals 0 to 25 meters and 0 to 50 meters are not in error by more than 1 part in 500,000; the probable error does not exceed 1 part in 1,500,000. The values for the lengths of the intervals 0 to 12 1/2 meters and 0 to 37 1/2 meters are not in error by more tha n 1 part in 250,000; the probable error does not exceed 1 part in 750,000.
The values for the lengths were obtained from measurements made at 20.2° centigrade, and in reducing to 25 ° centigrade, the thermal expansion of +0.020 millimeter per 50 meters per degree centigrade was used. For the Director National Bureau of Standards
Lewis V Judson Chief, Length Section Optics and Metrology Division
Test No.
Date:
2.4/G15354
September 10, 1954 Figure 56Continued.
difficult computation is necessary if standardization is not available for broken tape lengths. Under these circumstances it is necessary to make use of the formula for correction due to sag (catenary correction) which is:
C
24
l
3
in which: n=number of sections into which the tape is divided by equidistant supports l=the length of a section in meters w=the weight of the tape in grams per meter t=tension in grams The minus sign in the formula presupposes that the correction is to be applied to a tape standardized under conditions
of full support throughout. In other words, the length measured by a fully supported tape is shortened by the effect of sag. (2) Since the Bureau of Standards does not ordinarily standardize tapes fully supported, it is often necessary to reduce a tape standardized with 3 or 4 supports to the value with full support. This reduction is made by applying the correction for sag to one of the standardized lengths. It is generally easiest to use the standardization with three supports, at 0, 25, and 50 meters. The computation of the sag correction is greatly simplified by using tables found in USC&GS. Sp. Pub. 247 and TM 5236. Both tables required that t15,000 grams. TM 5236 gives the catenary corrections for various lengths
and weights of tape, while SP 247 gives the value of the quantity 
X1010for each 3o gram from 20
grams to 30 grams. d. To illustrate the application of the catenary correction, the standardized tape, for which the certificate is shown in figure 56, will be reduced to full support and then various odd lengths and supports computed. (1) From the certificate, the length of the tape interval, from 0 to 50 meters when supported at 3 points, 0, 25, and 50, is 50.00067. The weight of the tape is 25.8 grams per meter. From TM 5236, the catenary correction for a 25meter interval for that weight tape is 1.93 millimeters. For two 25meter intervals, the correction is 2 X 1.93= 3.86 millimeters. The length of the tape when supported throughout is then50.00067 +0.00386=50.00453 meters Starting with this standard length as supported throughout, the correction for an odd distance can be found. Assuming a length was measured as 37$ meters with 3 supports at 0, 25, and 37 /2, the correct measured length is: % times 50.00453=37.50340 minus the catenary corrections for 25 meters (1.93 mm) and 12% meters (0.24 mm) which gives a corrected measured length of 37.50123 meters. The catenary correction for 12 /2 meters could be computed from the formula or taken as 3g the correction for 25 meters. The fraction y comes from the fact that the correction varies as the cube of the length. (2) Comparing the computed value for the 37 2 meter length over 3 supports with the value on the certificate, a discrepancy of 0.11 mm is found. This discrepancy is negligible and occurs because of the markings or irregular stretching of the tape. By using this method, the correct standard length can be obtained for any odd measurement or support arrangement. e. The inclination (or slope) correction reduces all measurements to a horizontal plane. In order 100
to make this reduction to horizontal, the 'difference of elevation of the ends of the tape must be known. Also very important are the elevations of any intermediate tape supports if they differ greatly from the grade of the end supports. The required differences of elevation are usually determined by spirit leveling. This leveling will be discussed later in the text. (1) The inclination correction can be computed from the formula:
or
CG=
4 h2 h 21 813
h6 1615
in which CG= inclination correction (grade correction), = inclined length, and h=difference in elevation of the ends of the inclined length. (2) For short lengths or steep grades, use the formula:
CG= (1
2
For 50meter lengths and differences of elevation of less than 7.5 meters, TM 5236 can be used, or the series formula: CG
h 2 h4 21 813
No more than two terms are necessary. (3) Differences in elevations as abstracted from the level books and inclination corrections as computed or as taken from tables are recorded on DA Form 1915 (Abstract of Levels and Computation of Inclination Corrections) (fig. 57). In the second column of the form, the distances between the points in the first column are entered. The mean differences of elevation between the points are written in the third column. Be sure to indicate the units of measure for the differences of elevation by crossing out either meters or feet at the top of the column. The inclination correction is entered in millimeters or 0.01 foot for each length. The sum of the correction is obtained for the section and recorded at the end of the column. Elevations are computed for sufficient supports so that a mean of the
ABSTRACT OF LEVELS AND COMPUTATION OF INCLINATION CORRECTIONS (TM 5237)
PROJECT
LOCATION
DATE
442
U.. 5,9. INL AIN FERENCE OF MENCORRECTION (M or 0.01 I) (Meters or9+4
DISTANCE Mtr)
POINT
Adis,
1C.
MA ELEVATION or
ELEVATION Me(Mhter
REMARKS
1042.4L
A
2
IA
so
aff
~
13.0
1
0~
A3A
L
50
4.s. 14
AaA7
B3
25
0.23
L.
DATE
4~4
11LE
.~~+1.16
COMPUTED BY
104#4IO0 L
.23g
±L0A n8
6
i
.S6
.23 ORGANIZATION
____
L046,3
Q148.16
1047g1
CHECKED BY
DATE
11
~FRM57195DAIFEB
Figure 57.
Computation of inclination corrections (DA Form 1915).
101
elevations will provide a mean elevation for the section which is accurate to within 2 meters. Use of this mean elevation in computing the reduction to sea level of a section is explained in the following paragraph. The sum of the inclination corrections for the section is entered on DA Form 1914. Any odd tape lengths, such as 12%1
or 37%1
meters, or setups
and setbacks, should show their individual inclination corrections merely for the sake of convenience in checking. f. Each section of a base line must also be reduced from the measured length to the length at sea level. It is for this purpose that the mean elevation of the section is computed on the Abstract of Levels form. The formula for reducing the base to sea level is: C= S
h
h2
r+S S
h3
+ ...
in which C = correction to reduce a length, S, to sea level; h= mean elevation of the section; and r = radius of curvature of the earth's surface for that section. Only the first term of the formula is needed, except for firstorder base lines at high altitudes. The value of log r can be found in table IV, appendix III, using the mean latitude of the ends of the base and the azimuth of the base as the arguments. (1) For this example, the mean latitude is 250 N and the azimuth of the line is 67°. The value of log r from the table is 6.80459. The mean elevation of the section, as found on the Abstract of Levels form, is 1044.9 meters. The approximate length of the section is 625 meters. The reduction to sea level is computed as follows: log 625
=
log 1044.9 =
2.79588 3.01907 3.1954110
colog r
=
log C C
= 9.0103610 =  0.1024 meter
(2) Inclination corrections always shorten the measured length. Sealevel corrections shorten the measured length if the base is above mean sea level, and lengthen the measured length if the base is below mean sea level. 102
g. As the subject of setups and setbacks is sometimes confusing, additional discussion is needed. Setups and setbacks can take either of two forms. They can be partial tape lengths, such as 20 meters, 15 meters, and so on, or they can be short measurements of the order of 5 or 10 millimeters measured with a scale made to bring the tape end onto the marking strip. Special care must be taken in the computation of temperature, tape, and inclination corrections for setups and setbacks.
40. Final Length and Probable Error a. The reduced length of the section is now obtained by applying the corrections to the recorded length. b. The adopted length is the mean of the reduced length of the forward and backward measurements of the section, provided the discrepancy between the measurements is less than 25 mm V/K (K is the length of the section in kilometers). The limit of 25 mm K applies to thirdorder base measurements. Closer limits are placed on first and secondorder bases. c. In the sample base computation, the reduced lengths of the forward and backward measurements are 624.6648 meters and 624.6784 meters, respectively. The allowable difference in measurements is 25 mm /0.625=19.76 mm. The actual difference is 13.6 mm, which is within the allowable limits, and the 2 reduced lengths are meaned to obtain the adopted length of 624.6716 meters for the section A to B. d. The value in the v column is for use in finding the probable error of the section and the base. The residual v is the difference between the reduced length and the adopted length. e. The last column on the form is headed P.E. and may be used for either v2 (residual squared) or for the probable error of the section. The probable error is computed, using the following formula: ;n(n1)
P.E.=0.6745 V n(n1)
in which v is the residual, and n is the number of acceptable measurements made of the section. Where a section is measured only twice, the probable error is merely 0.6745 times 1 the difference between the two measurements. f. The probable error of the entire base is the square root of the sum of the squares of the prob
able errors of the individual sections. The probable error is also expressed as a fraction with a numerator of 1, such as 1/1,000,000, meaning an error of 1 part in 1 million. For thirdorder base lines, the computed probable error should not exceed 1 part in 250,000. For first and secondorder bases, the computed probable errors should not exceed 1 part in 1,000,000 and 1 part in 500,000 respectively. g. All results are entered on DA Form 2851 (BaselineAbstract of Results) (fig. 58).
perature of standardization, unless both tapes have the same coefficient of expansion. (3) The tension must be the same for both tapes. (4) The true lengths of the field tape are then tabulated (fig. 59). b. The length correction table for the field tape (fig. 60) is computed and tabulated as follows: (1) The correction is computed for tape segments which are multiples of 10 feet, or 5 meters depending on the units of the tape. (2) The catenary factor, L3 , is computed. This factor is the cube of the segment length divided by the total length of the tape cubed.
41. Standardization of Field Tapes When the tape used in the field has been compared to a standardized tape, a length correction table is computed for use in the reduction of lengths measured with the field tape. a. In order to have a true comparison of the field and standardized tapes, certain reductions are made to the comparison measurements of the two tapes. (1) The standardized tape is reduced to its true length for the conditions of measurement. (2) Both tapes are corrected for the difference in temperature of observation and tem
L3=(segment length
3
(3) The tape correction, C,, is considered directly proportional to the length of the tape. In figure 59, the correction for 100 feet is +0.002, therefore the correction for 10 feet is +0.0002. Between
BASELINE
REDUCED LENGTH
PISULA
624.6716
DATE
LOCATION
USA
17 May 1954
OBSERVER AND ORGANIZATION
C.
PISULA 
FROM STATION (Master)
PROBABLE ERROR LOGARITHM OF
AMS
±4.59 mm
LENGTH (Meters)
1/136 094
2.79565176
ELEVATION
A
LOG ARCSIN CORR
M
TO STATION (Remote)
METERS RATIO
NO. OF
ELEVATION
B
M
0 REJECTIONS
OBSERVATIONS
2
0
TAPE STANDARDIZATION MANUFACTURER
TYPE
INVAR INVAR
K & E K & E
MFG'R. NO.
N. B.
5123 5124
DATE
S. NO.
7871
10 Sept 54 16 May 54
ELECTRONIC INSTRUMENT DATE CALIBRATED
MASTER (Type and No.)
DATE CALIBRATED
REMOTE (Type and No.)
REMARKS
MICRO
FILM NO.
This baseline does not meet thirdorder specifications, and should be used as a check base only. Tape K & E 5123 was standardized after use in the field. COMPUTED BY
AMS  R.A.
Smith
DATE
CHECKED BY
DATE
25 Jan 56
AMS  W.C. Aumen
25 Jan 56 BASELINE ABSTRACT OF RESULTS
DA FORM 2851, 1 OCT 64
(TM 5237)
Figure 58.
BaselineAbstract of Results (DA Form 2851).
103
100
and 200
feet the correction
is
+0.00500.0020+0.0030 and the correction for 110 feet is +0.0023. (4) The catenary correction, C,, is determined by multiplying the catenary factor (L3) by the total catenary correction for the tape. The total catenary correction is found by subtracting the suspended length from the fully supported length of the tape. In figure 59 this correction is 200.005199.929= 0.076 for the 200 foot length. The check at the 100 foot mark shows the computed catenary correction of 0.0095 is very close to the measured correction of 0.009. This check should always be made. (5) The tape and catenary corrections are combined to form the Ct+C corrections. (6) These corrections are applied to each segment to give a table of corrections for
use with any lengths measured with the field tape.
42. Broken Base a. Description. A broken base is a base consisting of more than one horizontal tangent. No portion of the base with considerable length should be inclined at an angle of more than 20 ° to the final projected length of the base and the maximum should be kept down to 120 if possible. b. Computation. To reduce the broken base to a single horizontal tangent, the law of cosines is used. a2 = b2+c 2  2bc cos A where a is the single horizontal tangent to be determined, b and c are the two measured segments of the base, and A is the angle at the intersection of the two measured segments.
RESULTS OF TAPE COMPARISON
LOCATION:
TIME:
Ellensburg, Washington
12:30 p.m.
STANDARD TAPE NO: OBSERVER:
R.
C.
DATE:
17 October 1957
TENSION: K&E 8159 Campbell
20 lbs.
FIELD TAPE NO: CHIEF OF PARTY:
8161 J. E. Norton
SUPPORTED ON A HORIZONTAL FLAT SURFACE INTERVAL
L NGTH
0 to 100 feet 100 to 200 feet 0 to 200 feet
100 .002 100 .001 200 .005
SUPPORTED AT THE ENDS OF THE INTERVALS INDICATED BELOW INTERVALS
IrNGTH
0 to 100 feet 100 to 200 feet 0 to 200 feet
99.993 99.992 199.929 Figure 59.
104
Results of field tape comparison.
LENGTH CORRECTION TABLE
FIELD TAPE NO:
LENGTH (feet)
K&E
8161
DATE COMPARED :
CATENARY FACTOR
COMPUTATIONS
Ct+Cs
Cs
(Ct
L3
(2)
(1) c 10 20 30 40 50 60 70 80 90 100
110 120 130 140 150 160 170 180 190 200
.000125 .001000 .003375 .008000 .015625 .027000 .042875 .064000 .091125 .125000
.166375 .216000 .274625 .343000 .421875 .512000 .614125 .729000 .857375 1.000000
17 Oct 1957
(3)
(4)
TOTAL CORRECTION W/TWO (2) SUPPORTS (6) 
(5)
+.0002 +.0004
. 0000 . 0001
+.0006
.
0003
+.0002 +.0003 +.0003
+.0008 +.0010 +.0012 +.0014
. 0006
+.0002
. 0012
.
0002
.000 .000 .000 .000 .000
.
0021
.
0009
. 001
.
0033
. 0019
. 002
. 0049
.
.
003
.
. 0051.
.
005
.
007
+.0016 +.0018
0069
0033
. (0095)
. (0075)
+.0020
. 009
.
+.0023 +.0026 +.0029 +.0032
.
0126
. 0103
. 010
.
0164
.
.
.
0209
. 018
.
0261
. 0180 . 0229
+.0035
. 0321
. 0286
.
029
+.0038 +.0041 +.0044
.
0389
.
0351
.
035
.
0467
043
0554
0426 0510
.
.
. .
+.0047 +.0050
.
0652
.
0605
. 060
.
0760
.
0710
.
Figure 60.
007
0138
014
. 023
. 051 071
Length correction table (field tape).
Section II. TACHYMETRY MEASUREMENTS 43. Methods Used Tachymetry is generally construed as the measurement of distance by optical means. Included within this category are the stadia methods and the subtense methods. Each method makes use of a base of some type. The stadia methods utilize a vertical base which depends on the length of rod subtended between crosshairs within the instrument. The subtense methods
utilize an outside base of fixed length, across which horizontal angles are measured.
44. Stadia Methods The stadia methods determine distance by the length of a vertical rod intercepted between two fixed crosshairs within the instrument. The method may be used for lower order mapping where the corrections applied are very rough. 105
It is used as the means of determining distances for leveling for which the stadia constant is determined regularly and with considerable care. The selfreducing tacheometer uses an intercept on the vertical staff which is always 1/100th of the distance.
45. Use of the Self Reducing Tacheometer a. The RDS tacheometer equipment consists of a tacheometer and a vertical staff. The tacheometer is constructed in such a way that the stadia crosswires always intercept that portion of the staff that is 1/100th of the distance, regardless of the inclination of the telescope. The telescope also displays the appropriate curve (of four) and the difference of elevation factor that is required to determine the difference of elevation at hand. By use of this method, it is possible to determine the difference in elevation, direction, and distance from the same observations. b. The distance is determined by reading the rod intercept between the lower and upper stadia crosshairs, considering the millimeters of the intercept to be whole numbers and multiplying by 0.1. The difference in elevation is determined by multiplying the rod intercept, between the lower stadia crosshair and the cutting point of elevation curve across rod, by the elevation factor displayed beside the elevation curve. In some cases, two elevation curves and two elevation factors appear in the telescope at one time. In such cases each rod intercept and its corresponding elevation factor will be handled as described, but the differences in elevation for both must be the same. The signs of the elevation factors must be altered to conform to the elevations in the direction of progress. c. The tacheometer should not be used in cases where the required tolerance in elevation is less than 2 meters.
meters. The directions for the use of a subtense bar state that the midpoint of the bar must be centered over the station, the bar must be level, and the bar must be perpendicular to the line of sight of the instrument. Figure 61 illustrates the conditions for a subtense measurement. b. A and B, figure 61, are stations, d is the subtense bar of known length, a is the subtended angle which is measured by the transit or theodolite, and D is the required length from A to B. Since the figure is an isosceles triangle, the perpendicular bisector (D) of the base (d) also bisects the vertex angle (a) and forms two right triangles in which an acute angle (%a) and a side (Yd) are known. By plane trigonometry, cot D %a=Yd or D= (yd) (cot %a), but the subtense bar is 2 meters long which makes 1 d= 1 meter, and the formula becomes D =cot %a in meters. This formula is useful in emergencies, but very accurate tables are required for the cotangent functions. c. The best method for solving subtense distances is by using tables XLII and XLIII in TM 5236, which have the subtended angle (a) as the argument and give the distance immediately in meters (table XLII) or feet (table XLIII). The tables are valid only with a 2meter subtense bar. The whole subtended angle is used as the argument in the tables. d. It is possible to obtain thirdorder accuracy by the subtense method by using the proper equipment and careful observation.
d= subtense .bar
46. Subtense Method a. Subtense distances, as the name implies, are measured by the angle subtended by a known length. The length of a subtense bar is usually 2
Figure 61.
Sketch, subtense measurement.
Section III. MEASUREMENTS USING LIGHT WAVES 47. GeodimeterModel 2 a. The Model 2 series Geodimeter measures distances indirectly by measuring the time required for a light beam to pass from the Geodim106
eter to a reflector and back to the Geodimeter. The Model 2 Geodimeter is primarily a first order baseline instrument. Its size and weight limits its use to drive stations and makes its use on lower order surveys impractical.
b. According to the manufacturer, the Model 2A Geodimeter will determine distances up to 30 miles with an instrumental error of 1 cm 1 ppm (part per million). The approximate distance to be measured must be known within 1,000 meters. c. As with all of the electronic distance measuring equipment, the atmospheric unknowns will probably introduce more error into the lines measured than the inherent accuracy of the in
=
strument. To reduce these unknown conditions to a minimum, only calibrated thermometers, psychrometers, and altimeters should be used.
48. Measurement. Reduction For the purpose of explaining the computation of the length of a line, it is assumed a Model 2A Geodimeter was used. A calibration was previously performed (fig. 62). The field observa
CALIBRATION CONSTANTS GEODI1,TER NO.11h
Calibrated by Army Map Service, 9 August 1961
1.1526
Zero Correction (Z)
m
.7960 M
Light Conductor Length (L) Transmitter Mirror Lens Focus
7.0 mm
Receiver Mirror Lens Focus
7.8 mm 5550 Ao
Photocell Wave Length
1.0003042
Refractive Index (00C & 760 mm Hg)
27.000
Calibration Temperature Unit Length, F1 @ OC & 760 mm Hg
7.4925333 M
Unit Length, F2 @ OOC & 760 mm Hg
7.4553801 N
Unit Length, F3 @ 000 & 760 mm Hg
7.2746155 M
Frequency, F1
10,000,000 CPS
Frequency, F2
l0,049,834 CFS
Frequency, F 3
10,299,559 CPS
Frism Eccentricity Correction (19 prism bank) Prism Eccentricity Correction
(54
.o480 14
prism bank)
Velocity of Light in Vacuum
.0450 14 299,792.5 IKS
Coefficient of Expansion of Aluminum Figure 62.
Calibration constants, geodimeter model
0.000022 1/00
2. 107
LC 2,
tions have been recorded on DA Form 2852 (Geodimeter (Model 2) Observations) (fig. 63), including the Mirror Constant, Approximate Length of Line, Transmitter and Receiver Focus, and Eccentricity of the Mirror and the Geodimeter. a. Determination of the Fine Delay (LC). For each frequency, the electrical length is reduced to a physical length. A space near the bottom of the observation sheet has been provided for this computation. The formula may be written: M) LC,,I,
LC=LCI(
M,
Where: LC = Fine Delay in Meters LC1 = First Light Conductor Setting
LC2 =Second Light Conductor Setting LCIm = Mean of First Light Conductor Readings
GEODIMETER
NO.
=
Watch the signs carefully. Obviously if the LC2 setting is larger than the LC1 setting, the resulting LC will be larger than LC1. b. Light Conductor, Coarse (nL). On the observation form, under the column headed LC, are numbers 768. These numbers indicate that seven light conductors were used or more accurately six light conductors and a portion of another. Since the exact length of the light conductors is shown on the calibration sheet, multiply the number of whole light conductors by this length (6X0.7960=4.7760) to obtain the number of meters. This value is entered on DA Form 2853 (Geodimeter (Model 2) Computations) (fig. 64).
(LCLC2)
PROJECT
Mean of Second Light Conductor Readings Mean of the two Mean Mirror Readings
=
OBSERVATION
GEODIMETER (Model 2) OBSERVATIONS
N
/4/
CALIFORNIA
GEODIMETER HEIGHT
1%,9 10.705
&ANAL , AMS
764
768 ./4 

5 26 0 250 29 7 29 /1290 29 7 32 0 29 0 .23 2 21 0 22 7 109 / 08 / /05 7 S26
_
4 SUM MEAN
2728 GEOD
2037
TIME
2702 MIRROR
TEMP 22.4 °C 22.2 PRESS
/oo.0

242
2508 MEAN
GEOD
MIRROR
0 2 3
22 / 28 26 520 28 4 22 176 23
MIRROR
2/1 24 9 27 0
GEOD
LC
c

73.0
Of 720
LC = 6800

°CF
2
3
MIRROR
vGEOD
MIRROR
2045 TIME 2057 2055 2104 2/03 TIME 21/2 21/1 22.2 °c 22.0C TEMP 22.0 °c 21.8 °c 21.9 ° 21.8 °c TEMP 21.8 c 2/1. c 0 990 M 898 m PRESS 98.0 m 88.0 98.0 m 87 0 PRESS 970 m 88.0m sf i70 575 575or5755,57F F 575 °5780°r5H 5885 .0 71.5 °f 71.0 f 72.5 ° 72.0 F M 720 F 71.5 °F 71.5 r 75 OFM 2202 26.85
2702 26.85 2702 25.08
x.04
LC=.7200
TEMP
MEAN 22.20 cc REMARKS
PRESS
W
TEMP
D
1172.38a FMEAN
751.51157.88
2188 OC
19 PR/SM
BANK
UNITS
x
LC =.5400 
75.6815750
D
°F
7/1.62 .
EMP
MEAN
2/.62
(300m, 1000 fl)
MM Ng
DA FORM 2852, 1 OCT 64
Figure 63.
108
GEOD
MIRROR
2120
2720
21.7 c 214 % 96.0 m 870 570 of 710 OF 7.0 O
8/25  78.80
,
x .04
=.5 66 W
USED
ARE

,8.s8o 81258.04
.04
I PRESS
GiVe units for temperature and pressure; give pressure reduCed for inSt ConStant
PRESSURE
23.33
.6523.65  22.10
n.7/17
__.6765
NOTE:
23.6
8so +
78 2 75?9 78 1 78 8 868 86 9 85 8 841 85 8 138_ 86 0 87 8 73769573 9 67 4 324 4 3250323 9 315 2 8/ /0812580987880
O55
H
LC
MIRROR
854 f.j
SIGN
O /55 4 22. 7 93 / 94(0 93 5 88 4 suM 93 28 23552 38 22MEAN
4
MURPRIY
MIRROR
4_
_ 

