fytmll Wlmvmity "pfcrarg THE GIFT OF [iitl^SA 2041 QB 145.A C 19 ne " Universit >"- ibrar y G |?i?iii3i'™P pliCation of ,he 'heory of lea 3 1924 004 410 522 x Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924004410522 Serial No. 9 DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, Superintendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION OSCAR S. ADAMS COMPUTER UNITED STATES COAST AND GEODETIC SURVEY Special Publication No. 28 WASHINGTON GOVERNMENT PRINTING OFFICE 1915 ADDITIONAL COPIES OF THIS PUBLICATION MAY BE PROCURED FROM THE SUPERINTENDENT OP DOCUMENTS GOVERNMENT PRINTING OFFICE WASHINGTON, D. C. AT 26 CENTS PER COPY CONTENTS. Page. General statement 7 Station adjustment 7 Observed angles g List of directions g Condition equations g Formation of normal equations by differentiation 9 Correlate equations xi Formation of normal equations 11 Normal equations 12 Discussion of method of solution of normal equations 12 Solution of normal equations 13 Back solution 13 Computation of corrections 13 Adjustment of a quadrilateral 14 General statement 14 Lists of directions 16 Figure 16 Angle equations 17 Side equation 17 Formation of normal equations by differentiation 17 Correlate equations 18 Normal equations 18 Solution of normal equations 19 Back solution 19 Computation of corrections 19 Adjustment of a quadrilateral by the use of two angle and two side equations. . 20 Angle equations 20 Side equations 20 Correlate equations 20 Normal equations 20 Solution of normal equations 21 Back solution ' 21 Computation of corrections 21 Solution of a set of normals including terms usually omitted 22 Discussion of the solution 22 Solution of triangles 23 Position computations, secondary triangulation 24 List of geographic positions 26 Development of condition equations for latitude and longitude closures 26 Equations in a net ^2 Adjustment of a figure with latitude, longitude, azimuth, and length closure conditions " Figure 34 Angle equations °" Azimuth equation Side equations Length equation 3 4 CONTENTS. Adjustment of a figure with latitude, longitude, azimuth, and length closure conditions — Continued. Page. Figure for latitude and longitude equations 37 Formation of azimuth equation 38 Preliminary computation of triangles 38 Preliminary computation of positions, primary form 40 Formation of latitude and longitude condition equations 50 Latitude equation 51 Longitude equation 51 Correlate equations 52 List of corrections 55 Normal equations 56 Solution of normals 58 Back solution 68 Computation of corrections 69 Final solution of triangles 71 Final computation of positions 76 List of geographic positions 91 Adjustment of triangulation by the method of variation of geographic coordi- nates 91 Development of formulas 91 Adjustment of a quadrilateral with two points fixed 94 Lists of observed directions 94 Preliminary computation of. triangles 95 Preliminary computation of positions 96 Formation of observation equations 98 Table for formation of normals, No. 1 100 Table for formation of normals, No. 2 101 Normal equations 101 Solution of normals 101 Back solution 102 Computation of corrections 102 Adjusted computation of triangles 103 Adjustment of three new pdints by variation of geographic coordinates 103 General statement 103 Figure 104 First method 105 List of directions 105 Lists of fixed positions •. 106 Preliminary computation of triangles 107 Preliminary computation of positions 110 Formation of observation equations 114. Table for formation of normals, No. 1 118 Table for formation of normals, No. 2 r 119 Normal equations 119 Solution of normals 120 Back solution 121 Computation of corrections 121 Final computation of triangles 122 Second method 125 Formation of observation equations 126 Table for formation of normals, No. 1 127 Table for formation of normals, No. 2 127 Normal equations 127 Solution of normals 128 COKTENTS. 5 Adjustment of three new points by variation of geographic coordinates— Con. Second method — Continued. Page Back solution 129 Computation of corrections 129 Final computation of triangles 130 Final computation of positions 134 Computation of probable errors 138 Adjustment of a figure with latitude and longitude, azimuth, and length condi- tions by variation of geographic coordinates 139 Table of fixed positions 139 Pre limin ary computation of triangles 140 Preliminary computation of positions 144 Figure 157 Formation of observation equations 158 Table for formation of normals, No. 1 162 Table for formation of normals, No. 2 166 Normal equations 168 Solution of normals 169 Back solution 174 Computation of corrections 175 Final computation of triangles 178 Final computation of positions 182 Adjustments by the angle method 196 Adjustment of verticals 197 General statement 197 Figure 197 Computation of elevations from reciprocal observations 198 Computation of elevations from nonreciprocal observations 199 Fixed elevations 200 Assumed and adjusted elevations 200 Formation of observation equations 200 Table of formation of equations 201 Computation of probable error 202 Formation of normal equations by differentiation 202 Table for formation of normal equations 204 Normal equations 204 Solution of normal equations 204 Back solution 205 Development of formulas for trigonometric leveling 205 General statement 205 Development of formulas 207 Examples of computation by formulas ." 214 Recapitulation of formulas 216 Notes on construction and use of tables 217 Tables 218 Notes on the developments 219 APPLICATION OP THE THEORY OP LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION By Oscak S. Adams Computer United States Coast and Geodetic Survey GENERAL STATEMENT In this publication the aim has not been to develop the theory of least squares, but to illustrate the application of the method to the problems arising in the adjustment of triangulation. The general idea has been to collect material in one volume that will serve as a working manual for the computer in the office and for such other members of the Survey as may desire to make these special applica- tions. It has not been deemed necessary to insert the derivation of formulae except in the case of a few special ones that are not usually found in the textbooks on least squares. For the general theory reference should be made to such books as the following: CrandaJl: Geodesy and Least Squares. Helmert: Die Ausgleichungsrechnung nach der Metode der kleinsten Quadrate. Jordan: Handbuch der Vermessungskunde, volume ]. Merriman: Textbook of Least Squares. Wright and Hayford: Adjustment of Observations. Some of the simpler cases are treated first, such as the local adjust- ment at a station, the adjustment of a simple quadrilateral, etc. After these is given the development of the condition equations for latitude and longitude closures, followed by a sample adjustment including the condition equations for these closures, together with the equations for length and azimuth conditions. A method of adjustment by the variation of geographic coordinates is then developed and applied first to a quadrilateral, then to a figure with a few new points connected with a number of fixed points. The same method is applied to the adjustment of a figure with lati- tude, longitude, length, and azimuth conditions. A sample adjust- ment of a vertical net is carried through and lastly there is given the development of the formulas for the computation of vertical observa- tions, together with examples of the method of computation. 