19 O 3
Sum
D E L
768
__ 
F R E
2046
90.0 i
j
LC
MIRROR
28 0 20 2 22'2
/00
MIRROR TENDER ___
772 __
SIGN
0.0/0o n(IAs.P4 WIT TER MIRROR ECC
LC
MIRROR
/FFDANL4 OBSERVER
ECC
42.572 r
j7
Km.
RECORDER
mm
GEODIMETER
24 21 27 4 2 3 29 7
GEOD
2035
D E
1.2
0.9 mm ELEVATION
v
_2~
_
+
F R E
LC
MIRROR
LC
MIRROR
FOCUS
RECEIVER
56.030 m
m
15.508
/0
FOCUS
ELEVATION
MIRROR HEIGHT
JUNCTION, usC GS /934
SIGN
TRANSMITTER
21 SFP1? /96/
STATION
MIRROR STATION
E
AMBATO
DATE
USAM GEODIMETER
APPROXIMATE LENGTH
BASELINE
0.0480
ORGANIZATION
F R EL
(TM 5237)
MIRROR CONSTANT
LOCATION
Geodimeter (model 2) observations (DA Form 2852).
°c
PRESS
W
7S1.75
15725
F
71.2
OF
~ROJET
OBSRVATIN
(MODEL 2) COMPUTATIONS
NOGEODMETER
/ LOCATION
(TN a237)
GEODIMETER NO.
DATE
OBSERVER
U SA/vS
AMBATO
2ISept61
/41
CAL.IFORNIA ORGANI ATION
BASELINE
GEODIMETER
WHITTER
STATION
DETERMINATION OP PARTIAL UNIT LENGTH LIGHT CONDUCTOR, COARSE(iiL) FINE MLc)
LIGHT CONDUCTOR, ZERO CORRECTION
(Z) +
ECCENTRICITY. GEODIMETER ECCENTRICITY,
________
4.7760
4.770
.6765 1.12
.7/17
.
STATION
MIRROR
IJUNCrION USC(GS/R934
AMS /96/
CANAL
5.5720 5366
.
1. /526
1. " /26
0 10 0
I
.0/00
f
./0
0480

. 0480

.
o27
+
.0/27
+
co0 S

MIRROR
CONSTANT, MIRROR
.
FOCUS CORRECTIONS
.
TEMP CORR., LIGHT CONDUCTOR

.0005
6.5793
Ki
SUMOF AGOVE (8)

6.6145
K2
22. 20 5 40
PRESSURE (P) (mm H)
000 7
7. 2352
K3
HUMIDITY (%)
HUMIDITY CORRECTION
.00 QO004,1

cps
FREQUENCY. (f)' QUARTER
WAVE

1.00027780
REFRACTIVE INDEX (Na)
fi
/0 000 000
f2
lU 7.492731019
LENGTH (U)
75/. 75
42
1.000 2 7821I
____________
2/."62
2/. 88 71.68
75/
2
1
0/ 27
.
DETERMINATION OP QUARTER WAVE LENGTH (OC)
TEMPERATURE (t)
0480
42
1.00027858
1.000 27885
.o
0ooo4 2
/.000
278 /6
0ooooo41 100027844 f3 /0 2P9 559
/0 049 834 7
,

4555 74 213
U3
7274802954
DETERMINATION OP NUMBER OP UNITS
APPROX LENGTH (LA)
.LAKi N'izLAKl/U1
Li=(NjxU1 )+K1 L:
+
PHASE SIGN, fa
PHASE SIGN, fl
/0 000 9993. 4207 /333 7 9994.3898
/3404
9997.08,47
i.
U2
0.037/56283
m
AN=1f2 L;i/U1U
725,
2
/2=2+A 4 /2
+N/405
Nl
N 3 =N 1 (1.03)
DETERMINATION OP SLOPE LENGTH UNIT
SLOPE LENGTH NQ
=
/0 527. 287/ /0533 8664
LENGTH (UxN)
Wave
(UxN)+K
r087.4 1
lenth
316.288
of light in micron..
)
+ gk
Hum. Corr. 
i 7
)4. +10 5. 1
M.
'=NiU)K Im. U 1
6 949
L12.
9993.3855
=NxU)K N2LAK2/U2 LAs
m.
..
PHASE SIGN, f3
x 108 0 + t/273
3IC
/0526.
N.AL
8853 i+ t
Q
6399 iO533. 875/
/0 527. 2708 /0533.
1447
.)76
U = 299792500
4 (f) (Ns)
1'*o
M4/4pJA N. '64
A OPTE7 A S
DA FORM 2853, 1 OCT 64
Figure 64.
Geodimeter (model
2)
computations (D.
Form
285). 109
c. Light Conductor, Fine (LC). See a above, for determination of this value. d. Zero Correction. This value is sometimes called the calibration constant and is determined during calibration. The value is the distance from the back centering point of the Geodimeter to the electrical center of the Geodimeter and can only be determined by calibration procedure. This value will change for each instrument and will change if any major components are replaced or repaired. e. Mirror Constant. The reflex used with the Geodimeter must have its constant computed. This value is furnished on the sheet of Calibration constants or may be computed. Since the refractive index of glass is about 1.57 as compared to air, the thickness of the prisms from the front to the apex at the back must be multiplied by 0.57 to get the Mirror constant. The correction is negative as the light travels farther when going through the glass due to the index of refraction being 1.57 for glass as compared with 1.00 for air. f. Focus Corrections. The focus correction is the algebraic sum of the Transmitter and Receiver Focus as calibrated, minus the sum of the Transmitter and Receiver Focus as determined during observations. Watch the sign. The formula may be written: FC= (Tf
+R f,) 
(Tfo+Rfo)
Where: FC=Focus Correction Tf,=Transmitter Focus Calibrated Value Rf,=Receiver Focus Calibrated Value Tfo= Transmitter Focus Observed Value Rfo=Receiver Focus Observed Value g. Temperature Correction of Light Conductor The difference in observation temperature and the calibration temperature will cause an expansion or contraction of the light conductor tubes. This correction can be determined by the formula: nL(0.000022) (to tc) Where: nL=Light Conductor, Course (b above) 0.000022 =Coefficient of Expansion of Aluminum (meters per degree Celsius) to=Observation Temperature (Celsius) tc=Calibration Temperature (Celsius) (tot,) determines the sign of the correction.
110
h. Temperature (t) (°C.). The mean temperature (OC.) of the air for each frequency is brought forward from the observation sheet. If temperature is in °F., convert to OC. using table XV, appendix III. i. Pressure (P) (mm Hgq). The mean pressure is brought forward from the observation sheet and if necessary converted from altimeter readings to millimeters of mercury. Conversion tables are furnished in table XVI, appendix III, or computed from the following formula: P,(2880.0065h
Po\
288
5 25 6
.
,
Where: P 2=Barometric Pressure Po=Barometric Pressure (760 mm Hg) h=Altitude (Meters)
at sea level
j. Relative Humidity. The relative humidity may be determined from tables supplied with the psychrometer or altimeter or by a Humidity Slide Rule (Short and Mason) using the wet and dry bulb temperatures as arguments. k. Refractive index (Na). Before proceeding further it is necessary to compute the refractive index so that the length of a quarter wavelength at each frequency can be determined. The formulas are:
N 1+273.2
5.5X10 8 e 1+ t 273.2 1+273.2
P) 760
and: Ng=l+
(16.288 12 2876.4+3 \82
)
(0.1361) 4 _ 107
)+5\X
Where: N =Refractive Index Ng=Refractive Index for group velocity t=Temperature in degrees Celsius P Pressure in mm Hg e=Humidity in mm Hg X=Wavelength of light in microns (10 6m) (1) Ng values are furnished in table XVII, appendix III, for a series of values of X. The photocell wavelength (X) must be furnished on the calibration data and will be different for each instrument and/or each photocell.
(2) Na is determined from the first portion of the Na formula NQ=1+
N1 1+
t
P 760)
Na is the refractive index without the humidity correction. (3) Humidity CorrectionThe humidity correction is taken from the correction for humidity graph (chart 4, app. II) using Temperature Celsius and Relative Humidity as the arguments. This correction is subtracted from Na to produce the refractive index (Na). (4) Frequency (f)The frequencies for fJ, f2, and f3 are determined during calibration and are given in cycles per second. These values are assumed to remain constant until a new calibration is performed. (5) Quarter Wavelength (U)The quarter wavelength is determined by the following formula: U_299,792,500 4(f)(Na) Where: 299,792,500±400 meters per second is the velocity of light in a vacuum as adopted by the International Union of Geodesy and Geophysics and the International Scientific Radio Union at their international meeting at Toronto, Canada in 1957. 1. Determination of Number of Units. With the value for a quarter wavelength at each frequency of modulation corrected for the temperature, pressure, and humidity at the time of observation, the number of whole quarterwavelengths at each frequency in the line being measured can be determined. (1) From the observation sheet determine the phase signs from the four signs of each frequency at the top of the page. If the four signs for the frequency are alike, the phase sign is positive; if they are unlike, the sign is negative. (2) From the observation sheet, enter the approximate distance on the computation form. This length must be within 1,000 meters of the correct length or an incorrect result will be obtained. If the
approximate distance cannot be determined by any other method, a short base with all angles turned should be observed by the field party to provide data for an approximate length. (3) The sum of the corrections for frequency 1 and frequency 2 (KI and K 2) are subtracted from the approximate length. LAK1 and LAK
2
(4) Nl and NZ are determined from the formula: N LAKI LAK 2 SLAK and N LA U1 U2 NI and NZ are rounded to agree with the phase sign of each frequency. For example, the phase sign of fJ is negative and Ni computes out as 1333.7. According to the phase sign, Ni must be an odd number, therefore 1333 is entered on the form. (5) LI and L2 are determined from the formulas: L = (Ni X U 1) +K, and L = (N2 X U) +K
2
and then Li is subtracted from L2. If L is larger than L2 the resulting value will be negative. (6) U2 is subtracted from U 1 and divided into L2L' to produce AN which is the value to apply to N; to obtain the total number of quarterwavelengths. (7) The resulting N 1 and N 2 should agree with the phase sign for each frequency: Phase sign ,
N is an odd number
Phase sign +, N is an even number A relationship exists between the N values and is 100, 100.5 and 103. N 3 is determined by multiplying N 1 by 1.03. m. Determination of Slope Length. The unit length is determined by multiplying the corrected quarter wavelength (U) for each frequency by the N value for that frequency. The quarter wavelength has been corrected for temperature, pressure, and humidity and will change for each frequency. The slope length is determined by adding the internal corrections (K) for each frequency to the unit length (UXN) for each frequency. 111
n. Electronic Distance Measurement Summary. (1) The slope lengths for a series of measurements over a line are entered on DA Form 2854 (Electronic Distance Measurement Summary) (fig. 65). This form is provided for listing of the measurements, determination of statistics, and reduction to a geodetic distance on the reference spheroid. After listing the measurements, rejection of doubtful observations should be made using rejection limits determined by Chauvenet's Formula (table XI). The arithmetic mean distance is determined and residuals (v's) computed by subtracting the observed distance from the mean distance. The formulas for computing the probable errors are given on the form. The ratio is the mean distance divided by the probable error. (2) The observed slope distance must be reduced to a geodetic distance on the reference spheroid. Care should be used when abstracting the elevations of the instruments as observations are made over several days and the height of the instrument will vary. o. Horizontal Distance. The horizontal distance (H) is computed using the Pythagorean theorem: H H=
/ (SLOPE
DISTANCE)2  (DIFFERENCE IN ELEVATION)2
(1) The difference in elevation (d) is obtained from the following formula: d= (ha+HIa) (hb+HI,)
Where:
112
2 Red.= (to)( 06265) tan [ H when H is a horizontal distance and:
(t o) (206265) sin T when T is a slope distance and " is the observed zenith distance. When using slope lengths, the formulas for h2hl are as follows: Red.
h 2 hl=T sin
(
2'),
for reciprocal observations h2 h 1 =T sin (90° +k), for nonreciprocal observations (3) If nonreciprocal observations are used, a value for (0.5M) or 0.429 should be used in the computations. (4) The, use of altimeter elevations to determine differences of elevation is permissible if more accurate methods of determination are not required. The altimeters must be carefully calibrated both relatively and absolutely, and fairly stable air conditions must exist between the stations. p. ChordArc Correction. The ChordArc Correction (K) is applied to the horizontal distance to change it from a chord distance to an arc distance on the surface of the spheroid. This correction is computed using the formula: H3 K 24p2 or by the approximate formula: K= 1.027H3 X 10 15
ha= Elevation of occupied station hb=Elevation of distant station HIa,=Height of instrument at occupied station HIbHeight of instrument at distant station
Where: H=Horizontal distance p=Mean radius of curvature from table XX (app. III), using azimuth of the line and mean latitude as the arguments.
(2) The elevations of the stations are normally determined by either differential leveling or trigonometric leveling. When computing the elevations by trigonometric leveling, the reduction to line formula. is:
a. Method of Use. The Model 4 Geodimeter measures distances indirectly by measuring the time required for a light beam to pass from the Geodimeter to a reflector and back to the Geodimeter. The approximate distance must be known to within 1,000 meters. According to the
49. Geodimeter Model 4
DISTANCE MEASUREMENT SUMMARY
.DROJECTELECTRONIC
YUMAA
TRAVER.5E
YUCCA

(IN 5237) OATE
ORGANIZATION
L
TYPE
PE8 . 196,4
USAMS
ARIZONA OF LINK MEASURED
LIBASE
BASELINE
F
TRAVERSE
0
Dos.
LUSCf6S
/.9
'934
ITNE
/0533. 86604 .
2
.

.__8751 4
832
.
S__
.. ____
8786
8786
___
___8772
IOSERVER MP NY
INST. ELEV.
42. 572
J48180
DISTANCE REDUCTION
8,555
DIFF. OF ELEVATION (d
.52. 458
MEAN ELEVATION
0024
Az. OFLINE (y
900
0057
MEAN LATITUDE (#a)
33
.0c//
86 81
5675
0078
. .
.8751
7
,0111
+
8853
56,0o30
RESIDUAL (y)
METERS
SKRR WTE
STATION KLKV.INTKLV
STATION ELKV.
H.1.
8ANK PRIS4 .5.608 ".
(Slane) NO
0.703 M.
and Me.).
INST. (7yp.
REMOTE STATION
JUNiCrION
N. I.
/4/1
AMS 1961
CANAL
OI TRIANGULATION
TRILATERATION INST. (7jp*dNo.).
MASTER STATION
MEANRVADUSE
M)
00"
00'
c 00' 0
34
OF
00 94
SLOPE
+ .
0024
HORIZONTAL DISTANCE (H)
.
00/1
ECCENTRICITY
*1.
0003
METERS
/05338775
DISTANCE (T)
0.00o35
CHORDARC CORRECTION (E)
4.
0. 00/12 00865 ,
CORRECTION
.8780
.
000$
SEA LEVEL' REDUCTION (C)

11
.8703
.,.
00 72
GEODETIC DISTANCE (1)
/0533.7887
12
.849.5
+.
50
8.5
13 122
14.8823
. .5
87,82
168822
.
/7
17.88
88 27 19~okse,vdb
___
.

 .
007
0047
o.O4 2 oo'S2
20
/0533.
+S3.84
8776
H3
K
:/T H=T5
Km02Hx0
i'i.I
METERS
/8(o6o
C:H
h+
H
2T _ZT
2
3 _ ST dom.
S=H+K+C
~C
cngicgtogpk.
o
REMARKS
the
/ on 21 Sept. /96/. ons
9 fakens
.000550 32
+0t.9'/
PE(ODS) = t 0.674EV PE(Nn) = t 0.6715
CLAR KE
REFERENCE SPHEROID
Observatioas /0 Iffr4 /8 takeni oej 22 Sept. 1961.
22
DISTANCEK
METERS
"
10
ao080 coSO. 0048
METERS
ZY
2 2
*
 1)
/n(n 1)
I//600 000 n =NO. OF OBSERVATIONS ZVa = SUM OF RESIDUALS SQUARED
1COMPUTED (F Ji,,Qc a AM 5 BY
DATE
DEC . 3
CHCKD
BYATE
J.L.an
uw
AMS
1JAN. '04
DA FORM 2854, 1 OCT 64
Figure 65. 757381
0

65  8
Electronic Distance Measurement Summary
(DA Form 2854).
113
manufacturer, the Model 4 Geodimeter will determine distances within 1 cm. t 5 ppm (parts per million). The Model 4B is listed as having a night range up to 3,000 meters, but it has been used successfully on 8,000 meter lengths under ideal conditions. The Model 4D employs a mercury vapor lamp and the manufacturer claims a night range of 25 miles and a daylight range of 3 miles. (1) A quarter wavelength for any Geodimeter may be determined from the following formula: x 4
C
4(F) (Na)
Where :=Quarter wavelength C =Velocity of light F = Modulating frequency Na = Index of refraction
(2) The Model 4 Geodimeter has been designed to have a quarter wavelength of 2.5 meters for Frequency 1, at a temperature of 6 degrees Celsius and a pressure of 760 mm of mercury. If a color sensitivity of 5,650 angstroms is adopted for the photomultiplier tube, the refractive index for light waves at this temperature and pressure is 1.0003104. These values were used by the manufacturer to produce the tables furnished with the instrument, but the manufacturer used a value of 299,792,900 meters per second for the speed of light. The velocity of 299,792,500 meters per second has been adopted internationally. (3) In order to use the velocity of light of 299,792,500 meters per second with the tables furnished with the Model 4 Geodimeter, it is necessary to change the modulating frequencies from F 1 =29,970,000 cycles per second; F 2=30,044,920 c/sec, F 3=30,468,500 c/sec as furnished by the manufacturer to the following: F 1=29,969,947 c/sec, F 2= 30,044,872 c/sec, F 3=30,468,445 c/sec. If the frequencies are not changed from the manufacturer's values, then the distance measured can be multiplied by 0.9999986657 to correct the computed distance for the velocity of light difference.
114
(4) When the color sensitivity of the photomultiplier tube is known, a new refractive index should be computed and applied as a refractive index deviation. Computation of the refractive index and the deviation is taken up later in this section.
b. Geodimeter Reductions. The Model 4 Geodimeter measurements are recorded on DA Form 2855 (Geodimeter (Model 4) Observations and Computations) (fig. 66) which also serves as a computation form for field and office use. For purposes of explaining the computation procedure it is assumed that the headings have been completely filled out including elevations, the approximate distance, and the calibration date. A sheet containing the most recent calibration (delay line) data must be furnished the computer. A sample sheet containing the calibration data is shown in figure 67. c. Determination of Meters From Delay Line Data. Phase 1, 2, 3, and 4 of Frequencies 1, 2 and 3 with the sign of each are recorded in the field. The signs for F 1, F 2, and F 3 for the Reflex and Geodimeter are determined by the sign of the initial setting for Phase 1 of F 1, F 2, and F 3 . These signs are determined by the direction the null indicator moves in relation to the movement of the delay line control. The four phase readings are meaned in each of the six columns. Using these mean values as the argument, the meters are interpolated from the calibration sheet and are entered on the form to the nearest millimeter. Subtract the meters in the Geodimeter column from the meters in the Reflex column for each frequency. The resulting value must be positive in each case. If a subtraction is impossible, add U 1, U 2, or U3 to the reflex meters under F1, F 2, and Fs as needed. The signs to be entered in the ReflexGeodimeter colurin are determined from the signs at the top of the six columns of readings and are paired for each frequency. For example, if the signs for Frequency 1 are both positive or both negative then the ReflexGeodimeter sign is positive (even); if the signs are unlike, the ReflexGeodimeter sign will be negative (odd). However, if U 1, U2, or Us is added to the meters so a subtraction can be performed, then the sign of the reflex minus Geodimeter will change. The L 1, L2, and L3 are the ReflexGeodimeter values. However, L 2 and L3 must be larger than L 1, and therefore U2 and/or Us are added if necessary and the signs changed again if they are used.
T
PROJECT
I/V/ADiA DATE
APPROX.
28 FEB5./962 INSTRUMENT
NO.
3.6 REFL ECTOR
47 REFLECTOR
REFLEX
+
SIGN
SIGN
,'97i+ /59S 1
1 2
/9P2 9 /93.
3_
4_ MEAN
+
REFLEX  GEOD. (+ U1 if required)

(L 3 
3565.
64
+
S2
8/

S2
M
E
E
M
HUNDREDS
REFLEX  GEOD. (+ U 2 if required) L
0. 357
OF METERS
APPROXIMATE
(La
9 9 9 9 9 8 6 6 (Dl+ D 2
000
U 2 X N2
M
L
2
66
_3
REFLECTOR CONSTANT
6
CORRECTION
+
0.
INDEX CORR. (RC)
O

ECCENTRICITY CORRECTION*
a'
1 +0397
cC
0.493
3566. +
5(00
METERS
/SMS
E
5
3564.28
M
2/184
3
M
356646 9 M
TEMPERATURE. DRY
WET
28.78 2.8
0_5.0
02.2
28.68
076
28.68 28.73
0 ,
METERS
60o 6
22
4.5
03.2
03.2
076 MEAN
7%
45%/
MEAN
06.2
REFRACTIVE
MEESCOMPUTED
039
HMDT
MEAN
54
02.7
25 2
INDEX
x1
7
06
1.0003104
REFRACTIVEINDEX (Ne')(
1.0002859
(RC)
.0000245
REFRACTIVE INDEX DEVIATION NOTE:
If the refractive index correction is used omit the temppressure correction. RC= DX RD
DATE 4
M
/ 42.9 /4.97
3 =
FACTOR (From lNom agram)
36)] 107 A A
t
J2S4 2 . 4 9 L 0
c
/ 426
U3 X1 N 3
METERS
AND MM/HG FOR PRESSURE.
COMPUTED BY
I84
0
M
ALTIMETER
)
FOR TEMPERATURE
F
°
MEAN
0.229

r4
MEESASSUMED
N9=1+ L2876.4 + 3 (1J&.F N
N3 =LA U
0.439
2.
N2 =LA  U2
3566.443
METERS
CHORDARC CORRECTION
GEODETIC DISTANCE
52.75
2.184
NI =LA U 1
L
__
SEALEVEL REDUCTION
so0
2.623
L3 (+ U3ifreq.)

534
237.00
M
METERS
3566. 762
HORIZONTAL 0IST. OR CORR.
237239 +
M
EES
08
.
SLOPE DISTANCE
4
2.85/
METERS
CORRECTION
53
MTRSGEODIMETER ETR PRESSSURE
030
0.
+
+
METEOROLOGICAL READINGS
.23

237
C
6
.6
}
2
3563.592 _
M_
55
E
X42 6
L 1 ) (21)

+
SIGN

REFLEX GEOD. (+ U3 if required)
TENS AND UNITS OF METERS
M
+D3)=3
CONSTANT
USE
M
2.85/
(+ U2if req.)
2
M
GEODIMETER

235
METERS
$0
°00
33
REFLEX
MEAN
o.647
/.04
METERS
OF LINE
MEAN LATITUDE
+ 
.52.75
86 .25
MEAN
$~
:3566.508
Dl
REFRACTIVE

4_
1508
HUMIDITY
53
50 METERS IF SIGNS OF L 3 AND L1 ARE NOT THE SAME
LA
TEMP.PRESSURE
+1
3

ELEV.
PHASE__SIGN
54
+
0.629
5.72 64._/96
L ) X (21) ADD
9'2+
88
2
F3
+
SIGN
S/
S508
X(4)
L 1)
+j
2
1. 508
_______
SIGN
GEODIMETER

5/ 25
75
AZIMUTH
440.66~ M
,M
52 49
ELEV.
INST.
439.10
REFLEX
PHASE
J. M. WRITE
550.0O INST.
ELEVATION
,
F2
N.'G. K
Inc._____
xLECMnm1A ANGLE N
C'OaauCION. SPRI ANGLE
47 0,
A
(TM
LOGGA ID
.
oJ1
9.isA
124q7
_
(3A
__1KT.
4
__2
2LA
2S
ome
.54.6 +2.
.20
BaD 2

52
II
g44g

g.8u~ 78
'56,
488 2.2
to.3
g.g32217
___
Chv
23 Kan.
~
2
.38
34
47.2.__
23Ta*
q
12
112
23
.3
F Ge. Tan 