7 8 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. STATION ADJUSTMENT The general rule followed by the observers of the Coast and Geo- detic Survey is to measure the angles at each station in the order of azimuth, thus giving rise to no conditions except the horizon closure. Occasionally, however, sum angles are observed and, when this is done, other conditions are introduced in addition to the horizon closure making it necessary to adjust the angles at the station by the method of least squares. If all angles were observed in the same way, the weight of each would be unity and the adjustment would be made without regard to weights. In the adjustment given below the angles were measured by the usual Coast and Geodetic Survey repetition method; that is, six measures of the angles with the tele- scope direct and six with it reversed for each set. A station has been chosen at which there are angles measured with one, two, and three sets in order to illustrate the method of weighting. Observed angles, Gray Cliff Observed stations Angle Weight V 3 Vl 3 Vl 3 V3 1 v t 2 »5 2 t>« 1 V7 1 »8 2 V, 1 ClO Adjusted final seconds* Boulder-Tower Tower-Tyonek Tyonek-Round Point Round Point-Boulder Bound Point-Birch Hill. . Birch Hoi-Boulder Boulder-Tyonek Tyonek-Birch Hill Bound Point-Moose Point Moose Point-Birch Hill. . . 65 06 26.61 30.9W9.3 19 46 27. 25 8 39 14.6 .6) •A\ .0M6.I .oj 14.61 18. 4 US.! 14.SJ 266 27 47.9 '"■ 23 20.6\ o1 „ a. or 1 - 8 23.1 200 04 22.21, 22. 3J' 84 52 56.2 75 02 35.0 22.2 64 32 12. 9 1 09, !}n. 27.5 15.9 46.7 22.4 24.3 57.4 38.3 11.3 11.1 List of directions , Gray Cliff Observed station Direction Adjusted final seconds* o / ft 00 00.0 65 06 29.3+Di 84 52 56.2+d+tij 93 32 12. l+Ki+»2+»3 158 04 23.5+Bi+Bs+Sa+oj 159 55 34. 7+n+Vi+Vl+Va+Vn // 29 9 13 3 24 6 Birch Hill 35.7 * These values result from the computation on p. 13. APPLICATION OF LEAST SQUARES TO TRIANGULATION. 9 There have been formed a complete list of directions without using five of the angles, each of which, then, gives rise to a condition, there being five conditions in all. The equations expressing these con- ditions are formed in the following manner: Angle Round Point-Boulder, observed, 266 27 47. 9+v t Angle Round Point-Boulder, from the list, 266 27 47.9- ■v, -v 2 —n Condition No. 1, 0=+0. O-H^+Uj+Wj+tii Angle Round Point-Birch Hill, from the list, 66 23 22. 6+v 9 +v lf> Angle Round Point-Birch Hill, observed, 66 23 21. 8+i; 5 Condition No. 2, 0=+0. 8-v 5 +v 9 +v 10 In the same way the other condition equations are formed. As a result there are finally: Condition equations 1. 0=+0. O+Vt+Vz+Vs+Vt 2. 0=+0. 8-i> 6 +i> 9 +i> 10 3. 0=~3.1+v 1 +v 2 +v 3 +v 6 +v l) +v 10 4. 0=+0.0+i; 1 +D 2 -r 7 5. 0=+3. 5+v 3 -v a +v s> +v 1<> FORMATION OF NORMAL EQUATIONS BY DIFFERENTIATION According to the theory of least squares, the most probable values will be determined by making the 2 p n v n 2 a minimum, subject to the given conditions. By the method of Lagrangian multipliers the formation of the normal equations can be much simplified. With the use of these the function u that is to be made a minimum is M=3 V+3 ^ 2 +3 v 3 2 +l V+2 v*+2 V+l V+l V+2 vf+1 v 10 2 -2 C^+v.+v^Vs +d 4 +0.0) - 2C 2 ( - v s +v,+v w +0.8) - 2C s (+v 1 +v 2 +v 3 +v a +v 9 +v i0 - 3.1) - 2C 4 (.+v l +v 2 -v 7 +0.0)-2C s (+v 3 -v s +v 9 +v 10 +3.5). The C's are merely undetermined multipliers, the values of which will be determined by the solution. The factor 2 is included to obviate later on the use of the fraction \ ; the minus sign is used for convenience. The function will be rendered a minimum if the partial differential coefficients with respect to v v v 2 , etc., are equated to zero. By this means ten equations will be formed, giving the ten v's expressed in terms of the C's. Differentiating with respect to v 1} v 2 , etc., in succession and equat- ing the results to zero, the following equations are obtained: 3u, -C.-Cj-C^O 3t- 2 -C^-Cj-C^O 3 „ 3 _ Cl -C l -C t =0 v l -C l =0 2u 5 +C 2 =0 2v s -C 3 =0 v 7 +C 4 =0 v s +C 5 =0 2v 9 -C 2 -C 3 -C B =0 v w -C 2 -C 3 -C,=Q 10 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Therefore "i =+* c,+i C 3 +J c 4 v 2 — H Ci+J C 3 +J C 4 v 3 =+i C,+J C 3 +i C 6 «« =+i ,„=+ C 2 +C 3 +C 5 Thus all of the v's are now expressed in terms of the O's. These can now be substituted in the condition equations forming five normal equations containing five O's and these equations may then be solved for the O's. If the normals are formed from these values, fractions will occur in practically all of the coefficients. This can be avoided by replacing C 1 by 6 0^, 2 by 6 C 2 ', etc. This is equiva- lent to using 12 Ci, 12 2 ', etc., in the original function instead of 2 C u 2 2 , etc., which, of course, is perfectly valid. The equations will then stand as follows: »! =+2 C,'+2 C/+2 C/ v 2 = +2 C/+2 C/+2 C/ v 3 =+2 C/+2 C/+2 C,' «« =+6 C/ «B = -3 C/ "6 =+3 C/ *7 = -6 C/ *« = -6 C,' ■»9 =+3 C/+3 C 3 '+3 C,' «io=+6 C/+6 C/+6 C 6 ' Dropping the prime and substituting these values in the first con- dition equation the following normal equation is obtained: 2 Ci+2 C 3 +2 CH-2 d+2 C 3 +2 C 4 +2 Ci+2 C 3 +2 C 6 +6 ^+0.0=0 +12 C, +6 C 3 +4 C 4 +2 C 5 +0.0=0 In a similar manner the other normal equations are formed, giving in all the following five equations : +12 Ci +6 C 3 + 4 C 4 + 2 C 6 +0.0=0 +12 C 2 + 9 C 3 +9 C 6 +0.8=0 + 6 C,+ 9 C 2 +18 C 3 + 4 C 4 +ll C 6 -3.1=0 + 4 Ci +4 Cj+10 C„ +0.0=0 + 2 Q+ 9 C 2 +ll C 3 +17 C 6 +3.5=0 This manner of forming the normal equations is called the method of correlates and is most conveniently carried out by means of a table of correlates formed as on page 11. After the determination of the O's by the solution of the normal equations, the v's may be computed from the equations of the v's APPLICATION OF LEAST SQUARES TO TEIANGULATION. 11 in terms of the O's. In the tabulated form below the first col- umn is multiplied by C 1} the second by 2 , etc. The sum of the first line multiplied by the - for that line gives v t ; so also for the other v's. Correlate equations 6 P 1 2 3 4 5 I v's* Adopted v's 1 2 +1 +1 +1 +3 +0. 618 +0.6 2 2 + 1 +1 + 1 +3 +0. 618 +0.6 3 2 +1 +1 +1 +3 -0. 050 -0.0 4 6 +1 +1 -1. 182 -1.2 5 3 -1 -1 +0. 585 +0.6 6 3 +1 + 1 +2. 133 +2.1 7 6 -1 -1 +1.230 +1.2 8 6 -1 -1 +3.234 +3.3 9 3 +1 +1 +1. +3 -0. 069 -0.1 10 6 +1 +1 +1 +3 -0. 138 -0.1 * These values result from the computation on p. 13. FORMATION OF THE NORMAL EQUATIONS After the condition equations are tabulated in correlates as above, the next step is the formation of the normal equations. In forming 1 n these the various products must be multiplied by - or by - in which p is the weight of the given v and a is some constant. (See the direct formation on p. 9.) It is most convenient to choose a so as to make most of the values integers, if this can be done without making the quantities too large. In this case 6 is the L. C. M. of the p's, hence it is chosen for a. The normal equations are formed by taking the algebraic sums of - times the products of the various columns. Normal No. 1 is, in symbols — ,6 4 p 1 ■ 2+2- • 1 • 3+2- ■ 1 • 4+2- r P — P - -p -■*+!+(*! -i-W) The algebraic sum of the sigma products in the formation checks or controls the formation of the normals. Each I line in the corre- lates is the algebraic sum of that line in the table. As is easily seen, the sum of the products of this column in the formation of the nor- mals should check the algebraic sum of the coefficients of the normal. On the first normal +12 + 6 + 4 + 2= +24, which is the same as the algebraic sum of the products in the correlates. The I column in the normals also includes the constant term. In the third normal +6 + 9 + 18+4 + 11 = +48. In the I columns of the normal +48-3.1 = +44.9. t ii is the constant term of the condition equation. 12 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Normal equations 1 2 3 4 5 7> J C's* 1 +12 + 6 + 4 + 2 +0.0 +24 -0.19700 2 +12 + 9 + 9 +0.8 +30.8 —0. 19531 3 +18 + 4 +11 -3.1 +44.9 +0. 71069 4 +10 +0.0 +18 • -0. 20547 5 +17 +3.5 +42.5 -0. 53917 * These values result from the computation on p. 13. DISCUSSION OF METHOD OF SOLUTION OF NORMAL EQUATIONS In the normal equations the coefficients in each equation occurring before the diagonal term are omitted, as the equations are sym- metrical with regard to the diagonal line. The set just given when written in full is as follows: 1 2 3 4 5 . 1 I 1 \+12 + 6 + 4 + 2 +0.0 +24 2 V+12 + 9 + 9 +0.8 +30.8 3 + 6 +V- rv+18 + 4 +11 -3.1 +44.9 4 + 4 + 4^ ^ 4 -9.72« 6 +5.53i> 6 +2.37fl 7 -3.50 Dg+1.13 v 9 ) Differentiating with respect to the v's in succession and equating to zero, there result after transposition the following equations: u^-d-2.03 C 4 i> 2 =+C 1 -C* 2 -3.35 C 4 „ 3 =+C 2 +5.38 C 4 « 4 =-Ci-C,+4.19 C 4 «b=+C,-9.72 C 4 „„=+Ci+5.53 C 4 « 7 =-C 2 -C 3 +2.37 C 4 D g =-d+C 2 -3.50 C 4 V=»+C x +Ci+l.lS C 4 ■ i' 10 = — C 2 fli 2 =+C 2 +C 3 By the substitution of these values in the four condition equations the following normal equations result: +6 Cj-2 C 2 +2 C 3 +4.65 C 4 -2.3=0 -2 Ci+6 C 2 +2 C 3 +2.86 C 4 +3.6=0 ' +2 d+2 C 2 +6 Cg-15.15 C 4 +2.2=0 +4.65 Ci+2.86 (7 2 -15.15 C 3 +206.0470 C 4 +1.4=0 These normal equations are formed most easily by means of the tabular form of the correlate equations given on page 18. * For triangles see p. 23. 18 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. The sum of the squares of each column gives the diagonal term in that equation in the normals. All coefficients before the diagonal term are omitted; each equation is read by starting at the top of the tabular form below, reading down the column to the diagonal term, and then along the horizontal line. Compare the full nor- mals given above with the tabular form below. After the diag- onal terms are determined column No. 1 in the correlates is multi- plied by column No. 2 and the algebraic sum of the products taken for the coefficient of normal No. 1 on No. 2; this is also the coefficient of No. 2 on No. 1. Column No. 1 times No. 3, with the algebraic sum of the products, gives the coefficient of No. 1 on No. 3 in the normals; also No. 3 on No. 1. Finally, the algebraic sum of the products of column No. 1 by column No. 4 gives the coefficient of normal No. 1 on No. 4. The algebraic sum of the products of col- umn No. 1 by the 2 column should check the algebraic sum of the coefficients of normal No. 1. To this should be added algebraically the constant term of normal No. 1 and the sum placed in the 2 col- umn of normal No. 1. (See the table of normals below.) In the same way the sum of the products of column No. 2 times column No. 3 is determined for the second normal, and by continuing the process all of the normals are formed. After the G's are determined by the solution of the normals the v'a are most conveniently computed by multiplying column No. 1 in the correlates by G t , column No. 2 by 2 , column No. 3 by G 3 , and column No. 4 by (7 4 . Then the algebraic sum of line No. 1 gives v t ; of No. 2, v 2 , etc. (See the computation of the v's on p. 19.) Correlate equations 1 2 3 4 2 • v's* Adopted v's 1)2 1 -1 -2.03 -3.03 -0.503 -0.5 0.25 2 +1 -1 -3.35 -3.35 +1.004 +1.0 1.00 3 +1 +5.38 +6.38 -0.501 -0.5 0.25 4 -1 -1 +4.19 +2.19 -0. 227 -0.2 0.04 5 +1 -9.72 -8.72 -0.015 -0.0 0.00 6 +1 +5.53 +6.53 +0.242 +0.3 0.09 7 -1 -1 +2.37 +0.37 +0. 663 +0.7 0.49 8 -1 +1 -3.50 -3.50 -0. 493 -0.5 0.25 9 +1 +1 +1.13 +3.13 -0. 170 -0.2 0.04 10 -1 -1 +0.099 +0.1 0.01 11 -1 -1 +0. 740 +0.7 0.49 12 +1 +1 +2 -0. 840 -0.8 0.64 3.55 Normal equations 1 2 3 4 1 2 C's* +6 -2 +6 +2 +2 +6 + 4.65 + 2.86 - 15.15 _i_on« r\Ain -2.3 +3.6 +2.2 1.1 A + 8.35 + 12.46 - 2.95 i mn annn +0. 6547 -0.0994 -0. 7401 APPLICATION OF LEAST SQUABES TO TBIANGULATION. 19 V3~55 - L r-= ±0.6. Solution of normal equations 1 2 3 4 ■n * +6 ft 1 -2 +0. 33333 +2 -0.33333 + 4.65 - 0.775 -2.3 +0. 38333 + 8.35 - 1. 39167 +6 -0. 6667 +5.3333 ft 1 2 +2 +0.6667 +2.6667 -0.50001 + 2.86 + 1.55 + 4.41 — 0.82688 +3.6 -0. 7667 +2.8333 -0. 53125 + 12.46 + 2.7833 + 15.2433 - 2.85814 +6 -0. 6667 -1.3333 +4 ft 1 2 3 - 15.15 - 1.55 - 2.205 - 18.905 + 4.72625 +2.2 +0. 7667 -1.4167 +1.55 -0. 3875 - 2.95 - 2.7833 - 7. 6217 - 13.355 + 3.33875 +206. 