Ren *
Ken
_
K
23
enrd~e
R
Fig ur8.
148
~~~ J
e n
_
__
_ e
3
_
3o&
6
8136
pol o 4.2___ °O
5
303
9.750
.. a__
7_
___ ___
___
_
__
_
3.
24
249
p&no Preliminaryd
120
3. 4
__
____2&32

k
3~s.tar4 ..
___
Rom 3__
16&39.
3.
236 1~ e
__2
3..0110
07kp
3. I6432(
____
co puaton of trage
o
edcint
etr
D
om11)
Ecc. must be used because in each case, one of the stations was not occupied. For the second computation of the preliminary triangle, the reduced directions at Ken should be used to obtain all angles at Ken, and the triangles should be corrected where necessary to close to 1800. (b) Attention is now directed to the change in the list of directions. Although the list was originally labeled Ken Ecc., the application of the eccentric reductions has changed the list so that the directions in the Corrected Direction with Zero Initial column are at station Ken. (9) A second illustration shows the reduction to center for an eccentric object observed. In this illustration, the lists of directions for all the stations involved and the reduction to center computation are shown in figures 87 and 88, but the preliminary triangles are not shown as no new principle is involved in computing the triangles. The two important points to be brought out in this illustration are the methods of obtaining the directions for the eccentric reduction computation and the application of the reductions to the appropriate lists of directions. (10) To obtain the directions (a's) for the reduction to center form, subtract 180 ° from the direction from Home to Home Ecc. as referred to Park on the list of directions for Home (222°55'20"180 °
= 42°55'20"). Since this direction (Home Ecc. to Home) must be 00 according to the rules stated previously, the procedure is to subtract the direction Home Ecc. to Home (42°55'20") from each direction on the list of directions for Home. The results of this operation are the directions from Home Ecc. to each station on the list and referred to Home as 0° . These are the directions entered on the reduction to center form. For example, on the list of directions for Home, the direction to Park is 00000'00"00 as it is the initial; subtracting 42°55'20" from it gives the direction 317°04'40", from Home Ecc. to Park as
referred to Home. This value is rounded off to 317°05 ' for use on the reduction to,
center form.
The direction from Home
Ecc. to Cedar is 47007'49'64255'20" ' =4°12 ' . The rest of the directions are
found in the same manner. (11) The eccentric reductions are now computed as explained previously, and the algebraic signs attached (fig. 88). The reductions are applied to the individual list of directions. The reduction for Cedar of +3'18 is applied to the direction to Home Ecc. on the list of directions for Cedar; the reduction for Gerst of +44':84 is applied to the direction to Home Ecc. on the list for Gerst, and so on. The application of the eccentric reductions automatically changes the observed station from Home Ecc. to Home on the various lists. (12), In all cases, a sketch of the eccentric conditions should be shown on the list of directions and on the reduction to center computation form.
59. Strength of Figure a. Introduction. (1) For computation purposes, it is often necessary to know the relative strength of the figures involved in the triangulation net. The strength of a figure depends on the size of the length angles in the triangles through which the length is computed, on the number of directions observed in the figure, and on the number of conditions to be satisfied in the figure. (2) The strength of a figure is designated by the letter R and is computed from the equation: R=(DC
1\ [SSDAB
in which D= the number of directions observed in the figure; C= the number of conditions to be satisfied in the figure;
SA and
8
B are the logarithmic differences
in the sines for 1" change in the distance angles A and B of a triangle. (3) In table XLI, TM 5236, the values tabulated are [bf+SA2B+ 8 in units of the sixth place of logarithms. The two arguments of the table are the distance angles in degrees with the smaller distance angle being given at the top of the table. 149
3753ORGANIZATION
PROJECT
3753
OF DIRECTIONS
ICLIST
S
IAB.(TM
5.237)
r
OBSERVER
INST. (TYPE) (NO.)
OBSERVED STATION
10
ECCENTRIC REDUCTION
00.00 i
o2 o7.4 a
Phrk5q
al
0
+
SEA LEVEL REDUCTION
n
CORRECTED DIRECTION WITH ZERO INITIAL
n
'..2
5&5TA2 21
Taknma
00 . 00.00 i
O
02
___43
7_7._ .
____
58
59
11.6
5q
29
285
~STATION
OBSERVER
INST. (TYPE) (NO.)
OBSERVED STATION
OBSERVED DIRECTION
I nsane Ecc
3
#
/
0
00
00.00
62
57
2S.g9
II
/
if
51.3

oo .o
53ORGANIZATION
CORIIECTED DIRECTION WITH ZERO INITIAL
SEA LEVEL REDUCTION'
f
,/
i
0
00
00.00
62
58
17.2.
88
'0
O'.0____
Wi OBERE
BSRE
____________
_______
JRCIN
O
/
f
0
00
00.00
12.
13
22.0
.2a
53.2
_46.
DATE
Nci. 1319I46
T2
BSRVDDIETIN
SAIO
OBEVDSTTO
SEA LEVEL REDUCTION'
ECCENTRIC REDUCTION: /
if
0
n
1
+
44.
_
STAT ION (TYPE) (NO.)
Wild TZ
OBSERVED DIRECTION O
(Acr0 Wal la oc e H o4rn e E..c.
/
it
00
00.00
2 36
if
00.00
13
22.0
21
38.o
ECCENTRIC REDUCTION /
12.1 8.
L i , 1 
I
if
Pairk
__
Ja~n
CORRECTED DIRECTION WITH ZERO INITIAL 0
.
0
00
f
00.00
ADJUSTED DIRECTION' I _____
44Ai454.8 
i
OF DIRECTIONS
_______18
SEA LEVEL REDUCTION*
if
ADJUSTED DIRECTION'
°DATE
No. 137946
44 4554.8_____ 116
/
00
ILIST
Iand
__________
12
___46
tan
OBEVRINST.
CORRECTED DIRECTION WITH ZERO INITIAL
0

ainlf6
______
OSREJETORGANIZATION
OBSERVED STATION
if
______
fSTAT
INST. (TYPE) (NO.)
TK
/
IONGet
F.T. A.
LOCATION
ADJUSTED DIRECTION
0
LSOFDRCIN
4./fw
Mryn
LOCATI ON
17____ lJsn u
137A46
ECCENTRIC REDUCTION
o
_________88
PROJECT
DATE
W,14 T2
____________
L 7L6a17 324 8 .
Figure 87., Corrected lists of directions (DA Form 1917).
150
10.6
LIST OF DIRECTIONS
Macry lad
Womne
ADJUSTED DIRECTION'
]ORGANIZATION A , :r. T
37553
LOCATION
00
O
4
crn
PROJECT
DATE
OBSERVED DJRECTION
e
OBSERVER
¢
MaySTATION ryanCe
LOCATI ON
j
A
__
if
PROJECT3
53
ORGANIZATION
A 4S,
PROJECT
LIST OF DIRECTIONS
Ic
(TM 5237)
LOCATION
STATION
OBSERVER
INST. (TYPE) (NO.)
\4W d
TKE
DATE
T2
OBSERVED STATION
137446
No.
0
0
ar
I
ID
00
00.00
Cedcir
41
Gerst
1742J15A.
Geiv~iald I.1ma Ec. 2.7gmnr.
ft
55
20
____
_
___
_
___
__
r_
To
3KiI
Ecc. o
0
I
ft
0
00
00.00
ADJUSTED DIRECTION f
.
07A q.
25 427 222
t
I
CORRECTED DIRECTION WITH ZERO INITIAL
SEA LEVEL REDUCTION
ECCENTRIC REDUCTION
6S
IsJan
tob
_
_
s
To
,Par To
Figure 87=Continued.
b. Use of Table XLI in TM 5236. To compare two alternative figures, either quadrilaterals or centralpoint figures, so far as the strength with which the length is carried is concerned, proceed as follows: (1) For each figure, take out the distance angles (to the nearest degree if possible) for the best and secondbest chains of triangles through the figure. These chains are to be selected at first by estimation, and the estimate is to be checked later by the results of comparison.
(2) For each triangle in each chain, enter the table with the distance angles as the two arguments and take out the tabular value. (3) For each chain, the best and secondbest, through each figure, take the sum of the tabular values.
DC (4) Multiply each sum by the factor D for that figure, where D is the number of directions observed and C is the number of conditions to be satisfied in the figure. 151
PROJECT
REDUCTION TO
37.353
LOCATION
TYEO
wCCW4TRIC
a~
AtS. Inc.
ORGANIZATION DATE
21
J n.
0
STATION:
I
Worn* Distance (d)
Sum = a
Loa a
(rl
m)
.7
6 Loa
C
03
(meters)
2
Cologasin 1"= 5. 31 4 43
STATION
TTO
ouxcT AT STATION
Log d= 0.44 S560
56
CENTER
(TM 5237)
),
2.19Lo R OO M
RI Mor RamucrouI
R3fON
Center
Ce
4,'4
1QS.
Gefdl.I
S1&
dM 4.10&l JSW15 1. 6IQIL a
05
omyfts
DA I F
71921
Figure 88.
152
.
Q.2B 1 t A14 2 h. 1 £hJ5 A A L 8 , 6474 AJ. ZJ8
{ 1 8L
Computation of a reduction to center (eccentric object) (DA Form 192?1).
f
The quantities so obtained, namely:
line unoccupied,
(DC) D
/
5j[SA+A
(j) For a foursided, centralpoint figure with one corner station not on the fixed
I'SB
will for convenience be called R 1 and R 2 for the best and secondbest chains, respectively. (5) The strength of the figure is dependent mainly upon the strength of the best chain through it, hence the smaller the R 1 the greater the strength of the figure. The secondbest chain contributes somewhat to the total strength, and the other weaker and progressively less independent chains contribute still smaller amounts. In deciding between figures, they should be classed according to their best chains, unless the best chains are very nearly of equal strength and their second best chains differ greatly. DC c. Some Values of the Quantity D (1) The starting line is supposed to be completely fixed. 41 = 0.75. (a) For a single triangle, 4 (b) For a completed quadrilateral,
104 10 10
= 0.60.
(c) For a quadrilateral with one station on
82 the fixed line unoccupied, 8 2
0.75.
(d) For a quadrilateral with one station not on the fixed line unoccupied, 72 0.71. 7 (e) For a threesided, centralpoint figure, 10 104 = 0.60. 10 (f) For a threesided, centralpoint figure with one station on the fixed line unoccupied, 8
123 12 = 0.75.
2 = 0.75.
(g) For a threesided, centralpoint figure with one station not on the fixed line 72 unoccupied, = 0.71. (h) For a foursided, central point figure, 14 5 = 0.64. 14 (i) For a foursided, centralpoint figure with one corner station on the fixed
line unoccupied,
11= 0.73. 11 (kI) For a foursided, centralpoint figure with the central station not on the 102 fixed line unoccupied, 1020.80. 10
(1) For a foursided, centralpoint figure with one diagonal also observed, 160.56. 16 (m) For a foursided, centralpoint figure with the central station not on the fixed line unoccupied and one diagonal observed, 12 4 = 0.67. 12 (n) For a fivesided, centralpoint figure, 18 = 0.67. 18 (o) For a fivesided, centralpoint figure with a station on a fixed outside line 164 unoccupied, 16
4
0.75.
(p) For a fivesided, centralpoint figure with an outside station not on the 154 fixed line unoccupied, 15 = 0.73. (q) For a fivesided, centralpoint figure with the central station not on the 132 fixed line unoccupied, 13 0.85. (r) For a sixsided, centralpoint figure, 22 0.68. 227 22 (s) For a sixsided, centralpoint figure with one outside station on the fixed 205 line unoccupied, =20 0.75. 20 (t) For a sixsided, centralpoint figure with one outside station not on the 195 fixed line unoccupied, 19 =0.74. (u) For a sixsided, centralpoint figure with the central station not on the fixed line unoccupied, 16
0.88.
(2) To illustrate the application of the strength table, the R, and R 2 for figure 102 will be considered. Let it be assumed that the direction of progress is 153
from the bottom line toward the top line. It will be found that the smallest R, called R 1 for this figure, will be obtained by computing through the three bestshaped triangles around the central point. The next best R, called R 2, will be obtained by computing through the two triangles formed by the diagonal. The R 2 is easily computed as follows: From the known side to the diagonal, the distance angles are 890 and 27 ° . Using these angles as arguments in the strength table, the factor 17.5 is obtained. Similarly, from the diagonal to the top line, the distance angles are 91 ° and 26°, and the corresponding factor is 18.8. The sum of the two factors is 36.3. If the central point of the figure is an occupied station,D
DC
0.67, and R 2 = 36.3 X 0.67 = 24,
as given opposite the figure. (3) The R 1 may be computed in a similar manner by using the distance angles in the three bestshaped triangles around the central point. d. Examples of Various TriangulationFigures. (1) Figures 89 through 102 show some of the principles involved in the selection of strong figures and illustrate the use of the strength table. (2) In every figure the line which is supposed to be fixed in length and the line of which the length is required are represented by heavy lines. Either of these two heavy lines may be considered to be the fixed line and the other the required line. Opposite each figure, R, and R 2, as given by the table, are shown. The smaller the value of R 1, the greater the strength of the figure. R 2 need not be considered in comparing two figures unless the two values of R1 are equal, or nearly so. (3) Compare figures 89, 90, and 91. Figure 89 is a square quadrilateral, figure 90 is a rectangular quadilateral which is 12 as long in the direction of progress as it is wide, and figure 91 is a rectangular quadrilateral twice as long in the di154
All
®
stations
occupied.
Any one station occupied.
not R 2 m6
5
45
Figure 89.
Strength of figure diagram.
All
)
=0.56(see above),
and R 2 = 36.3 X 0.56 = 20. If the control point is unoccupied, as shown in figure 102,
®
O
RI1 R2=1
stations occupied.
Any one
R1 =2 R2 =2
station not
occupied.
27 63
63
Figure 90.
Figure 91.
Strength of figure diagram. (
All
®
Any one station on ixed line not occupied.
R,22
stations occupied.
R2.22
Rl m27 82
Strength of figure diagram.
A stetions occupied.
Figure 92.
.27
RI 21
Strength of figure diagram.
All
stations
occupied.
I
R212
Figure 93.
Strength of figure diagram.
rection of progress as it is wide. The comparison of the values of R 1 in figures 89 and 90 shows that shortening a rectangular quadrilateral in the direction of progress increases its strength. A
All stations occupied.
Figure 94.
comparison of figures 89 and 91 shows that extending a rectangular quadrilateral in the direction of progress weakens it. Figure 92, like figure 90, is short in the direction of progress.
/
RI 164(approx.) (opprox.) R2 =176
®
All
(
One outside station on fixed line not occupied.
stations
occupied.
RI = 36 R2 .102
Strength of figure diagram.
RI 2 R2 .12 RI "3 R2=15
Unoccupied station at inte rsection of fixed line and d line to be determined.
RI
v4
R2
20
Strength of figure diagram. Figure 100. ®
All
®
One corner station not occupied.
)
Figure 96.
not
Strength of figure diagram. Figure 99.
Figure 95.
Unoccupied station on fixed line.
stations occupied.
Central station occupied.
not
Strength of figure diagram.
RI 13 R2 15 R I 16 R2 =16 R I 17 R2 17
Strength of figure diagram. All
stations
occupied.
RI =9 R 2 =9
(A strong and quick expansion figure.) ()
All
)
stations
occupied.
Any one outside stotion not occupied.
® Central
station not occupied.
Figure 97.
R = 10 R2 15
RI III R2 = 16
RI = 3 R2 = 19
Figure 101.
Strength of figure diagram.
Strength of figure diagram.
All stations occupied.
Central station. not occupied.
R1 = 5
R2 • 5
DC
S
2816 28
Figure 98. ° Strength of figure diagram.
R I =18 R 2 =24
0.43
Figure 102.
Strength of figure diagram.
155
Such short quadrilaterals are in general very strong, even though badly distorted from the rectangular shape, but they are not economical as progress with them is slow. Figure 93 is badly distorted from a rectangular shape, but is still a moderately strong figure. The best pair of triangles for carrying the length through this figure are DSR and RSP. As a rule, one diagonal of the quadrilateral is common to the two triangles forming the best pair, and the other diagonal is common to the secondbest pair. In the unusual case illustrated in figure 93, a side line of the quadrilateral is common to the secondbest pair of triangles. Figure 94 is an example of a quadrilateral so much elongated, and therefore so weak, that it is not allowable in any class of triangulation. Figure 95 is the regular threesided, centralpoint figure. It is extremely strong. Figure 96 is the regular foursided, centralpoint figure. It is much weaker than figure 89, the corresponding quadrilateral. Figure 97 is the regular fivesided, centralpoint figure. Note that it is much weaker than any of the quadrilaterals shown in figures 89, 90, or 91. Figure 98 is a good example of a strong, quick expansion from a base. The expansion is in the ratio of 1 to 2. Figures 99 and 100 are given as a suggestion of the manner in which, in second and thirdorder triangulation, a point A, difficult or
impossible to occupy, may be used as a concluded point common to several figures. (4) Many of the figures given are too weak to be used on firstorder triangulation, but for convenience or reference and to illustrate the principles involved, they are included with the figures which can be used.
60. Side Equation Test a. Experience has shown that the requirement for triangle closures is not always sufficient, and the agreement in length of the various lines in the figure as computed through the two best chains of triangles must be checked before leaving the station. This is done by a logarithmic computation of the triangle sides, and a comparison of the loglengths of those for which there is a double determination. In a quadrilateral, these will be the three exterior sides other than the known side. For firstorder, the loglengths should agree within about one and onehalf to two times the logarithmic difference for one second in the sine of the smallest angle involved in the computation of the length; for second order, 2 to 4 times the difference; and 10 to 12 times the difference for third order. b. When a quadrilateral (fig. 103) has unsatisfactory triangle closures, it will be necessary to make an inspection of the closures to determine what stations should be reobserved. (1) Usually, it will be found that the triangle closures (fig. 104) will show two triangles with large triangle closures which have a common line. This indicates that poor
TONY BILL
WALLY MILLER Figure 103.
156
Quadrilateralfor sideequation test.
PROJECT
DATE
JA.4
P'ETCAI Y LOCATION
DE
TRIANLE
COMPUTATION OF
v]ORGANIZATION
SPRICAL SIPuERRCAL
STATION
03S33v30 ANOLKR
CORRucttON.
ANnLR
ANOLR
*COL 23
0LUOGAR
PLN
Excxaa
G
6864
______4.306
1 10+12 2
MILLER
8+9 3
BILL
1+2
56 14 39.34 76 /3 21.69 47 32 02.17
TO0NY
:
AN. ipii
0.0o80 1827w .9,987 3215 9. 867 8664
4,374 1906
13
4,.254 7355
__12
0.90______
03.20
4.3741906
23 WALL Y
2+3
1 2
748
3
SILL.
4+6
86 46 2775 38 22 35.98 5550S 53.32
MILLER
0,00/ 1822 9,792.97/7 ___9.9177956
13
83445 4.2 931684
__4.16
12 __
__ _
705
_
0.73
_
__23
_
_
_
6864
___4.306
.5+6
i
WALLY
/0+1I 7+9
2
TONY BILL
3
45 25 13.12 31 1I 32.28
1473524*
___0.
7143256 9880452
___9.
103 22 55.49
__
__5.
__13
/68 3644 0840
___4.
___12
__
23
2 0840
___4.44
94 37 /5S32 45 0/ 29.41 40 2) /4.63
13
0*
__0.00/1414
8496732 9._______ 8112459
___9.
___
12
__4.293/7/2
10.829
______59.36
9.Akox
R.
A MS
IF8. (04
CHECKEDDY
T8nCL

FORM FEB
7439
_4254 4__
______
COMPUTED myDAE
DA S
__4,442
0.74
______00.82
1+.3 1 MILLER 11+t2 2 TONY' 4+5 3 WALLY
_
A MS
DT
FEFF&64
U. &. Gwanmmuw "UIffe" OPiI IS17 04"m66
571,918
Figure 104.
Triangles for test quadrilateral.
157
observations were made at one or both of the stations at the ends of the line. (2) Applying a trial correction to the observations at one of the stations to improve the closure and by recomputing, the triangle may give better side checks, which definitely establishes the fact that this station must be reoccupied. If the side check is not improved but made worse, then the same procedure should be followed at the other station. In the majority of cases, the above procedure will show which station should be occupied. (3) Now and then the closures will be so distributed with poor side checks that the above procedure will not be conclusive. In this case, it will be necessary to make a side equation test to determine what station or stations must be reobserved. This example of making a side equation test is for firstorder triangulation. (4) The numbers on the sketch are measured directions. For example, 1 at MILLER means the direction from MILLER to TONY; 2 is the direction from MILLER to BILL; 3 is the direction from MILLER to WALLY, etc. Referring to the angle at MILLER from TONY to BILL, we use the expression 1+2, the left hand direction is always given a negative sign. At WALLY, the angle from TONY to BILL will be expressed as 5+6. If we consider the triangles in which directions were measured at station MILLER, then the following relationship can be expressed: MILLERTONY MILLERBILL MILLERBILL XM ILLE R WALLY MILLERWALLY X MILLERTONY

(5) Since the equation was written around MILLER, it is therefore called the pole. Equations similar to the abovecan be written for the other stations when they are used as poles. The sides in any triangle are proportional to the sines of the opposite angles and, therefore, we may write the following equation for the above: 158
Sine (8 +9) Sine (10 +12)
Sine (4 +6) Sine (7 +8)
Sine (11 +12) XSine (4 +5)
1 
or, using logarithms, it will becomeLog Sine (8+9)+Log Sine (4+6) +Log Sine (11+ 12)  Log Sine (10+ 12)  Log Sine ( 7 + 8)  Log Sine (4+5)=0 (6) The above relationship would be exact if the measurements were perfect, but since there is always a small error in the direction measurements, there will be a residual. This residual is the indication of the errors so far as the equation is concerned, and is referred to as the constant term of the equation. (7) This constant term, divided by the sum of the tabular log difference for one second of the sines of the angles involved, will give a quotient which is the average correction to be applied to the angles in the equation so as to eliminate the residual. By experience of many years, it has been found that for triangulation, this quotient must be less than 0.7" for firstorder, and 2" to 4" for secondorder. (8) If there are large closures and it appears that the error is at one end or both ends of a common line, the test should be applied by using both stations as poles, and if the quotient is greater than 0.7" for one of the equations, the error in angular measurements will be found at the opposite end of the line from the selected pole. This is true because the angles at the station selected for the pole do not enter into the equation, and the source of trouble must be at the station at the opposite end of the line from the pole. (9) A side equation test is given for the quadrilateral MILLERTON YBILLWALLY. A study of the triangle closures shows that errors are present at two or more stations. Therefore, equations are solved with each station as a pole, below.
Select Pole at MILLER Angle
Value
11+12
473202.17 854627.75 450129.41
 10+ 12 7 +8 4 +5
761321.69 555053.32 402114.63
8
+9
4
6
Select Pole
Log Sine
Angle
Log Diff. 1"
9.867 8664 9.998 8178 9.849 6732
+ 19. 3 + 1.5 +21. 1
 11+12 7 +9 2 +3
+ 5.2 + 14. 3 +24. 8
1
at
Log Sine
450129.41 1032255.49 382235.98
+3  10+11 7 +8
943715.32 311152.28 555053.32
9.998 5860 9.714 3256 9.917 7956
Average correction
Select Pole at TONY
7+9 4+5 1+2
1032255.49 402114.'63 561439.34

1.7
+34.8 + 14.3 103.5
171
1 103.5
1.
65"
A summary, therefore, shows the following results: Log Diff. 1'
Log Sine
452513.12 943715.32 473202.17
5.0
171
56
5+6 1+3 8+9

+26.6
9.630 7072 +9.630 6901
86.2
56 Average correction 56  0.65" 86.2
Value
+21.1
+9.630 6901
9.716 3630 +9.716 3574
Angle
Log Diff. 1"
9.849 6732 9.988 0452 9.792 9717
+9.716 3574 9.987 3215 9.917 7956 9.811 2459
WALLY
Value
9.852 6476 9.998 5860 9.867 8664
+20.7 
1.7
+19. 3
Pole at MILLERaverage correction is  0.65" Pole at TONY average Pole at BILL average Pole at WALLY average
correction is 0.98" correction is  1.93" correction is 1.65"
+9.719 1000 9.988 0452 9.811 2459 9.919 8173 9.719 +9.719
1084 1000

5.0
+24.8 +14. 1 85.6
 84 84 Average correction 84 = 0. 98" 85.elect Pole at BILL6 Select Pole at BILL Angle
2 +3  10+12
5
+6
Log Sine
Value
382235.98 761321.69 452513.12
9.792 9717 9.987 3215 9.852 6476
Log Diff. 1'
+26. 6 + 5.2 +20. 7
+9.632 9408 4 +6 1 +2  10+11
854627.75 561439.34 311152.28
9.998 8178 9.919 8173 9.714 3256 9.632 9607 +9.632 9408 199
Average correction
 199 102. 9
= 1. 93"
+ 1.5 +14. 1 +34. 8 102.9
(10) The equations show that the angles at MILLER enter into the three which give an average correction greater than the specified value of 0.70" while the angles at MILLER do not enter into the test where the average correction is less than that amount. It is definite that the directions measured at MILLER are probably in error. A test should be made before actually reoccupying the station by assuming a change in angles. This can be done very easily by multiplying the tabular log difference for 1 second by the number of seconds the angle is changed and adding this value to the sine or subtracting according to sign. Suppose the direction at MILLER to BILL to be in error. The angle at MILLER from TONY to BILL may be decreased with a corresponding increase in the angle BILL to WALLY. If we assume a decrease of 3" from TONY to BILL and the corresponding increase from BILL to WALLY, then the side equation test will result as follows:
159
Pole at TONYaverage correction 42 42  0.49" 85.6 Pole at BILLaverage correction78 .102.9
0.76"
Pole at WALLYaverage correction92  0.89" 103.5 (11) The correction for pole at MILLER will remain 0.65 since the angles at MILLER do not enter the test. An examination shows that the change at MILLER has improved the corrections and station MILLER should be REOBSERVED. (12) The average corrections with poles at BILL and WALLY are still too large, while with the pole at TONY, a satisfactory value was obtained. The angles measured at TONY do not enter into the equation with the pole at TONY, but do enter into the tests for the poles at BILL and WALLY. It is probable that the angles at TONY are in error. If we assume a change in direction to WALLY from TONY of 2" in order to balance the closures of either side of the diagonal, i.e., decrease the angle from BILL to WALLY at TONY by 2" and increase the angle from WALLY to MILLER by the same amount, the test will give the following results: Pole at BILLcorrection will be08  0.08" 102.9 Pole at WALLYcorrection will be+10 +10 = +0.10" 103.5 Pole at MILLERcorrection will be14  0.16" 86.2 (13) The correction at TONY will remain 0.49" since the angles do' not enter into the test when it is selected as the pole. Tests show that changes at TONY 160
will result in improved corrections and TONY should be reobserved. In this example, both MILLER and TONY must be reobserved to obtain satisfactory results. (14) The side equation test will be effective provided that either the angles are approximately equal or that the error involves fairly small angles. If the bad angles are close to 900, the test will not be conclusive because the tabular log difference for 1 second of the sine is very small, and an error of several seconds in such an angle might still give a quotient for the equation which will be less than 0 70".
61. Station Adjustment a. Many times, during highorder triangulation observations, duplicate directions are observed to a station from two or more initials. These duplicated directions cause condition equations which can be properly satisfied only by a least square solution in which each observed direction is weighted according to the number of sets involved in its determination. This least squares solution is referred to as a station adjustment. In the case of a station adjustment containing only one condition, the solution is referred to as a weighted mean. The station adjustment provides statistically the most probable value for the direction to each observed object. b. An example of a station adjustment is presented with the intention of clarifying the computation, and to emphasize the value of its application. The mathematics involved in the computation are presented in USC&GS Special Publication 138 (pages 816) and are not repeated here. The station chosen for this example is within a complex firstorder triangulation network and has a total of 27 directions referred to five different initials (fig. 105). These directions were each determined by meaning a set of not less than 12 circle position readings observed using a direction theodolite on which the horizontal circle micrometer may be read to within ± 0.2". The readings were abstracted from the field books according to the methods outlined in paragraph 55. c. Following is the procedure for preparing the condition equations: (1) A preliminary list of directions (fig. 106) is made from the abstracts of directions. All directions are evaluated and those
BOB EARL
HOMER
JOHN
FRED
Figure 105.
757381
0
 65  11
Diagram of observed directions.
161
TA
PROJECT
KE
LOCATION
PRELIMINARY LIST OF DIRECTIONS
U SA M S
ORGANIZATION 'rl
T
UKSTATION
OBSERVER
(TM
5237)
INST. (TYPE) (NO.)
DATE
T3
WILD
OBSERVED ST'ATIO V
ORSERVEnISIRECTION
ECCENTRIC I;EDUCTION
n
J
00.00
_____0
AL
0
BOB
46 54 45.51
ADJUSTED
DIRECTION CORRETFED WITH ZEROINITIAL
SEA LEVEL REDUCTION'
it
r
00
0
MAR160
NO._/2345
F WILSON
.T
0
r
00
DIRECTION'
IF
I"
i
00.00
___
45.55
EARL
/4 32.49____ 31.09 /08 40 58.86
FRA NK
58.71 /37 42 2756
GEORGE
145
DON
10/
53
40.88 40.98
HOMER
180 02 S56,63
JOHN
195 26 49.88 50.60
___ ___
5749
____
_______
___49.85
[ 00 00
EARL
JOHN
NEWl
00.00
86 45 50.88
INJTIA L __
51.78______
1
ROY
/66
29.39
2/
_
NEW INITIAL
00 00.00
GEORGE
00
JOHN
49 33 o8.57
ROY
00° 00' 00.00 35 56 24.20
NEW
NEW
WILBUR
__
0.
00,00,00
41
01 45.21___
PETE
62'19 51.91 79 1 536,69______ 9S 56 55/7______ S
FRANK
30o2
f344
H1O M ER columns are for office
COMPUTED
BYS
__ __
FRED
use and should he left
~54.87
AMSJ
FEB 5T 1917
Figure 106.
__
/5 3754______ 36 06.92________________ blank
in the field.
DAY

162
iN/TI IL
JOIN LLOYD
ROY
DAI
INITIA __
45. /8
" These
___
MA
I
60
CH
KE
CHECKE
0IJIc~de
y BYM/
JX,". 4'AM
M
DATE
MAY 60
II. S. GOVERNMENT PRINTING OFFICE :l195 0420665
List of directions (preliminary) for station adjustment.
directions which cause excessively large condition equations are rejected and not shown on this list. (2) An adjustment list of observed angles (fig. 107) is made repeating all the angles from the preliminary list. Means are determined for all of the angles having more than one value and a column headed WT. (weight) lists the number of values used in determining each mean angle. Another column headed v shows the identifying number at each correction (v) which will be applied to the observed angles upon completion at the station adjustment. The final column headed adjusted final seconds is left blank at this stage and will be completed when the computed corrections are applied. (3) The list of directions for adjustment (fig. 108) shows a direction to every observed station, all referenced to a single initial with the v's associated with the angles used to determine those directions. This list is completed utilizing the minimum number of v's required to refer all directions to the single initial. Again, the final seconds column (adjusted directions) is left blank and will be completed when the computed v's are applied. (4) Every v which was not used in the "list of directions for adjustment" creates a
condition. Each unused observed angle and its associated v is listed in turn and compared to the corresponding angle from the "list of directions for adjustment." In this example, the list angles have been subtracted in each case from the observed angles in order that the algebraic sign in the condition equations will be correct. This algebraic difference is set equal to zero in forming the condition equation. Since there were five angles not used in the sample adjustment list, there are five condition equations (fig. 109). d. The correlate equations are prepared from the condition equations. In figure 110, column 2, headed alp, a is a selected constant, and p is the weight of the particular v. Normally, a is chosen as the least common multiple of all the weights, p, in order that the values of a/p be integers. In this example, a equals 6. e. The normal equations (fig. 111) are obtained by taking the algebraic sums of a/p times the products of the various columns in the correlate equations. f. The Doolittle method is used in the solution of the normal equations (fig. 112). This method of solving normal equations is covered in paragraph 65. g. After the C's are determined, the v's are computed by substituting the values of the C's into the correlate equations taking into account the weights in the a/p column.
163
TMTABULATION
PROJET
OF GEODETIC DATA
(TM 5237)
KEN TUCK Y
LOCATION,
ORGANIZATION
LOOKOUT OBSERVED STATIONS OSREANLS FROM o NLS
AL
808,

W.()VADJUSTED W.~FINAL
/01 14
DON
SEOD
___
mml. 45.53' AL
.OOSAE
4,O 54 45.51 2
I
45.53
2
2
.31.79
2
.958.62
32.49
31.09 MN
AL

EARL
108 40 58.86 58.71 MN. 58.78
FRANK
AL

AL
 GEORGE
31.79
137 42
______
/
4
2
5
41.13
2
6
57.04
3
7
50.09
45 50.88_____ 5/. 78 _____ MN. 51.33 2
8
51.47
2/
9
28.78
/
/0
08.96
45.2/ 45.18 MN. 45.20
2
ii45.20
I
2756
2760
145 53 40.88 40.98
______
MN. 40.93 AL  H~OMER
/80
02 56.63
57.49 MN. 5706
/95 26 49.88
AL  JOHN
50.60
49.85
MN. 86
EARL JOHN
50./I
EARL  ROY
/66
GEORGE JOHN
49 33 08.57
JOHN  LLOYD
41
JOHN
29.39
0/
PETE
62
/9 .791
JOHN  ROY
79
35 36.69
JoNN FRED
95 S6
/2
51. 91 3731
I/3
S55.17 54.87
__________
2
2
14
55.02
/5 3754
I
15
375/
JOHN11HOMER 344 36 06.92
/
/6
06.95
.35 S6 24.20
/
17
24.20
MN. 5.0
_____
JOHN FRANK
302
ROY W ILBUR
TABULATED BY
DATE AMS
DA
.1962
1ORM
Figure 107.
164
______
APR.0 GO921961
1EKED I J.12a .
DATE
BY
4o

U.S. OVRMN
AMS
APR.6~0
MI0 19Z OFFICE: 7891789
List of angles for adjustment.
wAORGANIZATION V
PROJECT
TM
LOCATI ON
LIST OF DIRECTIONS
AIA
USAt 1SM
(TM 5237)
SATION
KENTUCKY
OBSERVER
LOOKOUT
INST. (TYPE) (NO.)
WILD
F WILSON OBERE
SATO.
BSREDDIETIN
0
808
46
54 45.53
/1
EARL
/08.40 58.78 /37 42 27.56 /45 S3 40.93
FRANK______
GEOR_____E JON LOD236
PEE257
G. te
DAFR
n'p
A44IS
+ V3
_
58,62
_______40
46 42.02 + A, + )2
46
42.00
02
2740
23 58
45.11 61. 60
+ V4 +'
Vs
__
+ V6 + V____
+ V1 f' g. V7 + 14_
+ V7 +
+ 1 *3
be left blank In the field. DATE

4 .31.79
+ V7 + y_
.291 23 45.13 3 /0 58 51.00
arc for ofliee use and should
54 45.53
_______
_______
5704 50.09 35.29
FRED
*These columns
Vi
00 00.00
02 26 28
2 75 02 26.80
COMPUTED BY
+
00.00
DIRECTION'
2760 41.13.
R~OY
WI LB UR
00
60
ADJUSTED)
42 53
02 5706 95 26 50.11 28 3.531
HMR180
CORRECTED DIRECTION WITHZERO INITIAL
____0
31.79 + V2
DON
14
ECCENTRIC
00.00
00
______MAR.
SEA LEVEL REDUCTION'
REDUCTION
OBEVDSATO.OSREDDRCIN
AL
DATE
NO. /2345
T3
I
APR. 6~0
DATE
CHECKED BY
J.4
ka.w
M71917
AMS
APR. 60
U. S. GOVERNMENT PRINTINGOFFICE :1957 0420665
Figure 108.
List of directions for adjustment.
165
TM
PROJECT LOCATION,
TABULATION OF GEODETIC DATA
I(TM
.,IVI
KETUK
5237)
ORGANIZATION
SM
STATION
LOOKOUT______ EARL JOHN
.8
ANGLE
O83.
L IS r ANGLE CoN, I TlON
CONDITION GEORGE
JOHN
0
/
/66~ 21 /66
2

ANGLE
V7
V
7
. V8
29 39 + Vy
V3 + V7 + V,3 0=+/.37 + VgV7 + Vv 
21 28.02

/5 37 45' +0.0 9
+
______
0
____

V4  V7 V4 +V7 +V1
344 3( 06.95 + V6
5
A
DATE R.(0CHECKlnED
0:0.03
AMaP.60zA4S
GPO 92,961
Figure 10.9.
+ Vo_______
344 36 06.92 + V,6
COND1 TION
DA , FR x1962
a
/5 ,37.54 + Vis
Of
4
ANGLE

166
302 302
oBS. ANGLE
BY
*
//Sr
_________LIST
ETAUULATED
0.00 +
V3 V3
49 33 091/8  vs + V7 00.61 + VS V 7 + V 0
CONDITION
JOI4N HOMER

49 33 08.57
08. AI GLE LIST. ANGLE
FRANK
9+V
08S. ANGLE
CONDITION JOHN
l
86 45 51. 33
08S. ANGLE LIST ANGLE
EARL ROY
5

V6

V7
+
V7 +
V6
DATE.
BY
AR6
U. S. GOVERNMENT PRINTIN~GOFFICE:
Condition equationsstation adjustment.
1957 0

21182
LOOKOUT___
Sta.
Equations
Correlate
Vs
___AdOPrED
al. T 3 2
3
3 4
3
2
45
1c
V 0.000 __0.000
+2 /
6 3
6
3
7
2
/
3
i'
_
/
0.164 *0.034
+1
+0./96
t0___
I#
6
/0
3
+/
+1
IS 8 9
I / 1/
0.0/7
__
+#0'./43
+0 II3
/4
6
_
/5
3' 6
/6 /7
6 6
0.00 0.16 *0.04 +0.20 0.02
0.021
+/
_
_
+o.392
I

+o.
0.00.
Sta.
.00
+8++ +1
2
0.034 0.03 IJ +1 +0.034 +o.o3
+
2i 3
1j
.
3
~
4
0.00
Equations
22
+2211k
Figure 111.
0,00
+1.37
0.0 JM 0.61 4~ + 0.09
Jj
__
___
Correlate equationsstationadjustment.
00

n
c
40+&04757
+11.00
+21.37 0.10227 + 10.39 +0.06540
+ /0.09 0.0o572 + 6.97 +0.00571
Normal equationsstationadjustment.
LOOKOUTI________
Solution /
2
+ 8.
3
+5
C,
/
0.00
614 +0.62
__0.000
Figure 110.
Normal
+0.39
0.000 ..
I
LOOKOUT
Sta
0.02
+o.14
0.000 0 .o0o
_
6_ 13
V 0.00
+1 0.614 o.61 +1
6
_2
___
of Normals
4
12.
+2.
S
n1
2.

0.00
0.62500 . 25000 +0.25000 +0.25000
=

0.00006
_
____
_
___
nc=
+/100
1.37500
+0.o4737
+2. 2. 2. #1.37 +21.37 13.87500 +0.75000 0.75000 0.75000 # 1.37000 +14.49500____ C2 =0.05405 +0.05405 +0.05405 0.09874 1.044(68 0.1022711. 2. 2. 0.61 +10.39
____+17
______
____
___
______~~
____
___
/c45946 10 1.45946 1.45946 0.68406 +6.85649____ 35 # 0.13953 + oo6540 o.45553 +0.06540 3 =
___
____C
+0.09
+2.
________+14. _____
______
C4
______ ____
_
__ ____
___
=
__
____
________~~~~~~~~ __________
_____
+10.09
#1.25582 +0.06861 +145802T
_____+13.25582
______
0.09474 0,00518  1.0999/ 41. 0.03 46.97 10.13684
Cs
=

0.05790
+10.07894
40.0057/ 0.99429
0.00572_
_
_
_
____
+0.0057/1
Figure 112. Solution of normal equationsstation adjustment.
Section II. QUADRILATERAL ADJ USTMENT (LEASTSQUARES METHOD) 62. Introduction a. The most common figure . occurring in triangulation is the quadrilateral with both diagonals observed, because it embodies both
strength and simplicity. A thorough understanding of the computation and adjustment of such a quadrilateral is basic to the understanding of a net adjustment. For this reason, a quadrilateral 167
adjustment will be explained and an example computation shown. Because this is to be background material, the quadrilateral in the example will be adjusted by the least squares method. DA Form 1925, Quadrilateral Adjustment.(Least Squares Method), has been set up for this solution. b. Field observations always contain small errors which cannot be distributed in any way except by the laws of probability. Because of these errors, certain geometric properties of a quadrilateral are not satisfied by the field observations. It is the purpose of the leastsquares adjustment to find the most probable values of the field observations that will satisfy the geometric conditions. If the following conditions are satisfied, all the geometric properties of the quadrilateral will also be satisfied. The conditions are as follows: (1) In the four triangles in the quadrilateral, the sum of the three angles in each triangle must total 1800 plus the spherical excess. (2) The length of the common sides of adjacent triangles must be the same no matter which of the two adjacent triangles is used to compute the length. c. By solving certain equations known as conditions equations, small corrections are determined which are applied to the observed angles. These corrected angles will then satisfy the above conditions. d. The two types of equations occurring in a quadrilateral are angle equations and side equations. In a completed quadrilateral (fig. 113), three angle equations and one side equation are needed. The number of angle and side equations in any figure can be determined by the formula in the section on triangulation adjustment.
63. Direction Method a. This illustration will be made using the direction method in which an angle is considered to be made up of two directions. The corrections, therefore, will be found for each of the two directions making up the angle. b. The adjustment is begun by numbering the observed directions on the sketch (fig. 113), starting at the left end of the fixed line looking into the quadrilateral. Any directions not observed are not numbered, and such directions are shown by a dashed line on the sketch. The numbers on the lines are used as the subscripts 168
Red
Lincoln Figure 113.
Hicks Quadrilateralsketch.
on the symbol v used to designate the corrections to the observed directions. The designation of the corrections is then vl, v2, v 3, and so on. For convenience, the symbol v is usually not written, thus the subscript is written as (1) and the symbol is understood. c. Each angle is designated by the two directions forming the angle. Always considering the angle as being measured clockwise, the first direction is negative and the second direction is positive. For example, the angle at Lincoln from Burdell to Red is designated 1+2, while the angle at Hicks from Lincoln to Burdell is 4+5. In this way, not only the angle, but also the correction to the angle is designated. For instance, the correction to the angle in the first example above is  1 +v 2 and when numerical values are found for vl and v 2, they are algebraically added as (v 1 +v 2). d. DA Form 1918 (fig. 114) is used for the computation of the triangles in this example. e. In order to be certain that all the triangles in a net are written on the triangle computation sheets, some sort of order should be established for writing the triangles. The system outlined here will establish a pattern which will continue through all subsequent computations. In the quadrilateral LincolnBurdellRedHicks, start with the fixed line HicksLincoln and proceed clockwise around the figure. The first station encountered is Burdell which will be number 1 in the first section of the triangle computation sheet. At Burdell, find the first clockwise line to a fixed station which will be the line to Hicks. Station Hicks then is number 2 in the first triangle. Still
PROJECT
DATE
12 43
1.2
Aug if
COMPUTATION
aSrnaORGANIZATION2
LOCATION
OBSERVED ANGLE
STATION
23
CORRECTION.,
SPHEICA E~LSRRA
EXCESS
ANGLE
NICKs " LINCOLN
_
11512 &,r.e/ NI ua / 2 . 5 3 LIMccUN 143
49. 000 55./ic1s
108 Of31.
L
43.2
1

2
2

3
1
4d+6
2
24o3
,HICKS Ij.lvcouI
3
+6.4
34.6 0,LI.L59.977851144
304
2.5
5.
29 09.2 43.. 57 48 45.3 ~
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SPERICL PANRFUNCTION: LOGARITH I, ANGLE
S2ZlL 51, 0!0. 28 5Z 0 Q l
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OF TRIANGLES
0"DATE
14.0
00.21 0.2
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CHECKED BY

v DATE
918
7oM1
Figure 114.
Computation of triangles for quadrilateral. 169i
at Burdell, find the next clockwise line which will be the line to Lincoln. Lincoln is number 3. The triangle thus defined is BurdellHicksLincoln. Now move to the next clockwise station which is Red. This is number 1 in the second triangle. The first clockwise line is to Hicks. Hicks is number 2. The next clockwise line is to Lincoln which is again number 3. The triangle RedHicksLincoln is now defined. Still at Red, the first clockwise line is to Hicks; the next clockwise line not previously used is to Burdell. The triangle now completed is RedHicksBurdell. All the triangles with RedHicks as the first clockwise line are now written, so shift to the next clockwise line which is RedLincoln. Lincoln is now number 2 in the fourth triangle. The next clockwise line is RedBurdell. The fourth triangle is RedLincolnBurdell. Notice that station Burdell is number 1 in the first triangle, and station Red is number 1 in the second, third, and fourth triangles. f. In the lefthand margin of the form, the designations of the angles are shown. The observed angles are from the list of directions. (In this example, the angles are from the list shown in the section on triangulation adjustment.) The sum of the observed angles is recorded for each triangle. Only the seconds of the sum need be written on the form. A bar over the seconds indicates that the sum is less than 180 ° . For example, in the first triangle in figure 114 the three angles total 179°59'53'.7 and the seconds are recorded as 53.7. The three angles in a triangle should total 1800 plus the spherical excess (E). (Spherical excess computation is discussed in paragraph 57.) 1800+e minus the sum of the three angles is the error of closure of the triangle. The algebraic sum of the corrections to the three angles (each angle correction made up of directions corrections) must equal the error of closure of the triangle. This statement leads to the condition equation known as the angle equation, which can be stated as: The algebraic sum of the v's in a triangle must equal the triangle closure. As an example, the angle equation for the triangle BurdellHicksLincoln is(1)
(3)(4)
+
(5)(11)
+
(12)=
+6.4
remembering that (1), (3), (4), and so on are the subscripts of v's. g. An angle equation can be written for each of the four triangles in a quadrilateral, but values for the v's which will satisfy three of the 170
equations will also satisfy the fourth, because the fourth equation is a combination of the other three. When choosing the threeangle equations to be solved, triangles with small angles should be avoided if possible. In the example, therefore, the angle equations for the second, third, and fourth triangles are selected. The three angle equations are as follows: 0 =  8.9  (2) + (3)  (4) + (6) (7) + (8) 0 =  6.5  (5) + (6)  (7) + (9)  (10)+ (11) 0 =  4.0  (1) + (2)  (8) + (9)  (10) + (12) The eq'uations are numbered in ascending order of v's for convenience in the solution. The equations are shown in correct order on the example form. h. The condition of side agreement is satisfied by including a side equation in the solution. One way to set up this equation is to select a station as a pole and write the product of the ratios of the lines running to that pole as equal to 1. By the laws of sines, the sines of the angles opposite the sides can replace the sides. By replacing natural sines by logarithms, the expression can be reduced to one which can be solved by addition and subtraction. i. In this example, the side equation is written using the pole at Lincoln in order to include the small angles at Burdell and Red. The ratio of the lines intersecting at Lincoln isLincolnBurdell LincolnRed LincolnRed XLincolnHick LincolnHicks LincolnBurdell
1
Substituting the sines of the angles opposite these sides and using the symbol designation of the angles, this ratio becomessin (8+9) Xsin ( 4 + 6 ) sin (11+12)1 sin (10+12) sin (7+8) sin (4+5) Replacing natural sines by log sines: log sin (8+9)+log sin (4+6) +log sin (11+12)log sin (10+12) log sin (7+8) log sin (4+5) = 0 When written on a side equation form, the log sines of the angles in the numerators are entered in the lefthand column and the log sines of the angles in the denominators are entered in the righthand column. j. For each angle, the tabular difference is entered in the column headed "Tab. Diff." (fig. 115) beside the angle. The tabular difference is the
QUADRILATERAL ADJUSTMENT
PRO243
(Least Squares Method)
LOCATIONORAIAINDT
Ca'lifAorniiq
0
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SKETCHANLEQAIS
10
supdel
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9
12
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0=
SIDE EQUATION ANGLE
SYMBOL
LOG. SINE
TAB.

2
+j
2
4
3
1
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1.03
1 I
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ANGLE
CORRELATE EQUATIONS
__________
1 1
SYMBOL
o6(,+355(7)4.S(ce)+.v40(9)fo.97Io).00eI)+4.o34i
99(4)I. 83(5 a
0= +76¢+1
Div
7 _06
10
,11
1
NORMAL EQUATIONS
1
'3
2
+6
16
2
4
E
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2
2
100
+2
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1
3
4
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DATE /jCHECKED
3
21.1
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23.11
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4

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d =± .6~ C=NUMBER OF CONDITIONS B
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COPUE
.
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PROBABLE ERROR OF OBSERVED DIRECTION 2C
x
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1.
4.0
+
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OF NORMALS 71
__________SOLUTION
2
01
1 3
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4.
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12
fAFORM DAI FEB 5712
Figure 115.
Computation of Quadrilateraladjustment.
171
change in the logarithm for 1 second change in the angle. Express the tabilar difference in units of the sixth decimal place of the logarithm. Notice that the tabular difference for an angle over 90 ° is minus. When this tabular arrangement of the side equation is used, the constant term of the equation is found by subtracting the sum of the log sines in the righthand column from the sum of the log sines in the lefthand column and pointing the difference off in units of the sixth decimal. If the righthand sum is larger than the lefthand sum, the constant term has a minus sign. The constant term is written with unchanged algebraic sign on the same side of the equation as the v terms. The tabular differences are used as the coefficient of the v's as shown in the Symbol column. If a v appears on both sides of the form, the coefficients for that v must be combined in the equation. The tabular differences at the righthand side reverse the algebraic sign for this operation. The side equation in the example, before coefficients of v are combined, is as follows: 0= +7.64+(0.06) (4)+(1.83) (4) +(1.83) (5)+(0.06) (6) +(3.55) (7)+(1.00) (8) +(3.55) (8)+(1.00) (9) +(0.97) (10)+(5.00) (11) +(5.00) (12)+(0.97) (12) When the coefficients of the v's are combined, the equation is as follows: 0= +7.64+ 1.89 (4)1.83 (5) 0.06 (6) +3.55 (7) 4.55 (8)+1.00 (9)+0.97 (10)5.00 (11) +4.03 (12) k. At this point, a check should be made to insure that the sum of the coefficients of corrections to directions (v's) radiating from any one station equals zero. 1. After the condition equations are formed, they are tabulated in correlates. For each equation, there is a numbered column in the arrangement of the correlates. On the horizontal lines at the sides of the correlates are the numbers of the v's. The coefficients of the v's in each equation are written in the correct column on the appropriate numbered line. (Coefficients in equation 1 are written in column 1, and so on.) The column headed "e," contains the quantities obtained by adding algebraically the coefficients on 172
the same horizontal line in the four columns. The columns headed "v", "Adopt v", and "v2", are filled in after the solution of the normal equations. The correlates are used to form the normal equations. m. The normal equations are arranged using a column for each condition equation (corresponding to the numbered columns for the correlates) plus a column headed "7" and a column headed "2]". The q column contains the constant terms of the condition equations, while the values in the 2,, column are used as a check corresponding to the values in the 2, column of the correlates. The normal equations themselves are numbered on the horizontal lines. (1) The normal equations are formed by finding the algebraic sums of the products of the values in one column of the correlates multiplied by the values in the various other columns in the correlates. The products are found only for values on the same horizontal line. In other words, a value on line 1 is multiplied only by other values on line 1 and not by values on any other line. (2) The first normal is found as the summation of the products of the values in column 1 of the correlates, multiplied by(a) The values in column 1 of the correlates. (b) The values in column 2 of the correlates. (c) The values in column 3 of the correlates. (d) The values in column 4 of the correlates. (e) Plus qj, the constant term of the first condition equation. (J) The values in column 2,of the correlates (after the summation of this multiplication, add 11). (3) The values obtained in (2) above, are tabulated on the form for the normals as follows: (a) in column 1, (b) in column 2, (c) in column 3, (d) in column 4, (e) in column q, and (f) in column 2,,; all on the first line. The value in the 2 column is [,q+column 2 times 2]j. As a check, the sum of the values in columns 1, 2, 3, 4, and should equal the value in the 2,column. (4) The second normal is found as the summation of the products of the values in column 2 of the correlates multiplied by: (a) The values in column 1 of the correlates.
77
(b) The values in column 2 of the correlates. (c) The values in column 3 of the correlates. (d) The values in column 4 of the correlates. (e) Plus
2,
the
constant
term
of the
second condition equation. (f) The values in column Z of the correlates (after the summation of this multiplication, add
2).
(5) The values obtained are tabulated in the .same order as explained for normal equation 1. (6) Notice that item (a) in (4) above, for normal 2 is exactly the same as item (b) for normal 1. If both these items were written in the normal equation, there would be a repetition of the value in line 1 column 2, and line 2 column 1. A repetition of this type occurs in each normal for all values to the left of the diagonal term. The diagonal term is the value obtained when a column in the correlates is multiplied by itself in the formation of the normals. Each normal, therefore, can be formed by finding the diagonal term and the terms to the right. The terms to the left of the diagonal term appear in the column above the diagonal term, in order, reading down from the top. The complete normal reads down the column to the diagonal and then across the line to the right. When checking the equation as explained for normal 1, remember to add down and across. Each term that falls in a numbered column in the tabulation of the normal equation is the coefficient of a constant (or C) corresponding to the column. The C's also correspond to the same numbered column in the correlates. (That is, C, is the constant for column 1 of the correlates and the normals.) n. The normal equations in the example could be written as follows: S
® 0i
o
6C012C2+2C38.61C44.0=0 2C1+6C2+2C 10.05C48.9=0 ±2C+2C2+6C0s6.75C46.5=0 +8.61C0 10.05C26.75C3± 83.4114C4+7.64=0
These are 4 simultaneous equations. (Notice that the terms to the left of the diagonal term are included.)
64. Solution of Normal Equations by Successive Substitution a. The solution of the example normal equations (par. 63) by successive substitution is as follows: Equation Q is solved for C1 in terms of C2, C3, C4, and a constant. (1) C,= +0.3333C20.3333C3 1.4350C4+ 0.6667 This value of C, is substituted in equation @: 2(0.3333C20.3333C31.4350C4 +0.6667)+6C2+2C310.05C48.9=0 which is: 
0.6667C2+0.6667C+2.8700C4
1.3334+6C2+2C 10.05C48.9=0 Collecting terms, equation @ becomes: +5.3333C2+2.6667C3 7.1800C4  10.2333=0 Solve this equation for C2 in terms of C03 C4, and a constant:  2.6667C3+ 7.1800C4+ 10.2333 C2=
(2)
C2= 0.5000C3+
5.3333
1.3463C4+1.9187
Substituting the value of C from ( and the value of C2 from @ in equation Q; reduced equation @ is found to be: 4.0000C03 6.0298C4 0.0500 =0 Solving this equation for Ca in terms of C4 and a constant: C3=+1.5075C4+0.0125 (3) Substituting the reduced values for C1, C2, and C3, from above formulas in ) and solving the normal equation resulting equation for C4 as a constant, the forward solution of the normals is complete. The reduced equation for C4 is: +52.2997C40.4714=0 Solving for C : (4) +0.4714 0.4714+0.0090 S+52.2997 The numnerical value of Ca is obtained by substituting the value of C4 into equation (3). The numerical value of C2 is obtained by substituting C03 and C4 into equation (2), and C1 is found from equation (1). This process is known as the back solution. 173
b. The solution of the normals can be checked by substituting the numerical values of the C's into the original normal equations. The v's are now found by substituting the C's into the correlate equations. For example, in this problem, v= (1)XC1 or
v=(1)(+ 1.2843)= 1.2843. The correlate equation for v2 is
v2
(+1)(C)+ (
21)(2),
or
v2= (+ 1) (1.2843)+ (1) (+1.9178) + 1.2843 1.9178
0.6335.
The value for each v is found in the same manner. Carry these values of the v's to at least 1 more decimal than is required in the final result. Substitute the values of the v's into the original condition equation as a check on the whole solution. After this check has been made, the v's can be rounded off to the desired number of decimals. The process of rounding off may disrupt the closure of the angle equations by 1 or 2 in the last decimal place. If this occurs, 1 or 2 of the v's should be arbitrarily raised or lowered 1 in the last decimal to insure the closure of the angle equations. 65.
Solution of Normal
Equations by the
Doolittle Method a. Although the discussion in paragraph 64 covers what actually is taking place in the solution of normal equations, the solution should be made by the Doolittle method. The Doolittle method, a form of successive substitution, is preferred as being the easiest for longer solutions due to the deletion of the diagonal terms in the equations. The form which the solution takes is shown at the bottom of the example form in figure 115. b. For this example, where there are four normals to be solved, the form has four horizontal spaces, columns corresponding to the normal equation arrangement, plus an additional column in which the numerical value of the C's can be written. The top horizontal space contains two lines, and the other three spaces have three lines each. On the top line of each space the normal equations are entered directly as formed from the correlates. On the bottom line of each space are the divided equations corresponding to (, 0, and ® as illustrated in paraequations graph 64a. The middle lines in the second, third, and fourth spaces are for the reduced equations found by the successive substitution of the C's.
T(i,
174
c. The Doolittle method simplifies the reduction of the normal equations greatly, as successive substitution rapidly becomes too laborious to be practical. In the Doolittle method, use is made of the coefficients from the reduced equations as well as from the divided equations. d. In the following explanation of the Doolittle method, the solution of the normals as shown in the example (fig. 115) will be used as reference, and equations (0, Q, 0, and ® are used here with the terms to the left of the diagonal terms removed. The first normal is written on the form on the first line. This equation is then divided by minus the diagonal term which is  (+)6. The result is a divided equation giving C 1 in terms of C 2, C3, C 4, and a constant. Normal equation 0 is written on the first line of the second space. This equation is reduced by the product of the coefficient of C2 in equation ( i (which is 2), times each of the divided coefficients of equation (i), algebraically added to each of the coefficients of equation ). Products are added only to coefficients in the same column as the divided coefficient making the product. For example, the diagonal term in equation ( (+6) is reduced by the product of (2) (+0.3333) which gives a reduced diagonal of The second term of normal 2 (+2) +5.3333. is reduced by the product (2) (0.3333) The which gives a reduced value of +2.6667. third term (10.05) is reduced by (2) (1.4350) to give 7.1800, and so on for n and 2,. As a check on the solution, the algebraic sum of the coefficients and n in the reduced equation should equal the value in the 2,, column with the possible exception of 1 or 2 units in the last decimal. Always change the value in the 2,, column to agree with the addition of the coefficients if the difference is in only 1 or 2 units. To check the division, the algebraic sum of the coefficients and q in the divided equations, should equal the value in the 2, column. Remember to include a 1 as the coefficient of the C found by dividing the equation. The reduced normal 2 is now divided by minus the reduced diagonal, [ (+5.3333)], which gives C2 in terms of C 3, C 4, and a constant. Normal equation ( is written on the first line of the third space. This equation is reduced by the product of the coefficient of C 3 in normal 1 (which is +2), times each of the divided coefficients of normal 1, plus the product of the coefficient of C 3 in reduced normal 2 (which is +2.6667), times each of the divided coefficients of normal 2.
Both products are added algebraically to the coefficients of equation ( in the same column as the divided coefficients making the products. Thus the diagonal term of normal 3 (+6) is reduced by the product of (+2) (0.3333) plus the product of (+2.6667) (0.5000) which gives a reduced diagonal of +4.0000. The second term in normal 3 ( 6.75) is reduced by (+2) ( 1.4350) plus (+2.6667) (+ 1.3463) which gives a reduced The third term (6.5) is term of 6.0298. reduced by (+2) (+0.6667) plus (+2.6667) (+1.9187) which gives 0.0500 for the reduced third term. The same procedure is used to obtain the reduced 2, term. The reduced equation is checked by adding across and then dividing by minus the reduced diagonal term which is [ (+4.0000)]. Repeating the process for nor
mal 4 produces a numerical value for C 4 of +0.0090. The solution of the C's and v's is made as previously explained. e. After the v's have been computed and checked through the equations, the adopted v's are applied to the angles on the computation of triangles sheet in the column headed "Corr'n". Applying the correction to the observed angle produces the spherical angle from which the spherical excess is subtracted to obtain the plane angle. The triangles are then computed as explained in paragraphs 57 through 61, and the geographic positions computed as explained in paragraphs 67 or 68. f. As part of the adjustment of a quadrilateral, the probable error of an observed direction is computed as shown on the form in figure 115.
Section III. GEOGRAPHIC POSITION 66. Introduction a. When the geographic position of a station is unknown, but the azimuth and distance to the station from a station of known position are available, the unknown position can be determined. There are many acceptable formulas, varying in accuracy, for this computation. In this manual, the USC&GS formulas were selected for all computations. These formulas are sufficient for all triangulation lengths in normal latitudes. b. Normally, two known stations are used to give a check on the position of the unknown station. The procedure generally followed is to solve a triangle, with the unknown station as 1 and the known stations 2 and 3, after correcting the observed spherical angles by some type of adjustment. Then using the azimuths between the known stations, and the spherical angles of the triangle, determine the azimuths to the unknown station from the known stations. Finally, the formulas are solved for the position of the unknown station and azimuths from the unknown station to the known stations. Due to the convergence of the meridians on the earth's surface, the back (or reverse) azimuth of a geodetic line is not exactly 180 ° different from the forward azimuth. The difference between these azimuths is known as convergence and is dependent on the difference in longitude of the ends of the line. c. It sometimes becomes necessary to compute the geographic azimuths and length of a line joining two stations which are fixed in position, but have not been directly connected by the obser
vations. In order to compute this line, an inverse or back computation must be made. The mathematical basis of the inverse position computation is exactly the same as that of the position computation. The computation is based upon the solution of the right spherical traingle formed by the line connecting the two known stations, and the line representing A4 and AX. 67. Direct Position Computation, Logarithmic Solution a. The formulas used in the solution of this problem are as follows:  0A=s cos a. B+s 2 sin 2 a. C 2 +(54) 2Dhs2 sin a .E 2 2 s kE+(3/2)s cos 2 a. kE + _2 cos 2 a sec22 A' 2k sin 2 1"; h=s cos a . B; S¢=s cos a . B+ 2 sin2 a. C hs 2 sin2 a . E; k=s2 sin2 a C;
J
sin AX= sin
tan
sec 4' sin a;
log AX=log s+Co ACog 8,+log sin a +log A'+log sec 4'; ) ('+ sin (AX)  tan (Aa)=
cos (€'€)
Aa=AX sin 1(4'+4) sec 1(A4) +(AX) 3F. Where: C (log AX) and C (log s) are the arcsine corrections, the arguments log AX and log s being indicated in parentheses. A logarithmic solution of a firstorder position computation is shown in figure 116. The dis175
PROJECT
33
U7TAH
LOCATION
147
TVSHAN
20
Oa 2 MT
NEBO
I
«a


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WHEELER
to I


WHEELERPEAK
PEAK

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ICHECKED BY
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REPLACES DA FORM 1922, 1 FEB 57, WHICH IS OBSOLETE.
Figure 116.
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522478407
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9+ 9/70. 7828
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_
sine
Sec 2//3
(8)
0 1
19

ineA~nr""
110 + 3
9 a'
sine a
S11111 3.9623636 corr.
_
0
8.509144 0
A'
TUSHAR?
3
it
APa' 37649909 7 !8.223______ Aresinl corr.
'Total 1+ COMPUTED BY
0.229
34
sec&0'0,194/05
1"
__
a
4/085 88~ /6 30.92
MT AEB0
3__
9.9676586 (1)=h
:~o~*592('& Sec20
10. 6876 6.1004
~
Aarc
93304
E
DA
(colog)F 13.900
37.
5.376/505
sinla
t2
180
Logarithms
S_
TvSHAR_
_
114,18 47.0/8 m+)3
_
DATEMYS
,/ S
00.00
45 56.235
39
a
#'
0/.4___
r
2 32 PEAK
(1)
8.5108661 (1)=h3 4575489
40
1/
(1)
B
.S3
2
Logarithmls
S5376/505 COS a
43 0
_49 29.300fr 5S9 109A016 WEELER
°0r
00
i
39 48 38,3/6
0
37
246 32 401
MIT. P4E80
to 2
/ 180


_
40536.08
48 04 05.50 68 109 41.58
2dZ
A
ORGANIZATION
797
+ 242
do 3
(°x) 11.506
.
"'7874
(8)
(8)
9.380
363/4423
4279.
"985,
+0,'?40 428022'5 6L(.844:8511
Theavy
POSITION COMPUTATION,
NOTE: Far log s Io a.9, omit terms belowheavy black line NOT inl boll tIpe or underlined.
.
FIRS r
(TM 5237)
ORDER TRIANGULATION (Logarithmic)
Position computation, firstorder triangulation (logarithmic) ( DAForm 1.922).
tances used are the sides of a triangle, and the azimuths are the angles the vertical section makes with the meridian, measured clockwise from the south. The solution is made using the Clarke 1866 Spheroid, and all factors are taken from USC&GS SP 8. For the International Spheroid, SP 200 would be used to obtain the necessary factors. Wheeler Peak, the unknown station, is number 1. Mt. Nebo and Tushar, the known stations, are numbered 2 and 3, respectively. The azimuths (a) from 2 to 3 and from 3 to 2 are entered on the form, and the second and third angles from the triangle computation are applied to these azimuths, giving the azimuths from the known stations to the unknown station. The log distances (s) are taken from the triangle computation, log sin a 21 (31), and log cos a 21 (31) are taken from the tables to seven decimal places, and the factors B, C, D, E, and F are taken from USC&GS Sp. Pub. 8 using the latitude (4) of the known station as the argument. The sum of log s, log cos a, and B is the log of (1), the sign being determined by the sign of cos a. In the Northern Hemisphere, the cosine is plus in the first (0° 90°) and fourth (270 ° 3600) quadrants, and minus in the second (9001800) and third (1800270 ° ) quadrants; in the Southern Hemisphere the reverse is true. The sum of log s2, log sin 2 a, and C is the log of (2), and the sum of log (h), log s2 sin 2 a, and E is the log of (4). Find the antilogs of (1) (2) and (4) and add algebraically to get (S¢). The sum of log (84)2 and D is the log of (3).
Log (5) is
the sum of log 2,log s2, log K, and log E. Log (6) is the sum of log (5), log 3, and log cos 2 a. Log (7) is the sum of log (6), colog E, log A2 arc2 1 "'and log sec2 4. 3 Note. If the distance (s) is small, (4), (5), (6), and (7) will be small and the characteristics must be carefully watched to avoid error.
Find the antilogs of (3), (5), (6), and (7) and add these algebraically to the sum of (1), (2), and (4) to get A0 in seconds. Apply A0 to the latitude (4) of the known station to obtain the latitude (4') of the unknown station. 4' is used to interpolate A', from Sp. Pub. 8, and log sec 4' from the tables. Add log s, log sin a, A', and log sec 4'. This sum is used as an approximate AA to find the arcsin correction, and takes its sign from sin a. In the Western Hemisphere, the sine of a is plus in the first (0090 °) and second (90°180 ° ) quadrants, and minus in the third 757381 0

65

12
(180°270 ° ) and fourth (2700360 ° ) quadrants; in the Eastern Hemisphere, the reverse is true. The arcsin correction is equal to the correction for AX minus the correction for s. Both of these corrections can be fouhd in Sp. Pub. 8, page 17, or computed from the formulas in Sp. Pub. 8, page 18. After the arcsin correction is computed, it is added numerically to the previous sum to obtain the log of AX. Applying AA algebraically to the longitude (X) of the known station gives the longitude (X') of the unknown station. The sum of log (AX) 3 and F is the log of (8), the sign being the same as the sign of AX. The sum of log A4 AX, log sin (4+40'), and log sec 0 is the log of  Aa (approx.), the sign being the same as the sign of AX in the Northern Hemisphere, west of Greenwich, and in the Southern Hemisphere, east of Greenwich, and opposite the sign of AA in the Northern Hemisphere, east of Greenwich, and the Southern Hemisphere, west of Greenwich. Add algebraically the antilogs of  a (approx) and (8) for Aa in seconds. Apply Aa to the azimuth (a) from 2 to 1 (3 to 1) to get the azimuth (a') from 1 to 2 (1 to 3). These steps are used on both sides of the form. The positions computed for station 1 should check within 1 or 2thousandths of a second for both latitude and longitude, and the azimuth should check within 1hundredth when the first angle of the triangle is applied to either side of the computation. b. Figure 117 is a logarithmic solution of a thirdorder position computation. The unknown station is Parson, numbered 1, and the known stations are Outer and Hard, numbered 2 and 3, respectively. The known azimuths are entered on the form, and the second and third angles of the triangle are applied to give the azimuths from the known stations to the unknown station. The distances (s) are taken from the triangle computation; log sin a, and log cos a to six decimal places, are taken from the tables; and factors B, C, and D are taken from USC&GS Sp. Pub. 8, using the latitude (.4) as the argument, and all are entered on the form. Log s, log cos a, and B are added The to find h, which is the first term in A4. sign of h is determined by the sign of cos a. The sum of log s2; log sin2 a, and C is the second term, and the sum of log h2 (enter in (60)2 space) and D is the log of the third term of  A4. Add the three terms algebraically to find  A4. Apply  A0 to the latitude (4) of the known station to 177
get the latitude (0') of the unknown station. Use 0' as the argument for A' in Sp. Pub. 8, and find the log sec gyp'from the tables to six decimal places. Adding log s, log sin a, A', and log sec 0m' gives the log of A. The sign of A is determined by the sign of sin a. Apply AX to the longitude (X) of the known station to get the longitude (X') of 5 688
PROJECT
PROJECT
C

6
2
NEW
HARD
."
OUTER
__'PARSON
2
'PAPARSON
t° .OUTER_ Fi rst
.35 18.742
40
2'
°0+
/
40
37
NEW
LOCATION
8
12_oUrE.R
2«
Anigle of Tlriansgle
X
OUTER
2
5983
A
.
8.595 it PARSON
iX'
9. 991 320
" ()19854/ (2) +_0.0012
'Logarithnms
B
8.510 807
(4
K
Logarithms S3.57652b
S__
a1°'«
(3) ,F 0.0004
8.5931
(5)
(2)=K _
70 3 7__ 7_08 47__
41573
coca«I
+
~+(6)
5
3.1
2s
3
(AX)
for AX+(
(4)
F
'Total COMPUTED8BY
iR.4.S. AMS
D
(8 DATE
. 91
MAY .56
4"40
37
18595'
Logaritbms
S 3.281346 cos 8.489222
0.1 /9 745
81n°a « 999959
_____
2)K()
. .
50/ 928
(,)a
(m+') "9.81347 2
1
3~
D
2.3873
3
2. 9500
E
rr
206 7 3/. 7635
W.c.A. AM S
(5)
3
0.47.7
POSITION COMPUTATION,
21.281
05.727 /96
4'4"
"01)40 37 /. s
3.281346
385 si
,999 793
A'
8.509 103
sec 4
0. 119 745
Arc sins
()+()
_
+ /.4.9/ 94
(cnlg)E =.92 Aaarc'1"
3
z
corr.
L
11.909 987'
i
5.11(o'+m) 198136(2/
sc4
(K)"
J)
Ixl+
.Aa
IF.
j8
toa
MAY 'NOTE: 56
27008
Munm
cost a
(6i) +
for
DATE
7.3 38 / 7Z 3 37
Logarithmus
E (5)
0
fo
(4)

s'

0.54.2 7
D.14'
sin)
(8)
7.200/11
L~ogs
_+1911 9699
_)
LO(3) +
6.56269
PARSON "
0:281 372
Sa
X ~
(2) +, o,0079
(')=.h
C111 .337 83
A
() (
296 554 8.509 /03
corr.
CHECKED BY
20.5/4 3 HARD
01919
(4)
REPLACES DA FORM 1922, 1 FEB 57.
,OCT 8412
Figure 117.
178

m
8.5/0804
Arcsin
tC4
for s 
4"
40 37 
0'
B
.576 526
D
I
()6.5445
[E
0.477~ sec
3
3/.763 05. 727
40 36 18.7
A'
,L4" //9.8525 (rolog)E
1)40:arcal
D2.3872
(7)
37
ina«9
(5)

6
73
S
]_E_
4o53
7153 05 1.33 7 31
C
(a
2 0 78
73 .36 33.9614 +
Los '4) .s x"
cos
(')=h
the unknown station. Add log AX, and log sin ~(0)+0") for the log of Da. Apply Da to the azimuth (a) from the known station to the unknown station to get the azimuth (a') from the unknown station to the known station. The positions computed for the unknown station should not differ by more than 1thousandth of a second
52 n92 I. 2806
Fur logs a to3., ornit teris.belowhesvy blackline NoT in
heavyhsoldityper udrlined
Third
(TM 5237)
ORDER TRIANGULATION
Position computation, thirdorder triangulation (logarithmic) (DA Form 19232).
(Logarithmic)
in either latitude or longitude. This method meets thirdorder requirements when the triangle sides do not exceed 25,000 meters (approx). c. Examples shown are for the Northern Hemisphere, west of Greenwich, and the following variations should be noted: (1) For computations in the Southern Hemisphere, the sign of the cosine is reversed. (2) For computations in the Eastern Hemisphere, the sign of the sine is reversed. (3) Aa is applied according to the following rules: (a) In the Northern Hemisphere, if a is less than 1800, Aa is minus; if a is greater than 180°, Aa is plus. fib) In the Southern Hemisphere, if a is less than 1800, Aa is plus; if a is greater than 1800, Aa is minus.
y 2 =yo+[y+(yfa10)]Va+K (Th2 A"[Q 1/
A solution, using natural functions, of a firstorder position computation is shown in figure 118. The solution is made using the Clarke 1866 Spheroid,
CIUPUTATIN
2
a_
ORAIAINDATE
NVevada To
2d L
3
326ZLIA .±8 1L _M , 6
RRE
&
a2To
I