0470 - 3.6038 - 3.6465 - 89.3498 +109.4469 ft +1.4 +1.7825 -2.3428 +7.3257 +8. 1654 -0.07461 +199. 8070 - 6.4712 - 12.6044 - 63. 1191 +117.6123 - 1.07461 Bach solution 4 3 2 1 —0. 07461 -0. 3875 -0. 3526 -0.5312 +0. 0617 +0. 3701 +0.3833 +0.0578 +0. 2467 -0. 0331 -0.07461 -0. 7401 -0.0994 +0.6547 Computation of corrections 1 2 3 4 5 6 -0.6547 +0. 1515 +0.6547 +0.0994 +0.2499 -0.0994 -0. 4014 -0. 6547 +0. 7401 -0. 3126 -0. 7401 +0. 7252 +0.6547 -0. 4126 -0.5032 -0.5 -0.5008 -0.5 -0. 0149 -0.0 +0. 2421 +0.3 +1.0040 +1.0 -0.2272 -0.2 7 8 9 10 11 12 +0.0994 +0. 7401 -0. 1768 -0.6547 -0.0994 +0. 2611 +0.6547 -0. 7401 -0. 0843 +0.0994 +0. 7401 -0.0994 -0. 7401 +0.0994 +0.1 +0.7401 +0.7 -0. 8395 -0.8 +0. 6627 +0.7 -0. 4930 -0.5 -0. 1697 -0.2 20 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. ADJUSTMENT OP A QUADRILATERAL BY THE USE OF TWO ANGLE AND TWO SIDE EQUATIONS * (See fig. 1 on p. 16.) Angle equations 0=-2.3-(l)+(2)-(4)+(6)-(8)+(9) 0=+3.6-(2)+(3)-(7)+(8)-(10)+(12) Side equations Symbol Angle Logarithm Tabular differ- ence Symbol Angle Logarithm Tabular differ- ence -7+9 -5+6 -2+3 61 47 35.0 20 50 56.7 32 09 01.2 9. 9450972 9. 5513374 9. 7260280 +1.13 +5.53 +3.35 -4+5 -1+3 -7+8 26 40 23.5 133 53 46.3 31 03 42.5 9. 6521506 9. 8576926 9.7126180 +4.19 -2.03 +3.50 9. 2224626 9. 2224612 0=+1.4-2.03(l)-3.35(2)+5.38(3)+4.19(4)-9.72(5)+5.53(6)+2.37(7)-3.50(8) +1.13(9) -2+3 -11+12 -8+9 -5+6 32 09 01.2 91 32 03.8 30 43 52.5 20 50 56.7 9. 7260280 9. 9998442 9. 7084309 9. 5513374 +3.35 -0.06 +3.54 +5.53 -7+8 -4+5 -1+2 -10+11 31 03 42.5 26 40 23.5 101 44 45.1 25 15 16.2 9. 7126180 9. 6521506 9. 9908094 9. 6300613 +3.50 +4.19 -0.44 +4.46 8. 9856405 8.9856393 0=+1.2-0.44(l)-2.91(2)+3.35(3)+4.19(4)-9.72(5)+5.53(6)+3.50(7)-7.04(8) +3.54(9)+4.46(10) -4.40(11) -0.06(12) Correlate equations 1 1 2 3 4 X »'sf Adopted v's -1 -2.03 -0.44 - 3.47 -0.495 -0.5 2 +1 -1 -3.35 -2.91 - 6.26 +0.996 +1.0 3 +1 +5.38 +3.35 + 9.73 -0. 502 -0.5 4 -1 +4.19 +4.19 + 7.38 -0. 227 -0.2 5 -9.72 -9.72 -19. 44 -0. 013 -0.0 6 +1 +5.53 +5.53 +12.06 +0.240 +0.3 7 -1 +2.37 +3.50 + 4.87 +0. 659 +0.7 8 -1 +1 -3.50 -7.04 -10.54 -0.500 -0.5 9 +1 +1.13 +3.54 + 5.67 -0. 159 ^0.2 10 -1 +4.46 + 3.46 +0. 113 +0.1 11 -4.40 , - 4.40 +0.717 +0.7 12 +1 -0.06 + 0.94 -0.830 -0.8 Normal equations 1 2 3 4 1 2 3 4 1 2 C's* +6 -2 +6 + 4.65 + 2.86 +206. 0470 + 9.45 - 8.80 +208.2153 +276.0980 -2.3 +3.6 +1.4 +1.2 + 15.80 + 1.66 +423. 1723 +486. 1633 +0.2328 -0. 8398 +0. 16435 -0. 16302 * For triangles, see p. 23 f These values result from the computation on p. 21. APPLICATION OF LEAST SQUARES TO TEIAKGULATION. Solution of normal equations 21 1 2 3 4 1 I +6 Ci 1 -2 +0. 33333 + 4.65 - 0.775 + 9.45 - 1.575 -2.3 +0. 38333 + 15.80 — 2. 63333 +6 -0. 6667 +5.3333 d 1 2 + 2.86 + 1.55 + 4.41 - 0.82688 - 8.80 + 3.15 - 5.65 + 1.05938 +3.6 -0. 7667 +2.8333 -0. 53125 + 1.66 + 5. 2667 + 6.9266 - 1. 29875 +206.0470 - 3.6038 - 3.6465 +198. 7967 ft 1 2 3 +208. 2153 - 7. 3238 + 4. 6719 +205. 5634 - 1.034038 +1.4 +1. 7825 -2.3428 +0.8397 -0. 004224 +423. 1723 - 12.2450 - 5.7275 +405. 1998 - 2. 038262 +276.0980 — 14.8838 — 5.9855 —212. 5604 + 42.6683 C t +1.2 +3.6225 +3.0015 -0. 8683 +6. 9557 -0. 16302 +486. 1633 - 24.8850 + 7. 3379 -418. 9920 + 49.6240 - 1. 16302 Back solution 4 3 2 1 -0. 16302 -0. 00422 +0. 16857 -0. 5312 -0. 1727 -0. 1359 +0.3833 +0. 2568 -0. 1274 -0. 2799 -0. 16302 +0. 16435 -0. 8398 +0. 2328 Computation of corrections 1 2 3 4 5 . 6 -0.2328 -0. 3336 +0. 0717 +0. 2328 +0. 8398 -0. 5506 +0. 4744 -0. 8398 +0. 8842 -0. 5461 -0. 2328 +0. 6886 -0.6831 -1.5975 +1.5846 +0.2328 +0. 9089 -0. 9015 -0. 0129 -0.0 -0. 4947 -0.5 -0. 5017 -0.5 -0. 2273 -0.2 +0. 2403 +0.3 +0.9964 +1.0 7 8 9 10 11 12 +0. 8398 +0. 3895 -0. 5706 -0. 2328 -0.8398 -0. 5752 +1. 1477 +0.2328 +0. 1857 -0. 5771 +0. 8398 -0. 7271 +0.7173 -0. 8398 +0. 0098 +0. 7173 +0.7 +0. 1127 +0.1 -0. 8300 -0.8 +0.6587 +0.7 -0. 1586 -0.2 —0.5001 -0.5 22 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. SOLUTION OF A SET OF NORMALS INCLUDING TERMS USUALLY OMITTED „ A set of four normal equations is solved below with inclusion of the terms omitted in the Doolittle method of solution. Solution of normals 1 2 3 i 1 S +6 -lCi -2 +2 (1) +2 -2 +4.65 -4.65 -2 +0.33333 + 2 - 0.33333 + 4.65 - 0.775 -2.3 +0.38333 + 8.35 - 1.39167 +6 -0.6667 +5.3333 -1 Ca +2 +CK6667 (1) -2.6667 (2) +2.86 +1.55 -4.41 + 2 + 0.6667 + 2.6667 - 0.50001 + 2.86 + 1.55 + 4.41 - 0.82688 +3.6 -0.7667 +2.8333 -0.53125 + 12.46 + 2.7833 + 15.2433 - 2.85814 + 6 - 0.6667 - 1.3333 + 4 - ICa -15. 15 - 1.55 (1) - 2.205 (2) +18.905 (3) - 15.15 - 1.55 - 2.205 - 18.905 + 4. 72625 +2.2 +0. 7667 -1.4167 +1.55 -0.3875 - 2.95 - 2.7833 - 7.6217 - 13.355 + 3.33875 +206.0470 - 3.6038 - 3.6465 - 89.3498 +109.4469 - 1C( +1.4 +1.7825 -2.3428 +7.3257 +8.1654 -0.07461 +199.8070 - 6.4712 - 12.6044 - 63.1191 +117.6123 - 1.07461 DISCUSSION OF THE SOLUTION The quantities in heavy type are the ones omitted in the Doolittle method of solution of normal equations. They sum up to zero with the possible variation of a few units in the last place of the solution. This shows that the method is one of curtailed substitution. It can also be seen that the quantity in the I column is the direct sum of all the quantities in each horizontal line including those in heavy type. All of the quantities in heavy type occur in the regular solution. This is of value in the control of the solution. If an equation fails to check the 2 column after it is added up, the error can generally be located by adding back through noting that the coefficient is changed in sign because it is multiplied by — 1. Note the product of equation No. 1 on No. 4; — 1.55 and + 1.55 are the products of No. 1 on No. 3 and No. 2, respectively; —4.65 is the coefficient of No. 4 on No. 1 with sign changed. The method is the same in all cases. Care should be taken with such coefficients as No. 2 and No. 3 on No. 1. They have the same value with opposite sign. If a mistake should be made on them the 2 column control would not catch it. Care should be taken not to make a mistake in the r; column and a compensating one in the I column. There is most danger of this in the addition. The control would not catch this and it would take much labor to correct it later. APPLICATION OF LEAST SQUAEES TO TEIANGULATION. 23 After each equation is added, it should be added horizontally to check the I column. If the check fails an error has been made and it must be found before proceeding. A slight variation in the last place of the solution is of course unavoidable. After the division of each equation by the reduced diagonal term, a horizontal addition should be made (including, of course, —1) to check the correctness of, the division. No time is ever lost in using care in the solution of the equations. It takes so much time and labor to rectify a mistake later that every means should be employed to detect and correct it in the solution. The larger the set, the more important it is to be on guard against errors. It is possible to carry a set through with almost absolute assurance that the solution is correct. If, in a given equation, the solution fails to check and the check of adding back through is satisfied, a mistake has been made somewhere in the solution columns and a compensating mistake in the I column. This can be caught by building up the omitted columns to the left of the given equation. They should each sum up to zero. If any one does not, the mistake in addition has been made in that equation in the column of the one being eliminated. Solution of triangles * Symbol Station Observed angle Correction Spheri- cal angle Spherical excess Plane angle Logarithm -8+9 -1+2 - 4+ -6 A, Ai A3—A1 Az—At 30 101 47 43 44 31 tt 52.5 45.1 20.2 +0.3 +1.5 +0.5 52.8 46.6 20.7 0.0 0.1 0.0 52.8 46.5 - 20.7 3.772745 0. 291568 9.990809 9. 867787 4. 055122 3.932100 +2.3 0.1 -10+11 -1+3 -5+6 A2—A1 A, A, A, Ai-Ai A t — A? 25 133 20 15 53 50 16.2 46.3 56.7 +0.6 +0.0 +0.-3 16.8 46.3 57.0 0.0 0.1 0.0 16.8 46.2 57.0 3.772745 0. 369936 9. 857693 9.551339 4.000374 3. 694020 +0.9 0.1 -10+12 -2+3 -7+8 A2—A3 A, Ai A, A\—Az A\—At 116 32 31 47 08 03 20.0 61.2 42.5 -0.9 -1.5 -1.2 19.1 59.7 41.3 0.1 0.0 0.0 19.0 59.7 41.3 3. 932100 0. 049306 9. 726023 9. 712614 3.707429 3. 694020 -3.6 0.1 -11+12 -4+5 -7+9 A\—Az At A, A, Ai—Ai At-Ax 91 26 61 32 40 47 03.8 23.5 35.0 -1.5 +0.2 -0.9 02.3 23.7 34.1 0.1 0.0 0.0 02.2 23.7 34.1 4. 055122 0. 000156 9. 652151 9. 945096 3. 707429 4. 000374 -2.2 0.1 * For the method of solution of triangles see United States Coast and Geodetic Survey Special Publi- cation No. 8, p. 6. 24 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Position computation, STATION A a a. Secondl angle / a AO. a' 1st term 2d and 3dl terms / -J0 Az to Ai Ai and As Ai to Ai As to As First angle of triangle 156 + 101 1 20 44 26.6 46.6 258 + 05 8 13.2 05.9 180 78 30 00 13 43 00.0 19.1 52.8 60 + 1 5( 01. 089 56. 720 As X AX X' 7.8 9.9 1.6 149 34 9 19. 237 15.880 60 60 5 5f it 6 29 57.809 8 cos a B h 3. 932100 9.990544 8. 508600 0.313737 As 3.932100 9.314765 8. 509299 «2 sin' a C 6420 8109 5750 149 ft" D 25 3. 5122 2.3224 03. 357 -57. 0380 + 0.3184 1. 756164 AX sin 1(4+4 9. 50279 0.3183 0.0001 5.8346 -56. 7196 9 sin a A' sec^' AX ') 2. 744981 9.941572 2. 744981 // -555.8800 —AO. 2. 686553 -485.89 STATION ^! a Secondl angle J a AO. «' A 4>' h(.4+4') 1st term 2d and 3d\ terms / -A 33 05. 749 g cos a; B . h 3. 694020 9. 972328 8.508601 0.313313 A t 3. 694020 9.538954 8. 509299 sin! a C 8804 4466 5750 149 h* D 29 3.484 2.322 11.442 +55.2425 + 0.0979 1. 742273 AX sin H4+4') -Ja 8.99020 0.0978 5.806 +55. 3404 a sin or A' sec^' 2. 488262 9.941507 1 AX 2. 488262 -307. 7953 2.429769 If -269. 01 * For an explanation of the forms for computing differences of latitude, longitude, and azimuth ; United States Coast and Geodetic Survey Special Publication No. 8, pp. 6-11. APPLICATION OF LEAST SQUARES TO TRIANGULATION. 25 secondary triangulation * STATION As a Third! angle/ a. Aa a' 4 A4> #' 1st term 2d, 3d, and' 4th terms Ai to As As and At A\ to As A3 to Ai 60 58 1 56. 416 58. 608 CO 50 57. 808 + 1 60 57 57 +118.0821 + 0. 5258 +118.6079 since A' ■ sec ' cos a. B 4. 055122 9. 507767 8. 509295 Ai A* s 2 sin 2 a C X AX 8. 11024 9. 95248 1.65837 9. 72109 0. 5261 0.0003 -0.OO06 4. 055122 9. 976241 8. 508600 0. 313737 2.853700 -714. 0029 AX sin i(,+4') 2.S5370O 9. 941676 2. 795376 336 - 47 180 108 4.1144 2.3221 6.4665 25 -ft s 2 sin 2 a E 08.4 20.7 47.7 24.3 00.0 12.0 -.1 57.360 54.003 03.357 2.072 8.063 6.640 6.775 STATION A, a. Third! angle/ a At A ifo+tf') 1st term 2d and 3d! terms / .A3 to As Ai and As As to Ai At to As 60 60 60 56 02 +111.9959 + 0.0639 +112. 0598 s since A' sec # 57.809 52.060 55 05. 749 cos a B AX 3. 707429 9. 865259 8. 508601 0.313313 2. 394602 +248.0858 As A< 3. 707429 9. 832475 8. 509298 2.049202 AX sin i(4+6') -Aa sin 2 a C 2. 394602 9. 941540 2.336142 n +216. 84 X AX 180 227 149 7. 41486 9. 73052 1. 65808 8. 80346 0.0636 149 h 2 D 29 4.098 2.322 6.420 19.1 41.3 37.8 36.8 00.0 01.0 03.357 08. 086 11. 443 - 1 26 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. List of geographic positions, Turnagain Arm, Alaska. Valdez datum Station Latitude and longitude Seconds in meters Azimuth Back azimuth To sta- tion Distance Logarithm A, A t O 1 It 60.56 57.809 149 25 03.357 60 55 05.749 149 29 11.442 1789.4 50.5 177.9 172.4 O 1 It 78 13 19.1 108 57 11.9 110 18 41.9 135 33 58.7 227 06 01.0 O 1 II 258 05 13.2 288 46 47.7 290 14 12.9 315 27 11.4 47 09 37.8 ^2 A, A x A, Meters 8552. 6 11353.3 4943. 3 10008.6 5098.3 3.932100 4.055122 3.694020 4.000374 3. 707429 DEVELOPMENT OF CONDITION. EQUATIONS FOR LATITUDE AND LONGITUDE CLOSURES After the conditions arising from the closure of triangles and from the equality of sides or lengths computed hy different routes have been satisfied, cases frequently arise where azimuth, latitude, and longitude conditions must be satisfied. There is given now a devel- opment of a form of condition equations that will bring about a closure in geographic position. Discrepancies in latitude and longitude arise whenever a chain of triangulation or a traverse closes on itself. The discrepancies may be distributed throughout the whole loop or in a selected portion of it, depending upon the circumstances. Of course the most rigid adjustment would require the discrepancies to be distributed through- out the whole chain. At times, however, this would require more labor than the importance of the work would justify. Also some parts of the loop may be much better determined than other parts, in which case the more poorly determined part should be required to make up the discrepancies. The discussion of the form of equations to be employed to effect the closure without discrepancies will be based upon the position computation formulae employed by the United States Coast and Geodetic Survey. (See United States Coast and Geodetic Survey Special Publication No. 8, p. 8.) The amount to be distributed being, of course, small compared with the total change in latitude and longitude, the only term of the latitude computation formula that need be considered is the first one. No appreciable changes due to the adjustment will take place in the other terms. . The formation of the equations must always start from a line fixed in length and azimuth. If a scheme of triangulation should start from a fixed line and run to two points which are fixed in position but are not the ends of a single line, then the formation of the equa- tions for each of the two points must start from the fixed line. There are, of course, two elements that enter into the determination of the position of any point as computed from a known point; these APPLICATION OF LEAST SQUARES TO TEIANGULATION. 