~~~~First Angle of Triangle
,
an zssin
1 9 3 0 8A 0
a
a
f 
y.=sOa
2
sx10O y cor.=+fa
0~72.~i 4677
z
o . _
)L1. 40,
y'2
2
K (Va/1,000) + BY COMPUTED
DA,
4135.8 428
+0
6sin a
+
2302.89O09
n=(z'/10.000)2
/895.3
0.035
+ DAECHECKED 7 a
BY
2507. 76/
~
257x1 w . C.R
2757.812
40059. 6 74, A86 4. OSp. 329,030 7
y2 (Va/1,000)
2
AENOTE D A 1 1y R
1 5 / 950M8 76 0.040263424
46
1 + cosA
a'
ev
6
25
2.
611,8.3825
0.60634257 0,5S9 703393
(approx.)
0.16
+
)
119.
an'
+
#a
Arcsin=V (Va) cor + 15
.9033~ /f ?73125
yi
r
./
Hx'=(approx.X)
sin
cos"*
°a" A3V+F() (approx.)
2ec H
2.7402 2
2
b=(y10,000)
4. 3MMCs 7 950 2 154 954 /76
y=80"11
r
1
+ 1.5916.o2&L753.823 M yv6,4 4,13243269$a
14.Rg82d~ +
32g.033
i
COl sa ax
w 00.00
00
?3
Q R
jf
yecor.=+fa
AV