27 are the distance from the known point and the azimuth of the line from the known to the unknown point. In the triangle 12 3, let 1 and 2 be fixed in position, and let us consider what change in the position of 3 will be produced by small changes in angles A, B, and C. The length to be carried forward is 1 to 3. Starting with the length 1 to 2, we have log 1 to 3 =log 1 to 2— log sin i?+log sin A. The change in length, then, depends upon the changes in angles A and B and the change in azimuth of the line 1 to 3 depends upon the change in angle O. The angles A and B, therefore, are called the length angles and angle the azimuth angle. If we can derive a linear expression for the effect of each of these separately, the total effect will be the sum of the two. Let d x and d B represent the change of the log sin for a change of one second in the angles A and B; v A and v B the number of seconds change in angles A and B, respectively. Then the change in log sin A will equal c )> ( m which <£ B is the computed latitude of 3 and c is the latitude of 1), and the product divided by M; this will give the change in seconds in the latitude of 3 due to the change in length of 1 to 3. 28 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Next must be considered the change in latitude due to the change in the azimuth angle C. If s is the length in meters of 1 to 3, the length of the small arc through which 3 turns is equal to s(y c ) arc 1" (as ds=rdd for a circle about the origin in polar coordinates), v- c equals the number of seconds " change in angle C, and arc 1" A is included to reduce this angle to circular measure. Let 3 be the original posi- tion of 3 and 3' the position due to a small rotation of 1 to 3 about 1. 1 \ 3 to3'=^ c arc 1" kg. 3. . The azimuth of 3 to 3' is 90° + «. The change in latitude due to s(v c ) arc 1" is equal to -s(v c ) arc l"cos (90° + a) times the B factor in the position computation, = +s(v c ) arc 1" sin a B = + (v c ) arc 1" Bs sin a But X B — k c = s sin a A' sec ' Therefore s sin oc = -t~t ^-rr A sec 9 B arc 1" Therefore the change in latitude = ., ,, U B — A c ) (v c ). In a similar way the change in longitude due to a change in length is, *B ^ M 1 + dM-d B (v. <>] and the change in longitude due to the change in azimuth is, \s(i? c )arc .1" sin (90° + «) A' sec ' = + (v c ) arc 1" s cos a A' sec$' — s cos a'B = c (neglecting the small terms) _—J B -^c s cos a= — B Therefore change in longitude = % (<£ B — cj> c ) (v c ) . The usage is to point off the log change for one second of arc as an integer in the sixth place of logarithms ; therefore as a number the tabular difference 1 : 10 6 and 10 6 The total change in seconds of latitude in the triangle is The total change in longitude is, 10 6 M [ + d A (v & )-d B (v B ) hF } A' sec 4>' arc I 7 B~ (4>B-c)(V c )* * The upper sign behu? used for a right azimuth angle, the lower sign for a left. APPLICATION OF LEAST SQUABES TO TSIANGULATION. 29 In this way the change could be determined for each triangle in the chain and the sum placed equal to the discrepancy, but this would require a very great amount of work. If any change takes place in the first triangle while the remaining triangles are for the moment supposed to remain fixed, this length change and azimuth change will affect not only this triangle, but will persist in each succeeding triangle. As a consequence the change of length and azimuth in the first triangle will be felt in the computation of every point after it in the chain. Let cj> n and X n be the computed $ and X of the end point. The change in the first triangle will apply not merely to B — c , etc., but to n — c , etc. Therefore the change in the final position due to the changes in the first triangle is, for latitude, ^[+^)-^)]±|^ «.-*,) (, c) * and for longitude, %Fy[+a>J- w]t An sec |; arc *" (^-fc) (to* In the same way the change in the final position due to changes in the second triangle can be determined, and so on through the whole chain. Each triangle will have an A, B, and O angle, A being the length angle next to the known side; B, the one opposite the known side; and C, the azimuth angle. The equations will finally stand 0= +(4-V)+i- [^*>J-^*b(»b)] n is the computed latitude of the final point and n ', the fixed lati- tude ; so also for ^ and V. It is exact enough to take c and X n — X c to minutes and tenths of a minute, so that it is advisable to divide the equations by 60 since, as they stand, 4> n — c , etc., are in seconds. Also it is best to mul- tiply through by 10 6 M to remove this factor from the denominator of the first summation. •" Upper sign (or right azimuth angle, lower for left. 30 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. Then we have M '60 ^±/° 6 ^£^>-^>- * o= +^io 6 (4-V)"+^[(4-4)' J-(^'-^c)' *.(tfe)] + i . T 1Q . ^Asec^arcJ" (^ ^y^). * The J. and 5 factors change so slowly that for any chain they can be taken for the mean.<£ and also sec cj> n can be used in the same way. A table can then be prepared for functions designated as a t and a 2 and defined as follows: , -~« ir-B arc 1" the A and B factors and the being used at a convenient interval. A table has been computed for latitudes starting at 24° for intervals of 4° up to 56°. The minus sign is used with a 2 in order that the same sign can be used on the directions of the azimuth angle for both latitude and longitude equations. If the discrepancy to be made up by the adjustment is large, or if the chain extends over a' great distance of latitude, it would be best to compute the values of a x and a 2 using A n and n and the B for the mean . If the chain to be adjusted extends principally east and west, in place of n — c a summation of the first terms Qi) in the position computations should be used. I c h would then replace n — C i the sign being used that would conform to n —4>c- These quantities should then be used throughout in forming the equations. If the, latitude and longitude equations are to be included in the main adjustment and the equations all solved simultaneously, the computation of the positions through the chain must be made with one length carried through the figures by means of the observed plane angles; that is, the angles as observed each diminished by \ of the spherical excess of the triangle. This could be done by carrying the length through a selected chain of triangles and then computing each of the various positions over a single line. Both lines of the triangle could not be used because the observed plane angles must be used in carrying the length and, under ordinary circumstances, the triangle would not be closed. To obviate this difficulty, it is best to use only the observed A and B angles and to conclude the angle, using, of course, the concluded correction symbols on this * Upper sign for right azimuth angle, lower for left. APPLICATION OF LEAST -SQUARES TO TBIANGTJLATION. 