L.534 Z8 421
4YC~
o3'"3 o'U
27.
.
238.0
sin* f
y
*
Hx'=(approx.A")
1
3'
a8
Arcain Y V aY cor = 15
/055/000) ffS32 p66.
1
o
P R
0.s04232
H
!4.M5qq
TpjpC~g 
a'
~:A
.7Q 599
+ f /
20/a~y66
JRE
180
/7 /s
/ 27.46 1 )'/ / 9b=(y/10,000) 2/.)
9
'
are taken from
00.00
00
2I1HE
To 2
a 3
a533a~ B_ UaER
180 1jAp~
a'
factors
4S.. Lie. 3d4
3
1/AYOR)
all
DER TRIANGULATION (Far ceatbg mdiaclbscaIpAtad)
Fi*t
A
OGNAON
OjpCjjE
therefore
USC&GS Sp. Pub. 241. For the International Spheroid, USC&GS G58 would be used to obtain the necessary factors. The station of unknown position is Hayford, numbered 1, and the known stations are Pioche and Burger, numbered 2 and 3, respectively. The azimuths between the known stations are entered on the form, and the second and third angles in the triangle computation are applied to these azimuths to get the azimuths from the known stations to the unknown station. The distances (s) are taken from the triangle computation (if the triangle computation was logarithmic, enter both the log distance and the
a. The formulas used in the solution of the problem are as follows: PRJCPOSITIN
b 107)]H+[Ix' ( V(a)) "107]
sin 4±sin 0 1fcos A0O
68. Direct Position Computation, Natural Function Solution
LOCATION
6
Foa under 8,000 meters omit terms under ol yeor underlined.
/*91930
+
+
.3681,484 7
369/. i65
theheavy
blerk
linenot in
FE571923
Figure 118.
Position computation, firstorder, naturalfunctions (DA Form 1923). 179
distance on the form). Extract sin a and cos a from the tables (eightplace natural functions must be used to meet the accuracy requirements). Next, multiply the length by the proper functions to determine x and y. The x value takes the algebraic sign of sine a, while the y value takes the algebraic sign opposite that of cosine a. The algebraic signs of sine and cosine are explained in the log computation. Next, compute the b value by moving the decimal point in y four places to the left and squaring the result. Multiply b by /f to find the x correction factor. f is found at the bottom of the page in USC&GS Sp. Pub. 241, using the latitude of the known station as the argument. The minus sign before the correction factor means x is reduced numerically. The correction factor is in units of the seventh place of decimals; therefore, in the example on the lefthand side, x would be multiplied by 1.00.00008843 or 0.99991157 to get x'. (For other methods of finding x', see page 81 of USC&GS Sp. Pub. 241.) Now use the value of x' to compute a in the same manner as the computation for b, taking a to 4 decimal places. Multiply f by a for the y correction factor. The plus sign indicates y must be increased numerically. The correction factor is in units of the seventh place of decimals, so in the example, y is multiplied by 1.0+0.00008660 or 1.00008660 to get y'. yo is taken from Sp. Pub. 241 under meridional arcs (meters) using the latitude of the known station as the argument. y' is applied to yo to get the yl value. Now using yl as the argument, the value for V and K are found. Note the correction for the second differences of V. The correction is in units of the fifth decimal place. (No space is provided on the form for K.) Multiply V by a and K by (Va/1,000)2 for the two corrections to be applied to yl to get y2.
The value of y2 should
check on both sides of the form. The Va correction is always minus and the K(Va/1,000)2 correction is always plus. Use y2 to interpolate for the latitude (4') of the unknown station. After 0' is determined, A4 is taken out to tenths of seconds and entered on the form. Using 0' as the argument, the value of H is interpolated in the table. (Note the correction for second difference in H.) Multiply H by x' for an approximate AX". Next, compute the arcsine correction factor which is X$5 of the product of V and Va. The plus sign indicates the approximate AX" is increased numerically. Like the 180
other corrective factor, the arcsine factor is in units of the seventh place of decimals. Apply the arcsine correction to get AX", which is applied to the longitude (A) of the known station to get the longitude (X') of the unknown station. The longitude should check on both sides of the form. From a table of natural functions, interpolate the cosine of A4. The sines of 4 and 4' can be obtained from Sp. Pub. 241. Add 1 to the cosine of A4 before entering on the form. Sin 0 and sin 4' are added and the sum divided by the 1+cos A4 value. Multiply this value by AX" to get the approximate value of Aa". The small correction F(AX") 3 must be computed and added numerically to Da". The argument for F is 4,,. The F factor as tabulated contains the factor 1012. To effect this factor, move the decimal point in AX" four places to the left and then cube the result. It is sufficient to take AX" to the nearest second for the computation of the correction. Notice that this correction is not the factor type such as the x and y corrections. In this case, the approximate  a" is numerically increased by the quantity F(AX") 3 . Apply the final Da"
to the appropriate azimuth (a) to
obtain the back azimuth (a'). On the computation form, the sum of the azimuth from 1 to 2 plus the first angle in the triangle should equal the azimuth from 1 to 3. In this computation, emphasis is put on the fact that the plus or minus before a corrective term indicates, respectively, a numerical increase or decrease of the term to be corrected, irrespective of the algebraic sign of the term. b. Figure 119 is a solution of a thirdorder position computation. For lines under 8,000 meters in length, none of the terms involving the use of f or b has a significant effect. As both lines here are well under this 8,000meter limit, no corrective terms are needed. Seven decimal places are sufficient for sin a and cos a. Both x' and y' may be taken the same as x and y, respectively. V need be taken out only to the same number of significant figures as a or perhaps one more. In the example shown, V may be interpolated by inspection to the number of figures needed. The arcsine correction is not needed, and AX" is the product of x' and H. Fiveplace sines are sufficient for the computation of Aa". c. Examples shown are for the Northern Hemisphere, west of Greenwich, and the following variations should be noted:
70. Inverse Position Computation, Logarithmic
(1) For computations in the Southern Hemisphere, the sign of the cosine is reversed. (2) For computations in the Eastern Hemisphere, the sign of the sine is reversed. (3) Aa is applied according to the following rules : (a) In the Northern Hemisphere, if a is less than 1800, Aa is minus; if a is greater than 1800, Aa is plus. (b) In the Southern Hemisphere, if a is less than 1800, Aa is plus; if a is greater than 1800, Aa is minus.
a. Figure 120 is a logarithmic solution of the
inverse position computation on DA Form 1924 (Inverse Position Computation). The positions of the stations, arbitrarily numbered 1 and 2, are entered at the top of the form. Next, A0, Ak'iA05 (seconds), A, 2 and AX (seconds) are computed, completing the top block of the form. Log A'0 and log A are extracted from the tables, to seven decimal places. The arcsin corrections are interpolated from table XXVI in TM 5236, or from USC&GS Sp. Pub. 8, page 15, and subtracted from log A'0 and log A giving
69. Direct Position Computation, Natural Function Solution (Any Spheroid) The formulas used in the solution of the problem are those listed in paragraph 67a. Tables for the A, B, C, D, E, and F factors for all major spheroids were prepared by the Army Map Service and are published under the TM 5241series (i.e., TM 524118, Latitude functions: Clarke 1866 spheroid). PROJECT
56
LOCATION
GergaORGAN
a
2
ATOD
m2d L
56886(7M
a
2L
3d L
4
12.
.2
25
.MPF/.
Q.29s 7463193sin3aH a=(z'/1O,000)
2
sinea
X,
ca C(
Hx' =(approx. A)
, 4
Y2
V
777
8,350 0 O,34
4.7,9
3
2
4)k2~rr
28 /&

1473
sa
4

.3.8
00
00.00
186
II30.0
8/
/18
0 90 p
5734
~
DAT Si DT 3J
784
Axe
4.pq.9
In 748a. 73
sin +"
0. ,63Va 002
o
3. 478, 344 
ha"
+.48

3
°a
esoi
heavy underusned OEbold type oe or ne , atro/
orai sin4.
(approx.)
+F (°A") +.0
x (Vail,000)2+ ,~
)
H Hx' =(approx.Ah) 15)Jt(e1,002 Acri=V(A A~BnV(a 15('1.0)
_+fa

/816.36
24526
(JRM
em ne oeado
1923' Figure 119.
/ .'5
22
S"
V
£ CNCE YA CHECKE BY
2.3
20
' 8/
s
9941" 7
Y= Cs
Y2
3
DATE2
.31
A12,351
0.x07&Z3L Za9
Va
+ F(°)"')
 a"
COMPUTED BY
osin
(approxt.)
°a"
K (Va/1,000)2+
DA , B7
4g~3 MARY
1.
___
MAY
2
+6.
'
1+co°+Y1+
//2
b=(Y/10,000)2
Y ""
sin
sin
To 3
+
,o.99cor.
429.71I
plO
x=sglsia
mr15 '
.34&, 77A, 
3/
Arctin=+VA
'
' .3,, Y
LJT.i.SE
'°*
x46/ corfb
y' cor. =e+fa
Y.
*
)
O0743g ,13049
32.71
Y7=S cosn a
A7o
180
1
6*
X,/
b(l /10.000)2
Cosa.393
To 2
a
a'
.49
v
b(1000 =lo5492(g8a=
sinea
fqy MA/B14
S$i
/12
'I
1
3
D$ ATE
00.00
54
9
/
FirstAngle of Triangle
+
nc
2?is~.i
10 00
ATbo
B.m are added to
and colog
2X'
5237)
A
.5 . 2
and log cos 0km are computed.
A
ORDER TRIANGULATION (For cakat macdas caylulsu)
71 ird
I
/'IAQY
&o
~qg rr rAroTo _27
Log A'k1 , log cos
IZATJON
To 3
Log cos
¢,m is used as the argument to find log A' and log B in USC&GS Sp. Pub. 8. The cologs are computed and entered as colog Bmn and colog Am,.
POSITION COMPUTATION,
°a.
a'
log Ao1.
+ 3.75 bh
ev ev
lc
lc
lentl a o e
Position computation, thirdorder, natural functions (DA Form 1923). 181
jINVERSE
23'105
PROJECT LOCATION
POSITION COMPUTATION
Oreg9onI
ORGAN IZATI ON
it(«+2\ S S
1
AX,
A«,co A,,2/
'!
Sf
'~
cos
s, Cos
(a±

A« 2 V
A
X
O5
A«=AX
sin ¢,, sec 2 +F(AX)'
23
in which log AX=Iog (X'X) correction for arc to sini; log Ao 1=log (0'0)correction for arc to sin; and log s~log s,+ correction for arc to sin. NAME OF STATIONS
.59 OQ0.715~ SPENCER 30 38.293 PETER~SON
4S
*
1.
'4
2.
+31
A(=0
*~ 244 A0,
3.2
logA#
cor. arcsin log A4
.41.504
5'7f9
'7
9
AX
2.r58
log AX
6
(
log sin 4,m
a
log sec 
(oppst in sign t tA4)
3 log AX
L47
log b
584
~

+
A«
1409

«±
2
4,p 4 4004
p
g7
1253
12
40at
9202
«+~ «+

193 630 .4,774 0490
cor. arcsin+
2
.~
'1Ithose 't
I
10those .69 ftLb
5
i. 9 1
a~lto) ia~45
i
logsl
Aa
a_}.
A,.
lga
logcos
311.8 1 , ggS
 A«(secs.)
colog
log sin
0.00
b
g. 9 5 it111
atif
317.98
a
2.
log cos4,.,
7g916.237 log tan
log F logF
(6854p
log AX,
log
2. 502f36,4n
log a
. 68 2951
A
cor. arc sin
4,747 (6720
«+2
;5.________ 7______
(secs.)
o
1.5
I.48q 4114Z
B.
05' 4.1.248 58 05.537
_____
3. 2.78 9qg 9 9? 7 ggc
1
log s1 cos
78

log cosZ colog
jf
+j
I23"
AX~R________
14
(secs.)
X
X' 122
47 74
log s
050
_59436.15
gcg
m.
NOTE.For log sup to 4.0 and for A0, or AX (or both) up to 3', omit all terms below the heavy line except printed (in whole or in part) in heavytyeo typ underscored, if using logarithms to 7 decimal
~places.
180
DA
l
FERB57
leGs
DATE
CHECKEDBY
ov4W.cZIJ.
1924 Figure
182
q(6
50
a''(2to 1) COMPUTED BY
120.
Inverse position computation, logarithmic (DA Form 1924).
DATE
17AiOV 64
obtain the log [81 cos (a+±Owhich is opposite in sign to A4.
Log AX1, log cos
m., and colog
Am are added to obtain log [si sin (a+)]
(b) In the Southern Hemisphere, if a is less than 1800, Aa is plus; if a is greater than 1800, Da is minus.
which
71. Inverse Position Computation, Natural Function Solution takes the sign of AX. Subtract log [s,cos (a+t)a a. Figure 122 is an example of the inverse • position computations performed with natural to obtain log tan a+) from log s sin(a+ 2a functions. The positions of the stations, arbitrarily numbered 1 and 2, are entered on the form, (a+ ) 1isextracted from the tables using the and a4 and AX are computed. 4 is used to find yo (meridional are in meters), and sin 4 in ), and this value is used to interlog tan (a+ USC&GS Sp. Pub. 241, and 4' is used to find y,, sin 4', and H. A is changed to seconds, and and log cos a+ ). polate for log sin a+ 2 the arcsin correction factor is obtained from the 0.39174 factor equals formula; correction Log s, is obtained by subtracting log sin (a+\2 This factor is in units of the sin 4') (0 and checked by subtracting 2 from [s sina(+), seventh decimal place, and is added to 1.0 before being used. AAXis divided by the correction The log cos (a+2) from log s cos a+ factor to find Hx', which is divided by H to s, and to get log to log s, arcsin correction is added obtain x'. Now compute a and fa. Use y2 as log sin AX, Log s is extracted from the tables. the argument to interpolate K, and an approximate V, which is used to compute an approximate A4 4) , and log sec 2 are added, the sum being log 2Va is found and K 1000) correction The Va. AX and 3 log a, which has the same sign as AX. log F are added, the sum being log b, which also subtracted from y2 to obtain the approximate y2. F for argument The as AX. has the same sign Va is added to the approximate y2 to give an from extracted b are and for a is 4. The values approximate yl, which is used to interpolate a new (seconds). Aa" the tables and added to obtain V. A new Va correction is computed and the correct yl is found. y' is the difference between Then 2a is found and applied to (a+yo and yi. y' is divided by the fa correction giving a (1 to 2). As and 1800 are applied to factor to find y. (Remember the correction a (1 to 2) to obtain a' (2 to 1). factor is in units of the seventh decimal place and must be added to 1.0 before being used.) b is b.Figure 121 is a computation of lowerorder found and the correction factor for x is computed. accuracy made by following the instructions in x' is divided by the factor to obtain x. (This the note at the bottom of the form. factor must be subtracted from 1.0 before being Note. For log s up to 4.0 and for AO or AX (or both) up to 3', omit all terms below the heavy line except those used.) The distance (s) is computed by the printed (in whole or in part) in heavy type or those Pythagorean theorem, underscored, if using logarithms to seven decimal places.
c. Examples shown are for the Northern Hemisphere, west of Greenwich, and the following variations should be noted: (1) For computations in the Southern Hemisphere, the sign of the cosine is reversed. (2) For computations in the Eastern Hemisphere, the sign of the sine is reversed. (3) Aa is applied according to the following rules: (a) In the Northern Hemisphere, if a is less than 1800, Aa is minus; if a is greater than 1800, aa is plus.
Divide x and y by s to find sin a and cos a. Extract a from the tables of natural functions. Aa is computed by the same method used in the position computation, and the back azimuth is computed and entered on the form. b. Figure 123 is an example of an inverse position computation for lines less than 8,000 meters. All correction factors are dropped and x' is used as x, and y' as y. This computation is sufficiently accurate for thirdorder surveys. DA Form 1923 is used for the computation. 183
IINVERSE
PROJECT
5~8~
POSITION COMPUTATION
I(TMf
5237)
LOCATION ORGANIZATION
Ae.In.
I
/Aa\Acs;
sl sin f
2
)= AX,
O
A¢,
'Ac,
s, cos l a+2)=
AX
 Aa=AX sin
2O~
in which log AX,=log (a'a)correction for arc to sin; log Ao,=log (gy'0) correction for correction for arc to sin.
arc
4%, sec2+fF(X)'
to sin; and log s=log s,+{
NAME OF STATIONS
7
0
0.ma2
IOC
*'
1.
X,
2.1
a
*
=*w
26
3L
L2
A# (secs.)
_____
a(opoit 4,45
colog o s cs 13.
2.102 9. 310238
*s0
lo
2sign
i
A
A.,
to A¢)
log s, sin~ log cos a+j Z
Is,
log
AX
3 log AX
lo sec2 i 
 3 4log
.l

loggbFa
log tan alog
} sin 2lo
A5492log
log a
cos
_1__________a____3_

7342
Aa~~~/
_11
2
927590 8
2
/
75
A5 +
44 92.566rm.
4 0
to1)
COMPUTED BY
9
.s544
log a
45toepitd(na( 4,41 /6+ 2457
19
(
a+~2)
6
Aa
}
a+
arcsin
(secs)
3.42770
log s,3.~S44
bcor.  A
NOTE.For log s up to 4.0 and for A# or AX (or both) up to 3', omit all terms below the heavy line except those lcs underscored, o2 ifo using logarithmshtov7 decimal
lcs
451 DATE
CHECKEDBY
.
DATE
C. CcrAa
DA I OS51924 Figure 121.
184
2Lg

AX.,
~colog
109.
2.04/0728
cor. arc  sin
2O4Silog 10060000
cosj

log AX

log A#,
______
(sees.)
2 0466551
cor. arc  sin
log
39,7910
AX,
log A0
a'
__________________
4?
Inverse position computation, logarithmic, thirdorder (DA Form 1924).
aw.7flS
1
65
c. Examples shown, are for the Northern Hemisphere, west of Greenwich, and the following variations should be noted: (1) For computations in the Southern Hemisphere, the sign of the cosine is reversed. (2) For computations in the Eastern Hemisphere, the sign of the sine is reversed. (3) Aa is applied according to the following rules: (a) In the Northern Hemisphere, if a is less than 1800, Da is minus; if a is greater than 1800, Da is plus. (b) In the Southern Hemisphere, if a is less than 1800, Da is plus; if a is greater than 1800, Da is minus.
difference of longitude, is equal to AX+IX, where X is an unknown quantity. The computation procedure consists of computing the value of X directly, then determining Al. Using Al in the Sodano equations, the azimuths (a and a') and the geodetic distance (5) are determined. b. The procedure and formulas for computation are as follows: (1) The known data are listed on DA Form 2858, Inverse Position Computation (Sodano Method). of west point
,01=Latitude 0 2
a. A noniterative solution of the inverse problem (fig. 124) based on ilelmert's successive approximation method was developed at the Army Map Service by Emanuel Sodano. In this method it is assumed that A the reduced
886POSITION IWS
34
//dyIOIed
Y/Er
To 2
p
1
q

4l 00
00.00
2/4
/7
/5.8/
Fis Anl of Tranl
*
102
7 590.
3927746 0'
1
a11
.~
,,
42.2986
9294
4 ,7325csa 5
a35+x
sin
xssnaH = acose
tan 02
ORDER T327 TRIANGULATION (For calcuinlig nmim cm* tiic)
40 42 05,753 8 23 4,
ae
yo
9^
D:
32
_&
?.3* 5
215.494
.0 1,
6A
8.3sin
2
t
Ced
3
To
1
a
1
To
3
&
4i'M.
o
42
50,6 53 2507. 8/5
CHCE C23K
YDATE~ ) C. A" BY
00
o
.
00.00
"______________________
I)4m
x cor.=fb
C~
Hx' =(approx.
icor. ==/0 _±+a
__,o
Al')
529y
0.557A
ya Y
Va
, rsn0
3
180
3
ysi ,
jr
.
ZssiaH Y4 = 7s cos a
0,5g703393 . 9 32_T
csA
sin +co.0+asin0
DATE
+

oal a
in
+ F (w~)
0. 035
K (Va/1,000) + BY COMPUTED
+1
Ad' (approx.)
.99
S.A8
V
a
_____________________)
+102. 719 6/0
o
a in0
1A

D A tFE
2
Xoor.=fb
0.13 89660
a=x/0 cor.=_+fa
To 2
"
_/027S64Y 4,7,182AcsnV(a2Ari+VV) 4 9 '~x'=(approx. &") 04232139
a
Y
//
b (y/10,000)
x./307
Y,
2
ti
4782A
.
1Ce
2' 1
059...03.63
180
_____________
1
1e
3d1 1
2To
a
tan 132=
3
a
&
a
1e 2 tan
___
3
2
2d1
Va
______________
tan ,6i=
NdORGANIZATION,
LOCATION
a
of east point
01
72. Inverse Position Computation (Long Line)
PROJECT
=Latitude
AX=Difference of longitude, always positive (2) The parametric latitudes, and /32, are computed and the sines and cosines 10place obtained (preferably from tables).
sin* ain1 +Csq

in+
sAin
or sin 0
(approx.)
2Ae"
3
+ F (A?')
V 2
K (Va/1,000) +NOTE: For, souder 8,005mtr1
IS heavy bold type or underlined.
ne mttrm m trsode
h h
evybakt
I ev
o lakhentI
7923
Figure 122.
Inverse position computation using natural functions (DA Form 19938).
185
k16N_e'
e2 = (eccentricity)2a (3) The approximate spherical distance (0) between the two points is determined, and converted to radians (1 degree= 0.0174 5329 2520 radians). cos
a=sin 01 sin t3 +cos 2
2
k116e2N2+e2' where N=e
2
16 sin 2 21" 2 ke2(16e N +e' ) 2 16e 2N 2+ e'
k3=
e/2
02 cos 8t) 9
___7__ t
L
_
_
1L(/J
0oO)02
__
t eau ons
6)
L.ra)2.3 z31i
o /it
pe
o5000t*
74
i
previously stated (fig. 129). The simplest way to set up this condition is to select 1 station as a pole and write the product of the ratios of the lines running to that pole and equate the resulting expression to 1. By the law of sines, the sines of the angles opposite the sides can replace the sides. Replacing natural sines by logarithms reduces the expression to one which can be solved by addition and subtraction. In this problem, the side equation for the quadrilateral LincolnBurdellRedHicks was written with the pole at Hicks. Therefore, the expression for the ratio of lines isHicksLincoln HicksBurdell
HicksBurdell HicksRed HicksRed X HicksLincoln
Figure 128.
Condition equationsfor net adjustment.
NicBlack. A length equation is needed to hold the fixed lengths HicksLincoln and HicksBlack constant. (1) Generally, the triangles with the largest angles are used for the angle equations, and the triangles with the small angles are used for the side equations. Although four angle equations could be written from the four triangles, one of the four would be a combination of the other three and any solution which satisfies three of the equations automatically satisfies the fourth. The sum of the v's designating the angles in a triangle must equal the correction needed to make the sum of the observed angles equal 180°0+. For example, in the triangle RedLincolnBurdell: (1)+(2)  (3) + (5)  (9) + (10) =4.0 or 0=4.0(1)+(2)(3)+(5)(9)+ (10) which is equation number 1 in this problem. Number the equations in ascending order of v's to keep the correlate equations from spreading too far apart. (2) Side equations are now set up on DA Form 1926 (Side/Length Equations) to insure the condition of the side agreement 194
Substituting the sines of the angles opposite these sides and using the symbol designation of the angles as they appear on the triangle sheet, the expression becomes: sine (4+5)sine (8+10) sine (1) sine (3+4) sine (2) 1 sine (8+9) Replacing sines by log sines: log sine (4+5)+log sine (8+10) +log sine (2)log sine (1) log sine (3+4) log
sine (8+9) =0
The items are entered on the side equations form under the appropriate heading. The column headed "Tab. Diff." contains the difference of logarithms per 1" change in angle. The tabular difference is usually expressed in units of the sixth place of the logarithm. Notice that the sign of the tabular difference for angles over 90° is minus. Subtract the sum of the right hand log sine column from the sum of the left hand log sine column and express the difference in units of the sixth decimal.
1243
PROJECT LOCATION
SIDE/
C a/i for,,a
ORGAN IZATION ____
DATE
29 Fngf
SYMBOL
ANGLE
___
22.AE
E
7.
2
__
9.588725/9 +~5.00
Lo(;.
ANGLE
SINE
TAB.
Diy
__
_
1
OAE 2L0i
8S 9
L~~09 /W L L9778.3,57 . +1,92.31 2&A 3o 459.7 1,0.3z
,
__ __±5L4A1593
+ ~~
_ _+i
_Y_
SYMIBOL
ICKS~
9985/O 95OJ2 094n SZA&A859g27 q95
+ 1LQ
_____
TAB. Dir,.
Loa. SINE
Po AR
____
4 *5
EQUATIONS
L0
5
s
~ z 0
~ 6941+i~~L ~
+A23(3
50~+LL ~ ~ _
_