31 angle. This method gives a much more reliable determination of the discrepancy, as it furnishes a check on each position, and thus prevents a mistake being left in ' the computation. If the figure adjustment is carried out first, there is no need to follow this method as the triangles would then be closed. . In this, case it is the general custom of the United States Coast and Geodetic Survey to choose the best chain of triangles and to form the equations through them, using the angle method in place of the direction method. Equations with absolute terms equal to zero must be included for the various triangles in order to hold them closed; also, if a length equation is included in the figure adjustment, it must be retained with zero discrepancy to hold the length. If the figure ends on a fixed line and a length equation is not put in the figure adjustment, the dis- crepancy must be put on the length equation used with the latitude and longitude equations. After adjustment is made for these final discrepancies the cross lines are computed by two sides and the included angle. The best results are probably obtained by the solution of all the equations at once, but this entails so much work that the angle method is often used in chains of minor importance. We have finally: M £= 10° = 7238.24 60 0=+7238.24(^,-^)"+2[(^ 1 -^ )'* a («a) 0=+723S.2i(L a -X n >y' + I[(k n -l c )'d A {v 1 ) In the equations v K , v B , and v c would be replaced by their correc- tion symbols, care being taken to use v c = —v A — v B , if the azimuth angle has been concluded in carrying the position computation through the chain. If an azimuth equation occurs, the constant term must be corrected by _|_ (A n -X n .) xsine of the mean <£, this being the amount that the azimuth will change from the changes in the back azimuths due to the changes in longitude. It should be noted that whenever a discrepancy of position is adjusted into a section of a loop, an external condition is placed upon the chain, as at best only part of this discrepancy is due to errors in the chain, the rest being due to the remainder of the loop. It is necessary to hold some parts of the triangulation fixed; otherwise when a loop closure is put in it would frequently be necessary to readjust nearly all of the triangulation of the country. The result is, however, that some chains of triangulation, excellent in them- selves, get some rather large corrections due to the position closure. 91865°— 15 3 32 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. EQUATIONS IN A NET In the adjustment of a quadrilateral, use is made of the two kinds of condition equations that are necessary for the adjustment of any figure that does not contain external conditions such as length, azi- muth, or loop closure. In fact most figures can be broken up into successive quadrilaterals. In forming the length equation, use is made of the two length angles in the various triangles passed through. In fig. 5, on page 37, the length angles are lettered A and B. The angle omitted is the azimuth angle of the given triangle. The log sin of the A angle, is added to the first length and the log sin of the B angle to the final length. So with all of the triangles through which the length is carried. The discrepancy is found and the equa- tion formed in the same way as in the case of an ordinary side equa- tion. See the formation of the length equation on page 37. If the spherical angles are used a correction for arc to sine must be applied to each length. (See the table of these corrections in Special Pub- lication No. 8, p. 17.) An azimuth equation is formed by adding algebraically to the first azimuth the various azimuth angles up to the second line fixed in azimuifti. When passing from one end of a line to another, the azimuth difference due to convergence of the meridians, must be applied as determined in the computation of positions. The alge- braic sum of the v's upon these angles must make up the discrepancy between the computed and fixed azimuths. See the computation on page 38. The determination of the exact number of side and angle equa- . tions in a net and the manner in which they come in, is one of the difficulties encountered by a beginner in the adjustment of trian- gulation. This is especially true if the net is somewhat complicated. The best method for this determination is to plot the figure point by point. By plotting the triangle Tower, Turn, and Dundas, in the figure on page 34 one angle equation is determined. Add Lazaro by the lines Lazaro to Turn and Lazaro to Tower. This gives another angle equation, making two. Another angle equation and a side equation are obtained by putting in the line Lazaro to Dundas. This makes a total of three angle equations and one side equation for the quadrilateral, just as it should be. Next plot Nichols by the lines Nichols to Lazaro and Nichols to Tower; this is a closed' triangle and gives a fourth angle equation. Put in Tow Hill by the lines Tow Hill to Nichols and Tow Hill to Lazaro; this does not give an angle equation as it is not a closed triangle. Draw the line Tow Hill to Tower; this gives a second side equation. In this way one can continue through the whole figure. If a full line Nichols to Turn were in the figure, it would give another angle and another side APPLICATION OP LEAST SQTJABES TO TEIANGULATION. 