_.+500
_
PoleBLACK 7+&352.
467f/4I5L 7B ~ L,65
a
+
+2.295
COMPUTED BY
D AI
'
66
a2.~_
_
_
__
9.
sz2.
(6.266 0S5125 DATE
+0.26
+0.0
77t1 /d
28
f40u1 6467 f
_
__
15(5)6)OI~ 2.10(17
+
____
9IL3 i,
.89
28 . ±zJ5+ /6 117 AiQ5 Z
44 1A9 41A432
13
12
J38aL...
4 64
_
+000~2/
CHECKEDBY
DATE
,F1926 Figure 129.
Computations of side equations (DA Form. 1926).
195
This difference will be the constant term of the side equation. Observe that if the sum of the right hand column is larger than the sum of the left hand column, the sign of the difference is minus. The tabular differences are the coefficients of the v's shown in the symbol column. To form the equation, the coefficients of each v are obtained by adding algebraically the coefficients for each v from both sides of the equation. The signs of the tabular differences on the right hand side of the equation are reversed for this operation. For example, the first side equation in this problem is formed as follows: 0=
+0.15+(+0.69) (1)+(+1.32) (2) +(2.31) (3)+[(+5.0) (4)+(2.31) (+4)]+(+5.00) (+5)+[(0.19) (8) +( 3.55) ( 8)]+( 3.55) (+9) + (0.19) (+10) Remember that the numbers in the symbol column are only the subscripts of the corresponding v's. A simpler method of writing the angles in their correct columns is to write the angles opposite the side startedfrom in the left hand column, and the angle opposite the side going to in the right hand column. In this equation, the rule would work as follows with the pole at Hicks: Starting from side: HicksLincoln ______ Going to side: HicksBurdell ..... Starting from side: HicksBurdell ...... Going to side: HicksRed__________ Starting from side: HicksRed _________ Going to side: HicksLincoln ......
Angle opposite 4+5
 1
Right
8+10
Left
3+4
Right
 2
Left
8+9
Use the smallest angles possible in the side equations, since a change in a small angle has more effect on its sine than the same change in a large angle would have on its sine. It may be necessary to divide a side equation by a constant, if the coefficients or constant term of the equation are large in comparison with those of the other equations. The second side equation in this problem was 196
divided by 10. When dividing an equation by a constant, care must be taken not to forget to divide the constant term of the equation as well as the coefficients. (3) Length equations are written in the same manner as side equations with the addition of the logarithms of the fixed lengths (fig. 130). The log of the length from which calculations are started is written on the right, and the log of the length on which calculations end is written on the left. In precise work, the logarithms should be corrected for the difference in are and sine (known as the arcsine correction). A table for this correction may be found in TM 5236. h. The explanation for the solution of condition equations by the Doolittle method may be found in paragraph 65, and in USC&GS Sp. Pubs. 28 and 138. By use of the accumulative features of the modern calculating machine, the forward solution of the normal equations can be fitted onto a prearranged sheet which is illustrated in the problem (fig. 131). The writtenbackward solution or solution of C's is also eliminated by use of the calculating machine. When the calculated C's are substituted in the normal equations, the equations should equal minus rl. This check will prove the numerical value of the C's. The diagonal terms must be included when checking the C's by this method. After the v's are computed from the correlate equations, they are applied to the angles and directions. i. The corrected spherical angles on the triangle computation sheets are reduced to plane angles by subtracting % of the spherical excess from each angle. If the spherical excess is not evenly divisible by 3, apply the odd amount to the angle nearest 900. The triangle sides are now computed by the sine law which, on DA Form 1918, means that log of the length 23 minus log sin angle 1 plus log sin angle 2 equals log length 13, and similarly log length 23 minus log sin angle 1 plus log sin angle 3 equals log length 12. Check the length of sides appearing in two or more triangles for agreement. j. After computing the adjusted triangles and entering the final adjusted seconds on the list of directions, the geographic positions are computed (fig. 132). DA Forms 1922 and 1923 may be used for the computation of the geographic positions. These are triangulation position computa
PROJECT
4
I4*
LENGTH EQUATION'S (TM 5237)
LOCATION
Cai0oria___ ORGANIZATION
DATE
2
Egrs.
2___ SYMBOL
_
__
_
_
Lao. SINE
ANGLE
_
__
TAB. Dirr.
_
__
SYMBOL
__
IAug 55
_
ANGLE
LoO. SINq
TAB. Dist.
5
8
_
*
9
3dou.
414 _4
5910a031BM20 z I~6~ +15
7
2
74 9%S8QZ12 #AL65 7+4 _____
Q0zOA O!2"
z
/3()*.
Figure 130.
3.4i24582
________
/.32(.2)2.
2zS225 44132.
~5Z1A&A .i '~ ri32di8~A
(

uP
(
_
(0.~(
16
0 16 (
Computation of length equations (DA Form 1926).
tion forms for logarithmic and machine solution, respectively. The forms can be modified for short lines by omitting certain correction factors, making them suitable for both first and thirdorder triangulation. Paragraph 67 gives an example of both first and thirdorder position computation by logarithms. USC&GS Sp. Pub. 200 is used for logarithmic computation on the International Ellipsoid. For machine computation of geographic positions, USC&GS Sp. Pub. 241 (Clarke Spheroid) and USC&GS G58 (International Ellipsoid) are necessary. k. Although azimuth, latitude, and longitude equations were not used in this example, they are necessities in some problems and their use should be mentioned. An azimuth equation is required whenever two or more fixed azimuths occur in an arc or net of triangulation, unless the fixed azimuths all radiate from a common point as in the example given in this manual. When two or more stations of fixed position occur in an arc of triangulation and are not connected by a line, it is
necessary to relate the fixed positions by latitude and longitude equations.
75. Azimuth, Latitude, and Longitude Equations as Used in the Direction Method Since neither azimuth nor latitude and longitude equations were required in the example as originally solved, the problem has been modified slightly, only for the sake of illustrating an azimuth and a latitude and longitude equation. The modified problem is shown in figure 133 and consists of fixing the positions of stations Black, Nic, Lincoln, and Burdell, and the azimuths of lines BlackNic and BurdellLincoln. The fixed data is taken from the adjustment of the original figure by the direction method. The observed angles are from the original list of directions. a. The purpose of an azimuth equation is to find a means of correcting observed directions or angles, so that the angles can be used to carry azimuths through a network of triangulation between fixed azimuths without a residual error. 197
J .2 ...iL




16
i
021


__


_
.
Z2 +o
w 3
S 2
2
15&
i
5
2
__
~
ALO
_4
______
,258
72
0...
+ 
+o
~~~ 
Q+QQ,21
_
_
_ Figure 131.
l
681 L 541
+,?
+0A4 Qd..
__~
Solution of condition equations.
± +4
2
+t40
~ 2Z5 ~2. _tO23o43
_
6,19
+4

_+
I
LL62
Z06Z250.243.8LQ62.
_
1
4.0ilI
±O,89 8 229 +0.J646 iO2. 7.1?0 tQ~?2
±64
C______ 
198
0,223

+___
_
40..
1,55i '  258 + 0,6I6 1.6O 0,6160 +0.7&0+3za


L..
&
+2
0,8162
.z.2

AJ6h733

LA

42

_37
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7
~
+

i.*
77L 1
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j222 4
6,6,
+~%
wad Solkio
_
_
27
tO OA.OQ+
0,22.
_+
_
L22
.Z8±

....
_. ±1 42 _I3 113 _
a
+
z
 _
+a
2.2
~f 1.7603. 2
aa.
6
2
2
,09
31
JLL 1Q.2L
____
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m
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3.

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2

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2ffi
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1
a i 021
_
O.
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5
.
2

za5.
_
1Ja2 .
±
'+0
______

6

.E~~,8
1
_z . 0.132. 1..LL

i
O 1
_


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~4&
2L
1804 +. LO3
2
POSITION COMATIO,
PJET/_243 LOCATION
2
a
ORGANIZATION
14
To
2'"L
+49om~~L. S& "L _Q Z3
I
__________
To 2 140Ck5
"ds
4
p'/10,000)2
y.

ina
cosa
7,93, 6223
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00
lf
o
0.
H
.
0" a .
Ysmoa
"a Va Y7 Va

4232.63 8. 4 2 . 5 2q2 1
DA
do
0. 6Z7441/A
Y'
sin'
0.61 767017
le
1+ oeA*M /. ni °+ si °' 1
+
er mOPTD s.
99999gg1 ~
cos
°A~t
7
j

2 8 8 52

298.85
ult CNEKE
r
W . C. ij.
53.4500
3e
2L49 2 t.u
Hx'(approx. A),') "377Z~
2o.046 7
0.04077T9
H z.aoV()
O78
69
0.61795179 O~77
do
da"'

4223
2+4.02499999 7M
°o '0uao
5. 20+
_____+
V+ $ (approx.)
. , I
9
2 K (Va/1,000) +
oTls
s
.HM
02397 940
4,25797.1J21
1
Va
:blFvv
________
g (app78~
Z.= n
00.00
03 ,6416.00
/110,000)2
0.
A +4aF(Xh " a o.
K (Va/1,000)8+ s
To 3 LNCL
Mr i4~ yo.=+(a
11.4 4 221 A&?. 97 f I 84.06J
pYceor.=+fa Y'
s.
a = AJQ*
01 /8
43.97
IZS
22L.5M * 13i0
q__
04/Q07 759
Q
1Rr~l
8To
.15ss
Hx'(approx.eA.) Anala V (Va)
$z/O0)
Ye
a
00.00
1183 4291
Q,.15,592045 x
I~
kcs
__________
22 49
VintAngle of Triangle
To 2
Au M5
/9I
_______________34.
_a
±' 1 Br/I
4ma
aI 8
J__'I~L
1
To
a* °a
DAT
8 Live.L
&
m. uCptAU.)
9M A TRI NULATIS (For sahu~uI
2a
umkr0,rSomIM Malt
2
3 .
45
.m m dob. im~ thmn
sNmmlhr
FE;1923 Figure 132.
Position computations (net adjustment) (DA Form 19P23).
b. The procedure in setting 'up the equation is to start with a fixed azimuth and, by successive use of observed angles, compute the azimuth of the next fixed azimuth. The difference between the computed and fixed azimuth thus obtained is the constant term of the azimuth equation. The sum of the corrections to all the observed angles used in obtaining the computed value of the fixed azimuth must numerically equal the constant term. When the adjustment is made by the direction method, the corrections are the v's applied to the observed directions.' When computations are in the geographic coordinate system, the forward and back azimuths of lines do not differ by exactly 1800, but by the amount known as geodetic convergence. It is to obtain the convergence that preliminary position computations are made for the lines
through which the azimuth is carried. For this example, the convergence can be taken from the, position computation made in the adjustment of of the original figure. c. The numerical example is as follows: Fixed azimuth NicBlack.. Observed angle at Nic___
(+51)
Azimuth NicHicks Convergence at Hicks Azimuth HicksNic Observed angle at Hicks_ Azimuth HicksRed__ Convergence at
Red. 
123°21'40"~.99 +45 01 07 .4 168 22 48 .39  00 46 .36 180 00 00 .00
( 56+ 57)
348 22 02 .03 44 52 05 .1 303 29 56 .93 + 2 50 .50 180 00 00 .00
199
POSITION COMPUTATION, 21
POET1243
Calit'ornia
LOCATION
1
ORGANIZATION
3 ,L/NcrOM'
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a2
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50.50
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8.2070
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First Angle of Triangle
.387 4.6.2.
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O 21.652 05m'
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d
"2
ORDER TRIANGULATION (For calculatiug mchne cwonud)
(7M 537)
DATE7s,9.1
4
sin 027,o4 1+sin~9 1
o
m
Aa' (approx.)
.140 05O82
+F(A,)
Aa"
 1 O5 0 2
h aOTEbottyForunde 50(5)atefs omsttrms under the heavy blaek lise not is
DA F s_1 923 Figure 133Continued.
Azimuth RedHicks___ Observed angle at Red_
123 32 47 .43
(61+62)
RedLincoln.Convergence at Lincoln
+30 41 59 .7
154 14 47 .13 01 45 .08
Azimuth
180 00 00 .00 Azimuth LincolnRed angle at Lincoln_
334.13 02 .05
Observed
(+66)
50
2046
.1
Computed azimuth LincolnBurdell 

283 52 15 .95
Fixed azimuth LincolnBurdell        
283 52 22 .55
Constant term (computed minus fixed) Azimuth equation : 0=
200

6.60± (51) +f (56)
 6".60


(57)

(61) + (62)  (66)
(1) The azimuth equation is formed by setting minus the constant term equal to the sum of the corrections to the angles. Transferring all terms to one side of the equation and setting the equation equal to zero gives the form shown. (2) A source of possible error occurs whenever an observed angle is subtracted to obtain an azimuth. The error frequently made is neglecting to change the signs of the corrections for that angle when the equation is written. In the example, the observed angle at Hicks is designated (56 +57), but the angle is subtracted to obtain the azimuth from Hicks to Red so the designation is written in the equation as
2
a
&ACK
To
2d L
212 40 /Z~
3CK
&
2
a
RLACK
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1
~
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t 8.
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S.Z
s= 6350.335
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#0.
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08
a =(x'/ 10,000)
n'
I l
b=(Y/10,000)
3
54937453
536.1R91
a
03a
3488. 723.

YZ
3,.488.7Z13 212 768. 757 /1.72
4,212 767.
V6.12763 2
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1
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3
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0.122
+
+827 '
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3
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0,20/62338
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cf 01794633 1832. 22.34 ,8470. 732. 2
_
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0. 206
2
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l
179
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DATE..,
01 /5/46 /22 42 14.220 8. 684 lga 2 b'=(Y/10,000) 0. Ni f 3.250 icr=f '
6,12763
,a. '
04OA8
H
Hx'=(approx. °X" Arsin+V (Va)

75.1460
0.08 7,4 0. 617441M.
15

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22 ,51.30
'a'
A.2
4 2216~7. 975  8? . 730 422 767. 24,5
y,'
00.00
12243 22.366
I
.0cor
y cor.=+fa
Va
/3.7
A/ic
04 48. 6
B5061 _8 0. 04101358S x=ssin a H .y=a Cosa Hx'=(approx. AX"~) 2/7.6258 0.
07,356
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4
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2
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Fis Ani of Triangle
207
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l 180
ORDER TRIANGULATION (For calAaig madse comups afro)
a
+3d~
°a
a'
2
COMPUTATION,
POET123POSITION
's
si
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.9gg?0
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ha
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blaehlise not in SSJ hev N oTE: tyor underS1etS etersomit terms under the heavy
FORM71923
Figure 132Continued.
The same  (56+57) or (+}56 57). situation arises with the angle at Lincoln. d. The discussion of latitude and longitude equations in this manual will be limited to a description and a short example problem. For a complete development of the formulas, see USC&GS Sp. Pub. 28. e. The example is shown in figure 133. The fixed data is taken from the adjustment of the original figure by the direction method, as it was for the azimuth equation. The fixed data in this case is the positions of stations Black, Nic, Lincoln, and Burdell, and thereby the lengths and azimuths of the lines BlackNic and LincolnBurdell Jf. The first step in forming the latitude and longitude equations is the setup and computation of preliminary triangles. A single chain of triangles
is used for preliminary triangle and position computations, and it should be the R 1, or strongest chain (fig. 134). One angle in each triangle is concluded and the other two angles are observed The observed angles are usually the angles. distance angles, that is, the angles used to compute the unknown sides in the triangle. The concluded angle is known as the azimuth angle. The exception to the rule that the azimuth angle is always concluded occurs when one of the distance angles is not observed. In this case, the observed azimnuth angle must be used and the unobserved distance angle concluded. The important point to remember when using concluded angles is that the designation of the correction to the angle is the sum of the corrections (with their signs changed) of the other two angles in the triangle.
201
POSITION COMPUTATION,
PROJEC
~
3
2
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2 /4~S
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a
a
&
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a
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i _
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.
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kcor.=+fa
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b=(y/b=,00/) b=
2016339
DATE
Red
~
a'
b
46.69
0.
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A& .22 51 M
07 46,2592 I4CK
d1&
0.
3
180______________ 00 00.00
To 2 First Angle of Triangle
8.2070
5
382~~4
S
1
a'
comptation) TRIANGULATION (Forcalculatig macbins ORDER
2
CaionaORGANIZATION,
LOCATION

la'°
d+
sin
osA
(approx.)
+ F (a)
,
124.007
3
124",0' 7
or ae8,15meters smit terms unlderlbs heauy black esybdtype or underlard
linenot in
.,1923
Figure 132Continued.
g. In addition to the correction symbols on the angles, each angle is also designated by A, B, or C. The azimuth angle is always called C, while the length angle adjacent to the known side in the triangle is called A, and the length angle opposite the known side is called B. The data from the preliminary triangles is used to compute the preliminary geographic positions. The combined data from the triangles and position computations is then used on DA Form 1927 (Latitude and Longitude Adjustment) to form. the latitude and longitude equations. h. The triangles used in the preliminary computations are shown in figure 134. The A, B, and C angles are labeled on the sketch for use in the latitude and longitude equations. The preliminary triangle computations are shown in figure 135. 202
Notice that the C angle could not be concluded in triangle IRedNicHlicks, because the distance angle A was not observed. This is the case when the azimuth angle C must be used as observed. The triangle computation can be made using either natural or logarithmic functions. In this example, the computations were made using natural functions, as the position computations were also made using natural functions. Some argument could be raised that a complete logarithmic computation could be more efficient, since the tabular differences for 1" for the log sines of the A and B angles are required for the latitude and longitude equations, but this is actually a minor point and the choice of logs or natural functions should be left to the individual computer.
Lincoln
/
Burdeli
Red
Red
s8
/ \?
Black
Lincoln
Burdell
/
//
Blaocl
Nic Figure 133.
Sketch of triangulation net (modified problem).
i. Using the angles and distances from the preliminary triangle computations, the preliminary position computations are made (fig. 136). Notice that both sides of the position computation form check the position of station 1, even though no formal adjustment was made of the triangles. The check is possible because of the arbitrary adjustment made by concluding one angle in each triangle. j. On completion of the preliminary triangle and position computations, all the necessary data is available for forming the latitude and longitude equations on DA Form 1927 (fig. 137). On the form, the following entries are made from the available data: (1) In the right hand, upper corner of the form in the spaces 0,, and X,, the computed latitude and longitude (in degrees, minutes, and decimals of minutes) of the fixed end station. (2) In the left hand column headed "Station," the names of the stations at which the C angles are recorded. It is convenient for later use, if a notation is made beside each station as to whether the C angle at the station is used in a clockwise or counter
Figure 134.
Sketch of net (R 1 arc).
clockwise direction. A plus (+) sign indicates a clockwise and a minus () sign a counterclockwise C angle. (3) The computed positions of the stations in the lefthand column in the columns headed 0, (latitude) and X, (longitude) in degrees, minutes, and decimals of minutes. (4) In the columns headed A, B, and C, the correction symbols for the A, B, and C angles. The symbols for A, B, and C can be taken directly from the triangle computation sheet when C is plus, but when C is minus the symbols for C on the triangle sheet must be entered with opposite sign on DA Form 1927 by the following rules: (a) When C is plus, the combination of the symbols for A, B, and C should be zero. (b)When C is minus, the combination of the symbols for A and B should equal the symbols for C. (5) The tabular differences for 1" for the log sines of angles A and B in columns headed +A and 5B. The tabular difference for the log sine of angles over 203
PROJECT
DATE
1243 LOCATION
14

2 1
OBSERVED ANGLE
___
HICAs
N/Ic (45 3 L~i 90 13 loc&s BIK 12Au Nc wex
ol01
39
(TM 5237)
*~,
494. 1.9

3

.
____
QL~41.9 0~.o 08.3 0i.1 oi
06867 .77309 0
11g5i
07
42 46.0
6.o
0.1 0140.0

44
52
.
5.
0
.99?6
O9.o0 0.8609/2(6 0 05, 0.7054768
24'ic(2E0~o)
s43Aj
FUNCTION: eATRL
PLANE ANE
______6428.231
75
lj1
____
4
. 09.9
"
B
Pre/i IAEP
SPHERICAL SPHERICAL ALE XCS
CORRECTION.
NcBLACK
ms'2 
COMPUTATION OF TRIANGLES
C~ionoORGANIZATION29E STATION
___
Auy 55
05.1
13RdICS__8l0AL
Red
12
15.494
____
~~0.1
_____
ReacKLIlcoLN Red 2
8"46711 +u
A oO3
2
57
48
453
HCA'S
0
00.1
(342o 4 9 29 06.2
08.6 4& 062 4L


_
_
_
LIM OL N Red
______
__
Figure 155.
90° is minus.
~~
___0/___
____
~
_
_
_
0.510 7 Q6.L 0. 99966413 __
~
_
08.6
4 _____________ _______
~
Ad S52
_____ 953
Preliminarycomputation of triangles (DA Form 1918).
The tabular difference for the B angle is entered on DA Form 1927 with opposite algebraic sign than as given in the tables; therefore, if the tabular difference in the log tables is plus for the log sine of a B angle less than 900, the tabular difference entered on DA Form 1927 in the S column will be minus. 204
_
8073 064 A53.'Q O83a7
___13LIoLCS
__12
_
(6) The computed value of the position of the fixed end station and the fixed value of the position of the fixed station. The computed value is designated c,, and X,, and the fixed value is 0', )4. Both values are entered in degrees, minutes, and seconds. (7) The quantities a1 and a 2 from the table at the right hand side of the form, using the
PROJECT LOCATION a
2
*
a/
0
,. To
li
2M"492
To
AL
1
To M First Angle of Triangle
44
*
2. _02 a" '10 4 4 8 .,2 571 /
0
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Y=s Oosa
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~
I 132
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43
,2 .370
sin b
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0. /743e
1929902gp
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Y=g _=scosa
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32 9. 3 70
1. 20
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p
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68

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15
c 
./67
sin
16
sin b'
0674
.63
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I 9irj9qqg4
0.740
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~
.270
4089
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431Vr+F()) Y2.
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22.476
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x cor.=_2fb zo41780~ '
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I
9
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z=s sin a
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6428.237
0 72 4 6S.25 7 1
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00.00
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,
0~".6/689070
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 Ad,' COMPITI OAL~,~ BY.,..
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Va
sin b + sin b'
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a=(s"/1o
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00
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ORGANIZATION
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__a
ui.
OlINCUU
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cns °b
sin b + sin b +enA °ds (17 (approx.)
0/K0S .
.1
87923
K (Va/1,000) + NTE:
A ee nede Rn mtro emit teem, under the bevyp binek line net in
A /G923 IFEB [
Figure 136.
Preliminaryposition computations (DA Form 1923). eybl
computed value of on, from the top of the page as the argument. k. Now that all the data has been entered from the preliminary triangle and position computations, the computation of the coefficients for the latitude and longitude equations can be completed on the form. The columns headed " OO'," and "AnX," need no explanation except the reminder to watch the algebraic signs of the quantities. The quantities in the columns headed "Latitude Equation" and "Longitude Equation" are simply the products of the quantities in the columns already filled out. Watch the algebraic signs. 1. In forming the latitude equation, the quantities in the column headed "(On O) A" are the coefficients of the symbols in column A, the quantities in the "(On0,) (SB)" column are the coefficients of the symbols in column B, the
yeo
quantities in "(X,, k)al" column are the coefficients of the symbols in column C. troftelttdeqaini7282 The longitude
The constant
equation is formed using nnm. the
quantities in the column headed (XX)A"as the coefficients of the symbols in column A, the quantities in the column headed " (An Ac) ( 8B) " as the coefficients of the symbols in column B, and the quantities in the column headed "(OOca2"as the coefficient of the symbols in column C. The constant term of the longitude equation is 7238.24 (An An') . To help clarify the formation of the equations and the coefficients, the latitude equation in the example will be written out in full before any of the coefficients of the symbols are collected into the final form. n. The latitude equation in full is205
hI~9!AW
PROJECTM~~I!,*CE
To 3
a2
a2To Aa
r

4
1
___________
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I
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sin
427pg..
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2
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w.
1/...
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DATE
.234a
744578
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a=(x/10OOO)
0.46616
+ F (Ae~)
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m+
3
634
V
7s Y=
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a 41
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00.00
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ICKS "=8073064 3s
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5:4
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T
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00
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2
First Angle of Triangle
l
a 04.01
¢02
180 a'
~
(For calculating machine computation) 
~
/
2/2., 16
.2
3
Hx' =(appro. AA
2
cor
sin
1 + con
2 d~.

5
1/14 274 .254
0.61744117
m
p66p
si'~' Ab*
sin #+ sine
1 + cos Am ace (approx.)'
9?? 9 97.
10,9
(44)F
2
K (Va/1,000) +
V (Va)
+
AK
++ F
V
5G732Q51
0. 0409,535
H
a
17049,6
INOTE: Fer s under 8,000meters emit terms under the heavy bluekline not In heavy heldtype or undrlined.
71923.
Figure 136Continued.
0= (7238.24) (+{0.004)}+ (0.21) (52)
+ (15.14) ( 57)  (15.14) (58)
+}(0.86) (52) + (0.86) (57) + (0.88) (58) * (2.79) (56) + (2.79) (57)}+ (2.79) (59)
+}(2.79) (61)+ (1.18) (59)+} ( 1.18)(61) + (2.94) (56)+ (2.94) (57) + (0.28) (54)+I (0.28) (56) +1(6.12) (66)+ (6.12) (67)I (4.70) (54)
± (4.70) (56)+ (4.70) (66) + (4.70) (67) Combining coefficients of the same symbols with regard to algebraic sign, such as (+0.210.86) (52) and (+0.28+{4.70) (54), and rearranging in 206
ascending order of the symbols, the equation becomes0= +28.952960.65(52) +4.98(54) 5.13(56) + 14.43 (57) 14.28 (58) + 3.97 (59)  3.97 (61) + 10.82 (66) 10.82 (67). This equation would probably be divided by 10 when used in a leastsquares solution. When dividing an equation by a constant, care must be taken, not to lose any significant numbers by rounding off.
76. Adjustment of Triangulation by the Angle Method a. The figure previously adjusted by the rigid direction method will now be adjusted by the
machine computation) PUATIO, ODER RIAOULAION(For calculating I~OC~lO CO?
PROJECT
o
a4
04
2d
i.3456
Ca,
2
66
4 2/7
K
2
04fOO
0.5
6/30+(log
a
) +10.
i H
4f 2
066
65143 pj
0. a82
209 
1.85
=
Xt
006 go4/o42.mc ?
Vat 0
3
(Iog
T3O 3
A~alm
0"
£W3~
In01 +
3
3' 3d3m
oa
01450
'4 q~gzsAa
1'
3
To
O4
53.114s
.2
h=(y/lOOOO)
2 S77/
0
xcr=f
0 6.
6
256/.
0/4 122
.3
.5
0/~
H
1
7
870047 ~oi40679,0
'Y672%1
+Faxe (/535
a ) b~ =(y/10,070 2
DA~~~0
23.a
/35 87/4 To99037 2f/
1Va(y/10,00+)
2
655/
)
71ES87
Figur 16Cotinued
angle method (fig. 138).. As the name implies, this method of adjustment solves the most probable angle, and individual directions are not corrected. b. The angle method involves the solution of a series of triangles. Consequently, 1 diagonal in each quadrilateral must be omitted. Figure 139 illustrates the net with 2 diagonals omitted. The stations are numbered starting from a fixed line and each station added is given a consecutive number. Thus, building up the figure point by point and numbering each station as it is added, Burdell is number 1, Red number 2, Nic number 3, and Black number 4. Notice that Black is numbered, even though it is a fixed station. The angles in each triangle are lettered. a, b, and c. :Letter b is always assigned to the angle opposite the side of known length. Letter a is assigned
to the angle adjacent to the known side and opposite the side through which the length is to be carried. Letter c is assigned to the remaining angle in the triangle which is also the angle used to carry the azimuth through the figure. Combining the letters with the number of the new point in the triangle completes the designation of the angles. For example, the angles in triangle RedIlicksBurdell will all carry the number 2 since Red is the new point in the triangle. The angle at Red will be letter b because it is opposite side HicksBurdell which is known from the computation of the first triangle. The length is to be carried through side HicksRed; therefore, the angle at Burdell is lettered a. The remaining angle is at Hicks and is lettered c. c. The angles are taken from the list of directions (fig. 140) exactly as in the direction method of 207
PROJECT
AND
LATITUDE
LOCATION
1243
ADU3MEN
(OTUD (TMS297)I,
ORGANIZATION FROM
22A
DATE
i.
n~
___
BLAEK ___
Nac
T
__
+8uat
of
]t
OAIN
(,
BASE
AIO(Endc Longeqution
ion )
A
~~~~
C~
(,
1t
eqato
L4aLNRps.BASE
TO.
c
2ng. equation4 Lat. equatbon
B
,.Oe) a,,)
)84(
(11.91.)(b)
1Xn )aG
TABLES
L.ong. o0quat1on
(+11 orL)
(O,,4,c)a2
0
2
2.12.. 2..SZ 09~___
7IL
2.04 2.02
2
I2

}5 _ ____0
_
_ __________
293..±.
_______2 00(a11
74
Z
_il
~i.A
_
.
0.oi
..
1
2I
i
t
1ll
_
55
_ _
_
_
2
_1Z4
8_______________ 00j____
____
)1,____ _______
1
4
.35_
95___
'A4i
_________22
FOR
_____
_
17
.08 2.04
47 22
2
3.5~±.J).~f.±..56)4
4.06 165
DA
12
2.43 2.42
2.2 2.2 2.22 2.11
2.02 2.20 2.420 2.29
19_______________ 2.00 2.424
420
7Z3A,.O 4CZz12... +_
2.030 2.221 2.9 2.08 2.02
24 29
_____4 Y1
2.011 2.210
120
___10 __t_
20 1.2 2.14 2.10
) 51)
2 403*
1.72
2.47
.4 1.47
2.426 2.44
.42
2.34
26...L.25) 1.0 2.27
1.
2
2.27
46
1.85
2.112 2.19
42
2.44
2.27
46
a
24
2.22 2.441 2.271 2.49B 2.24 2.2
448
2.22 1.22
'99
2.67 4. 0
42
2.072
44 47
1.52 .42
2.2 2.24
. 8U
44
1.14
2.47
72
1.427
3.44
72
1.42
4.91
1927 Figure ~~ ~
~ogtd ~eutos(A ~
om1 ~ 137. )~
~
~
~
6
4op.to8o0aiuean
Lincoln
Burdel~l
Red
Black Nic Figure 138. 757381
0
 65  1420
Sketch of triangulationnet. 209
Burdell
Lincoln ai
I
4b
Block
ks cNi Figure~~~~~~~~ eR(igetrage) 13.Secho
210
oROECT
LIST OF DIRECTIONS
ORGANIZATION
~(TM 52?
3

LOCATION
SATION
Ca/ fornia
aLACK (U15Cd6s)
OBSERVER
INST. (TYPE) (NO.)
DATE
//K
uea)
LOCATION
L5
___
0
__
____ ____
10.2

ATION DATE
1
0
ECCENTRIC
DIRECTION OBSEEVED
SEALEVEL
Ene,)29448 .2
(u C#6
ADJUSTED
DIRECTION CORRECTED
REDUCTION REDUCTION'
Real z
0.
idel//(90n
C0S~nd57L 1. P.
A/CAs
00.00
4.7L
INST. (TYPE) (NO.).
OBSERVED STATION
00
DIRECTION'
____

Californlia
OBSERVER
00.00
L3T
?2O39
(.&'i, ,,j ,
Ni
00
0
5
29 '4fn"
Red
ADJUSTED
DIRECTION ECCENTRIC SEALEVEL CORRECTED REDUCTION REDUCTION' WITS ZEROINITIAL.
BSRVDDIETIN OBERE SAIO OSRE STTOOBEVDDECIN
WITS ZEROINITIAL
DIRECTION'
54 28.8

___
3J~ / 3Z


LOCATION
STATION
OBEVRINST.
Sc)c G'o~0
q R edS~'
00
136
Gnye

11.3
2/
0


.2
5
00
_

5.
00.00
_____ /Red 6yprfawso ___2/________S
+uc~
LOCATION
DIRECTION'
~O

" Jr
AZACK
ADJUSTED
ECCENTRIC SEALEVEL CORRECTED DIRECTION REDUCTION REDCCTION' WITS ZEROINITIAL
00.00
426
Y
(
DATE
ORSERVED DIRECTION
OBSERVED STATION
A/ic
HICKS (csc.sJ
OBSEVERCah~rn~ _______________ (TYPE) (NO.)
IL
.I6. .
STATION
Clfri_________
OBSERVER
OBEREDSTTONOBERE OBEREDSTTON
OREEE O
P/ks
+
0
if
00
00.00
It
/
~


Real(9Pos)2L.5Q28.
0
5
0
00
STATION
A/i
Caliorna ___________
INST. (TYPE) (NO.)
Cpi Smi4i0 PP OBSERVED STATION
BLACK
(99C#45)
OBSERVED DIRECTION If
0
0
00.00
ADJUSTED DIEECTION'
ViV
00.00
______
LOCATIONOBSERVER
Gs DATE
CORRECTED DIRECTION ECCENTRIC SEALEVEL REDUCTION REDUCTION' WITH ZEROINITIAL
DRETIN DRETIN
/
4,
L.JNCOLA(
INST. (TYPE) (NO.)
I
(2
~E DATEqsA
.EENTRIC SEALEVEL CORRECTED DIRECTION EUCTION REDCCTION' WITS ZEROIITIAL I
r
.

0 O
/
'n if
0
00
00.00
ADJUSTED DIRECTION' /
0.
LOCATIONSTIO OBSERVER
INST. (TYPE) (NO.)
OBSERVED STATION
RIic*s
(OscsG
LINCOLN
f6as)
Figure 140.
OBSERVED DIRECTION
0
00
4
DATE
ECCENTRIC SEALEVEL REDUCTION REDUCTI
00.00
52


CORRECTED DIRECTION WITS ZEROINITIAL
0.
00
ADJUSTED
DIRECTION'
00.00

List of directions for angle method (DA Form 1917). 211
adjustment. The triangles are entered on DA Form 1918 (fig. 141) as explained in the direction method. The symbols in the lefthand column are entered beside the corresponding angles as shown on the sketch. Notice that the angle at Hicks from Lincoln to Black is fixed, and the list of directions must be corrected to hold the fixed angle. Only four of the eight triangles in the complete net are written because the two diagonals were omitted. The missing triangles will be computed after the adjustment is completed. Since the net now consists of only four simple triangles, there will be only four angle equations to close these triangles, and one additional equation to preserve the fixed angle at Hicks (fig. 142). This latter equation is merely a statement that the sum of the four corrections to the four angles at Hicks must equal zero. There are no side equations, and only one length equation (fig. 143) as in the direction nethod. d. The designation of the angle will also designate the correction to that angle. As in the direction method, the sum of the corrections to the angles in a triangle must equal the triangle closures. For example, in the triangle BurdellHicksLincoln, (la) + (lb) + (lc) + 6.4, or written in the usual form, 0=6.4+(la)+(lb)+(lc). Equation 5 fixes the angle at Hicks. The length equation is formed as in the direction method. The six equations are now solved by the method of correlates (fig. 144) as were the equations in the direction method. The final adopted v's in this case are the corrections to be applied to the angles. The angles are corrected on the triangle computation sheet, reduced to plane angles, and the lengths solved. Although the solution of the condition equations is much simpler in the angle method of adjustment than in the direction method, the application of the corrections to the list of directions is more complicated. e. Examples of the method of correcting the directions are(1) Station Lincoln. Station
Observed direction
Preliminary seconds
Hicks.00'00"0 00"0 Burdell  251 50 28. 6 25. 6 Red3021114.7 Triangle 1 angle Burdell 108°09'34!4 to Hicks At Hicks correction is 0'0 At Burdell correction is 3'0 Total3!0 Average correctioniV 5
212
Final seconds
01'.' 5 27.1
(2) Station Burdell. Observed direction
Station Lincoln
Red 
Preliminary seconds
  
000'00':0
_
294 48 01. 2 337 10 32. 9
Hicks_  _ .
Triangle 1, angle Hicks to Lincoln Triangle 2, angle Red to Hicks Angle Red to Lincoln At Lincoln correction is At Red correction is At Hicks correction is
Final seconds
00"0 56.2 31.1
57'7 53.9 28.8
220 49' 28'9 (A) 42
22
65
12 W10 5. 0 1. 8
34.9 03.8 (B)
Total  6'8 Average correction 2!3
(3) Station Hicks. Station
Obse rved directioln
Lincoln00 Burdell___________ 49 Red91 Nic 136 Black  180
00' 00 29 21 40
00'' 0 55. 2 06. 2 11. 3 54. 5
Triangle 1, angle Lincoln to Burdell Triangle 2, angle Burdell to Red Triangle Nic
3, angle
Triangle 4, angle Black
Red to
Nic
to
Preliminary seconds
Final seconds
00:0 56. 8 08.6 12.7 54.5
58"9 55.7 07.5 11.6 53.4
490 00'
56'8 (A)
42
28, 11.8
91 44
29 52
08.6 (B) 04.1
136 44
21 19
12.7 (C) 41.8
40 0'0
545 (D)
180 At At At At At
Lincoln correction is Burdell correction is Red correction is Nic correction is Black correction is
+ 1.6 +2.4 +1.4 0.0
Total +5" 4 Average correction +1!1
(4) Station Red. Station
Hicks  ______ Lincoln_____ Burdell___ NicBlack_____
Observed direction
00 30 95 284 321
00 41 09 17 09
00:'0 59.7 11.0 14.0 32.8
Triangle 2, angle Hicks to Burdell Triangle 3, angle Nic to Hicks At Hicks correction is At Burdell correction is At Nic correction is
Preliminary seconds
Final seconds
00'.'0
00!6
13. 5 13. 2
14. 1 13. 8
950 09' 13:5 (A) 75
42
0"0 +2.5 0. 8
Total +17 Average correction +0' 6
46. 8 (B)
PROJECT
DATE
1243
LOCATI ON
20 Au 3 55
CQionaORGAN
COMPUTATION OF TRIANGLES (TM 5337)
2Onr
IZATION
STATION
OBSERVED ANGLE
SPHEICA
CORRECTION. SPEANL
SPERCSL
SPERICL PANEFUNCTION: PANE
LOOAIH
3~7
___23
11
24
SLIjeoLM /08 13 8ukU~ON3.9767005
+udl L&
7 55 093.
00
WcS49
2
L ±30
S
2&.9 0.O %
14
O.K
2&91
2,589734I9
~~ 34.3
8787 9.97781/57
2
Q~apjiK
4.
1_
__
_ __
7
__
23
2k
__
1
2
2
WICK
W95
_
_
_
_
11.0 42a 28. 11.0
42
Bqurt/elI 13 2dSrll___ 12 Red  HICKS
2
_
442.5
_
Q.'1.~
/3
0762 .998,24070
4
___3.90704529
f
0.2
.
_
_
_
23
2
44
3eat
75* 42 46.0
_/
5
IK
.5
52
05.1
40.2 *r0
0iL2
___23
2
2_
8LC gC :S
fO&A.
0L.
A±L7
_
_
42
9. 93994 9.8484806 7 77863558
3. 95844203 ~~0.1
_
_
0.2 IOL 0.L 110.0 0390. 44 19 43.2 1.4 AL.& 0. A.81 .45C 0,/ 0 40. 08.2. 0.0 0.
~ p
12
.513
NiC__
LCK
_
_
_
_
_
__ _
_
9a.8
____3.80809877
0.4
_00.5
DATE
BY
.59421L8L8
98443 3.80280349
_
BLC FlCS______ __
COMPUTED
I
02L 0.L
_____
3_
DA
_
____3.54203
1
__
O.0
LIO±oo~

Ak
_
.?
Nvic I4ICKS
12
_
.0725 537
361
_
JL5829437 1 ?.. 32%8851
AOL&.
31.7 ±32 t
07662783
_
___4.
09
3
0.1/_
46.4
5
__
_0.1
__
_
CHECKED BY
DATE
FEB571918
Figure
141.
Computation of triangles for angle method (DA Form 1918). 213
I
(6)
Station
Nic.
Station
Preliminary
Observed direction
Black00 00' 00"0 Hicks45 01 07.4 +
.3a
Final
seconds
seconds
00"0 08. 2
00"0 08. 2
J. The preliminary seconds are obtained by using the adjusted spherical angle from the triangle computation sheet. A direction which was not
+

___
used to obtain an angle for a triangle used in the adjustment will not enter into this computation.
0.
These directions will be corrected _
__
_
difference
between
the
observed
The
later.
direction
and the preliminary seconds is listed for each
direction at which it occurs. Figure 142.
Condition equationsf or angle method.
involved (including the initial direction), giving the average correction per direction. The sign of
(5) Station Black.
Preliminary seconds
Observed direction
Station
HicksRedNic
00'!0
00 00". 00"0
51 90
57 39
Final seconds
the difference must be such as to change the ob
00'[
served direction to the preliminary seconds. The average correction is applied to the preliminary seconds to obtain the final seconds. g. The final computation in the adjustment is the solution for the lengths of the diagonals omitted from the net when the adjustment was

43.3 09.9
10. 1
Triangle 4, angle Hicks to Nic At Hicks correction is At Nic correction is

10. 2
900 39' 10':1 (A)
0"0
begun.
+0. 2
This computation is best performed on
DA Form 1919 (Triangle Computation Using Two Sides and Included Angle) (fig. 145). After
Total +0"2 Average correction +W"1 PROJECT
S/LENGTH EQUATIONS
1243
LOCATION
(TM 5237)
calirornia
ORGANIZATION
DATE 29E
9rs.
2oAug
ANOL
SYMBOL
LOG.SINE
TAB. Dirr.
bZCKS8L.ACK
e09
249.
1& 0h
.
aJ.5.0
/
Figure 143.
214
+1.25
.
28_
SYMBOL
Lo.
ANGLE
ACCLNO
81
1
/a
OBZL
_L_

The sum of these
differences is divided by the number of directions
SINE
3.
14
ssr TAB. Di.
54
108 09 31.4 3 S31TLL 0.6? _ _ 9.8286511 ( )5L 75
46.022+i~L 9.
_______
Length equation for angle method (DA Form 1926).
_
_
~

#
~r
2
1

+L
ALA

/ 56
IL.
1
+

ormALd
_
323
i

±L3
0.


1JL
L
...c..
0l.
fi.23
~
L
f2
2.2 +~2%
~
L~
~
iM21
2

Figure 144L Souto
00
+0.4
. f 5
¢
_
6.ZL
Q,21678Q5
+ZI

_
....
+O~Q2
f21
0,=
1±
.L
+_____
Q C
+242 2 .83
1h~as


6.
J +0/2.
4/~~~2/

Qff42Z
#42/

~+
C.
1
ofeqaios
215
PROJECT
TRIANGLE COMPUTATION USING TWO SIDES AND INCLUDED ANGLE
d
1243
j(TM
5237)
LOCATI ON
ORGANIZATION
91
C.
Sph. excess
2'? ~Lg
08.6
Log a
~
3 ~
4
j
j90og LgiC
4
SLog ta (456g)7550j4q45
Log ai
0.1
44
45
CD
44
)o58sl~Log 2.5 (4,50±')
15 25.75 j(Ap+Bp)
CD
13 33 3.56 57 485 41.31 304 21 '1/ 29 e'j.q,
Log a
3.90704529
I
(A,,Bp,)
Sum=AD Diff=BD
0
Log tan
yl(A 0Bp)
Log tan
____
.2 .6690°j~pLog sph. ex. 0. I
Sphi. excess
9,8733 9. 382 23920
___
___
(Sketch)
[/ivaN
b
Red
C
Colog sin A0
39793643
Log c
COMPUTATION
_____ _______________CHECK
No.
STATION
23
1
__
3 13
&Ed
1.2
SPHERICAL ANGLE
HCs LINCOLN Rd
141CKS LINCOLN
2
12
30 g,
42 2g
57
48 493
022 086
LGR't~
2..
006AOo
085q 493 q.253487
0.1
.179
BunadeII
11.3
LINCOL.1N
50
20
45.1
6well au
65
/2
03.8
a.,
____
0.2

*The subscripts s and p
0.1i
LINCOLN ____
__
4
70605
E
____
27
Re
6YL
A142.
4
____
COMPUTED BY
PLANE
LINCN Red _ 14ICKS___
2,3LINCOLN
___2
SPHLRICAL
6S8Z5501
___1Red
1L.
g53j55
45.1
9.8644030
1oT%5f8Q.
736451
____3
_
____
_
_
_
_
_
_
on this form refer to spherical and plane angles respectively.
nn
DATE4
CHECKED BY
, FES571919 Figure 145.
216
3
g* 39350083
Log tan I (Ap+BD)
.b
2
.
0
C
gg~g
Log sin CD
DA
=snD
tan z (A 0B 0 )= tan 'p tan Z (A, +Bp):
(Call longer side a).
btan (45°+q0)
2qDEng,
Solution of triangles for angle method (DA Form 1919).
DATE 4
_
PROJECT
TRAGECMUAINUSING TWO SIDES AND INCLUDEDANL RAGECMUAIN (TM 5237)ANL
I
1243 Ca/i> (rAiQ
LOCATION
C
.ecs. btan
(450
ORGANIZATION
5 9o. C tan
+0))
(Call longer side a)
j~
C5
p
91
a(A 0 B)=tan
tjLog a 45.8 Log tan
gi (45'04) 44 35 52.9 A5 4452.4 .1 . 9003034.33 Log tan 518 5.07 49 2 63 7Log Lo tan a (AA5 +B) ,+
CD _____
DATE
.
tr./A
p5
3 .1tan z4(A 3 gpp41
(ApBp,)
Sum=A Dif B5
89
C,
Log gsimC Log a
06 288(5+)° %SLgb1 fi5 Log
9.0547118 p
.0 60
Q .q
CsnAl]
ob
5
Red'
//45
9+B) L
pgg461 0.
(40t)7~
C5 (A+Bj 5 ) I
o
2
___
sph. ex.
00
Sphi. excess
3 j_
__
_
(Sketch)l
Q.op5? C
Log a
gg 724 O384
Log sin C5 Colog sin A5
a
1,0626Log c
"/gCk
BLACK~
CHECK COMPUTATION No.
STATION
SPHERICAL ANGLE
SPERCAL
PANE
.3.6080gg,
23
Red
___1
28LACK~
/WkS
3 13 ___12
Re 4CK kedRLACK
36
50
24.6
5/
57 /1
49.6 11.0
8'?
O.O
1
Pd36
2
Nc
1.3 ___12
*The subscripts
LACK
92M37J4U
.6
1796L49
~i2 ___19W4516
4OOEi
52 04 26
22.2, I 74
38
205
41
RedSLACK RedNic
0Qf 01 20
3,80 8a349 22.2. 5.2z81BO9 JJ,3 ~8Q6259
__
20.5
91 4476 4LQ16ML&
_
s and p on this form refer to spherical and plane angles respectively.
COMPUTED BY
DAIFORM
2446
Q~~
S.I
__________
23
3
LOGARITHM
A
DATE
1f
s
'CHECKED BY
w. C. A.
DT
2A7
1919 Figure 145Continued. 217
these computations are made, any missing angles can be obtained by combining angles fixed by the adjustment and computed angles. For example, computing the triangle RedHicksLincoln to obtain the diagonal LincolnRed, angle C at Hicks is found by adding the adjusted angles (1c) and (2c). The side HicksLincoln is the original fixed length, and the side HicksRed is taken from triangle 2 of the adjustment. Solution of the triangle on DA Form 1919 gives the length of the diagonal LincolnRed plus the angles at Lincoln and Red. The check computation on DA Form 1919 is made by solving the triangle RedHicksLincoln by the sine law using the computed angles at Red and Lincoln. The second triangle in the
1243
PROJECT
POSITION COMPUTATION,
LOCATION
To 3
24Z
*
ICS8/
00
00.00
47.co
8
6
vi.
'
9
lQ bogyI0,000) 2
0.977o2 0. ISA91612
C
z~s sina
_
1783,77
.38
y cor. _+fa
Y.
/
V
K (Vanl,000) 2 + COMPUTED BY
DA
DA s£4)7
12A_436
10 147~
08
463922
2.3 (togse=
0.
2
Y=s
x7 _Z xcor.
2"13

Ace (approx.) + F(°a')
ao7 . .S '
28.3
2
4223
2_aj , . (
H'aprx.i6j
Arcsio+V (V)
5. 2 2(
S19. 4Z1i
1+cs
Asd (approx.)
2
A
3
1fq 4 .
I
243.453
+ F(A)a
K (Va/1,000) + DT DA
2.
si 7
6.19
V
CHECKED BY
02/
°37.7
Yi.
Y2
3
376700S.ecOb(1000 ,52
H
2272 .42
Coa
Yu
s,
2
a39(.s
a921.73
=xIO.0) 2
*.4nA
Ae
DAT
ICL
yor. e8.~3 =+fa
233,453
Z. NOTE: For r tinder 8,00 meters omittermsunder the heavy black tie not in 5 heavyholdtypeorunderlined.
1FORM 6193
Figure 146.
218
4a3.~ ~ 7,32s 3q
°oI
A
38
sin a
°a.4
sin+7
7L (0 49
*'
00.00
i;a
035
)/
Arcsin+V (Va) __
4 2
25.43
0.0417=5
__1_______________+Icos,
Y2
35
00
LICL
j3 ,
+
C
x'=(approx.A")
3
To
1 d i'8
sin a
2216,47

Y5.y
Z11783,7568
'H
Q53.45 180
.4
First Angle ofTriangle
4.7628
~. .zS
8
1
________
x cor. _fb
p3~ a=(zd/10,000)2
22
To
.. ,A
_
03

nas
LIN=N
a'lR 1
122 43 29.
11929,454 *
3 '"__
06
I
cks
To 2 &
4a14
22 49 4629AikS~
aJ~0I ~LAt 3/
3dj
04
180 To 2
_
161
Z~O
___________
1l;a~/
~
.
To 1~JI
2'
sin
5RDER
OT
4
A
a
TRIANUILATO (For cdalmidcin couptaw)
(T
ORGANIZATION
2
a
check is RedLincolnBurdell, the fourth triangle in the quadrilateral. In this triangle the length Lincoln Burdell is taken from triangle 1 of the adjustment. The angle at Burdell is the sum of angles (lb) and (2a) of the adjustment. The angle at Red is found by subtracting the computed angle LincolnRedHicks from the adjusted angle (2b). Similarly, the angle at Lincoln is obtained by subtracting the computed angle RedLincolnHicks from the adjusted angle (1a). h. The list of directions can now be completed by adding the angles from the check computations to the directions previously corrected. i. Position computations are now made as in the direction method, using the adjusted angles and lengths (fig. 146).
Position computations for angle method (DA Form 1928).
PROJECT LOCATION
a2
ORGANIZATION
.__
&
Ree
1
2To __a
1
Red
?,9
3
13
First Angle of Triangle
30
4292.
*3
21.
02
2
Cos a
1
A'd
(loga
Ufl7 721/ y= .Y=s5co/a s Co. a i
2
a=x/0OO a(/1.0)
iiot
To
01 SL6.8 48 49,3
32
1
/es341LO~
2i
Arcsin+V
cor
s n1
=+
~
0'
4 2/7 212.opA2
1 +
cos °¢
4
/
0.817 6732,1210
0.3/
°a
8Y COMPUTED
IAe.
.
I DATEAf Ji
A
j
J14
.616
(approx.)
+ F (°>') 
59
.954
3
04
CHECKEDBY
4.C
3
45.08 00.00
154/4 152.43
LIA/IYIN 9~ 535W
f

0.04035936
0.12
o
s
8586.76 4' 4217 2/0.358
sin
K
209. 303
si
SS 
9
V

NOTE: Foraunder
0'
+0.5
+ F (°a)
2
(Va/l,000) +
1 0 182
0. 6 179 f/7g
l
/.ggggggpg
°
sin
or sin
°a (approx.)
6./3644
'DATE
0.43
15
Nnf
t+ cos
~ 2/7
7)
Arcsin=+V (Va
f
4
2f
Hx'= (approx.
Va Y2
)
H
4 2 2 7 91741 2 l
l
737
41473341
xcr
82586 7(/
~
Y
2.7363
x
a
=
S23.107
A'd
a=
h=(y/10,000)2
0. 90046126
a=xIO002 =x/000
c r
02 50,198

3.024 414 75328
Y=s cos a
Yo
41_ 43.3o5
A122 d) A'
(log
38.53
0434 93623
V
170.9
°a"
1 .3(C..384
I70A98
008
05 2/1648 1
*'
cosa
~.7
(Va) 15
2
K
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