33 equation. The angle equation added would have to include the directions on this line as would also the side equation. This method shows at once where the equations come in and what new v's must appear in the equations. Lines sighted over in only one direction have no effect on the number of angle equations. If the closed part of the figure is plotted, omitting all of the extra lines — that is, putting in each station with only two lines from those already plotted, a closed framework of the figure will be formed. The first triangle requires three lines, those after the first require two lines. The number of angle equations in the framework of the figure is thus equal to the number of lines in the figure minus the number of stations plus one. Every full line added to this framework gives another angle equation. Therefore, the whole number of angle equations in a net is equal to the whole number of full lines minus the number of occupied stations plus one. The lines sighted over in one direction have the same effect on the number of side equations that the full lines have. If the full frame- work of the figure is plotted with two lines to each station from those already determined, no side equation will as yet appear in the figure. Every extra line put in gives a side equation. The first triangle fixes three stations; the stations after these require two lines to be used in plotting them. Thus the number of lines needed to plot the framework is equal to twice the number of stations minus three. The full number of side equations will then be equal to the number of all the lines minus twice the number of all the stations plus three. Let w = total number of lines. n' = number of lines sighted over in both directions. S = total number of stations. S' = number of occupied stations. Then The number of angle equations in a net =71/ — 8' + 1. The number of side equations in a net = 71 — 2 S + 3. These formulas should be used to check the number determined by directly plotting the figure. In figure 4 on page 34, 71 = 41 7*/ = 38 S = 18 S' = 17 Therefore number of angle equations = 38—17 + 1= 22. , number of side equations = 4 1 — 36 + 3= 8. For convenience in solution it is best to use triangles with the larger angles for the angle equations, reserving the small angles to be used in the side equations. This will keep the large coefficients 34 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. in the side equations from appearing on the same directions as are used in the angle equations and will aid in the solution of the normals. The small angles need to appear in the side equations, as their tabular differences are proportionally much less affected by the dropping of decimal places than are those of the larger angles. ADJUSTMENT OF A FIGURE WITH LATITUDE, LONGITUDE, AZIMUTH, AND LENGTH CLOSURE CONDITIONS Ham Lim <&. South Twin Ken Khwain /found J ^- Beaver v^r/j&jW^Sf Cat *" .a/4 — &s& 5 & ■'Mid I Snipe Lazaro Turn Dundas "Ibiver V Torn"/// Fig. 4. In this figure, in addition to the angle, side, and length conditions, there are included conditions for azimuth, latitude, and longitude. APPLICATION OP LEAST SQUARES TO TRIANGULATION. 35 Ahylc equations - (1)+ (2)- (4)+ ( B )-(10)+(ll) - (1)+ (3)- (5)+ (6)-(12)+(13) - (2)+ (3)- (9)+(10)-(12)+(14) - (8)+ (9)-(14)+(16)-(25)+(26) -(16)+(17)-(23)+(25)-(31)+(32) -(16)+(18)-(24)+(25)-(33)+(34) -(23)+(24)-(30)+(32)-(34)+(35) -(17)+(19)-(29)+(31)-(36)+(37) -(17)+(21)-(28)+(31)-(42)+(45) -(19)+(21)+(36)-(38)-(42)+(44) -(21)+(22)-(40)+(42)-(49)+(52) -(20)+(22)-(47)+(48)-(49)+(51) -(40)+(43)-(46)+(47)-(51)+(52) -(40)+(41)-(50)+(52)-(56)+(58) -(39)+(40)-(52)+(55)-(59)+(61) -(50)+(55)-(57)+(58)-(59)+(60) -(53)+(55)-(59)+(62)-(67)+(68) -(54)+(55)-(59)+(63)-(71)+(72) -(62)+(63)-(66)+(67)-(71)+(73) -(63)+(65)-(70)+(71)-(74)+(75) -(64)+(65)-(74)+(76)-(77)+(78) -(69)+(70)-(75)+(76)-(77)+(79) Azimuth equation 0=(-7.1*-0.2)-(l)+(2)+(9)-(10)-(14)+(21)+(40)-(42) -(52)+(55)-(59)+(63)+(70)-(71)-(75)+(76) Computation of correction to azimuth constant: log 0.277=9. 442 log sin. mean 0=9. 912 log correction =9. 354 correction = —0. 2 Side equations Symbol Angle Logarithm Tabular differ- ence Symbol Angle Logarithm Tabular differ- ence - 9+11 -12+13 -1+2 O 1 If 93 11 39.1 25 52 .38.1 24 17 25.4 9.9993248 9. 6399291 9.6142236 -0.12 +4.34 +4.67 -13+14 -1+3 -10+11 16 37 43.4 110 36 08.7 42 00 30.0 9.4566222 9. 9712965 9. 8255810 +7.05 -0.79 +2.34 9.2534775 9. 2534997 -22.2-5.46(l)+4.67(2)+0.79(3)+0.12(9)+2.34(10)-2.46(ll)-4.34(12)+11.39(13) -7.05(14) -7+9 +15-16 +25-27 -25+26 111 09 20.5 ■ 36 08 04.3 30 04 51.8 9. 7706188 9. 7000325 9. 4403482 -0.81 +2.88 +3.64 + 7-9 +14-15 -25+27 -8+9 21 40 38.8 9. 5674745 +5.30 89 23 18.6 9. 9999753 +0.02 48 16 10.2 9. 8729041 +1.88 9. 4403539 -5.7 -4.49(7)+1.88(8)+2.61(9) -5.30(14)+8!8(15) -2.88(16) -0.74(25)+3 .64(26) -2.90(27) * See computation on p. 38. 36 COAST AND GEODETIC STJEVEY SPECIAL PUBLICATION NO. 28. Side equations — Continued Symbol Angle Logarithm Tabular differ- ence Symbol Angle Logarithm Tabular differ- ence -16+18 -23+24 -30+31 38 29 18.8 27 51 39.2 9 31 16.8 9. 7940405 9. 6696203 9. 2185744 + 2.65 + 3.98 +12. 55 -24+25 -30+32 -17+18 12 35 33.3 126 31 06.3 15 56 21.5 9. 3384902 9. 9050754 9. 4387305 +9.43 -1.56 +7.37 8. 6822352 8. 6822961 0= -60.9-2.65(16)+7.37(17) -4.72(18) -3.98(23)+13.41(24) +12.55(31)+1.56(32) -9.43(25) -14.11(30) £/ -17+19 -28+29 -42+44 35 17 43.3 9 58 42.9 51 05 11.2 9. 7617712 9.2387486 9.8910323 + 2.98 +11.97 + 1.70 -29+31 -44+45 -19+21 16 21 23.9 23 37 23.0 43 39 34.4 9. 4496664 9. 6028386 9. 8390831 +7.17 +4.81 +2.21 8.8915521 8. 8915781 -26.0-2.98(17)+5.19(19) -2.21(21) -11.97(28)+19.14(29) -7.17(31) -1.70(42) +6.51(44) -4.81(45) -49+52 -20+21 -46+47 89 47 52.8 26 37 18.2 33 20 40.5 9. 9999973 9. 6513730 9.7401044 +0.01 +4.20 +3.20 -21+22 -46+48 -51+52 26 22 55.0 105 41 48.0 35 09 14.0 9. 6477280 9.9834943 9. 7602524 +4.25 -0.59 +2.99 9.3914747 9.3914747 0=+0.0-4.20(20)+8.45(21)-4.25(22)-3.79(46)+3.20(47)+0.59(48)-0.01(49) +2.99(51) -2.98(52) -39+41 -59+60 -50+52 106 37 20.7 27 49 50.7 37 39 24.5 9.9836521 9. 6691879 9.7859918 -0.59 +3.99 +2.73 -60+61 -50+55 -40+41 22 08 21.3 125 14 18.4 63 10 28.9 9.5761788 9.9120934 9. 9505530 +5.17 -1.49 +1.07 9. 4388318 9.4388252 0=+6.6+0.59(39)+1.07(40)-1.66(41)-4.22(50)+2.73(52)+1.49(55)-3.99(59) +9.16(60) -5.17(61) -71+73 -59+62 -63+54 59 25 24.7 58 43 17.2 18 17 51.0 9. 9349784 9. 9317900 9. 4968619 +1.24 +1.28 +6.37 -62+63 -53+55 -72+73 43 56 28.3 59 03 08.4 22 50 29.2 9.8413093 9.9333037 9.5890357 +2.19 +1.26 +5.00 9.3636303 9.3636487 0=-18.4-5.11(53)+6.37(54)-1.26(55)- +5.00(72) -3.76(73) •1.28(59)+3.47(62)-2.19(63)-1.24(71) -74+76 -63+64 -69+70 94 10 29.2 28 13 32.1 61 25 13.8 9. 9988462 9. 6748099 9. 9435708 -0.15 +3.92 +1.15 -64+66 -69+71 -75+76 50 11 06.3 102 20 31.6 33 30 23.0 9. 8854273 9. 9898451 9. 7419627 +1.76 -0.46 +3. 18 9. 6172269 9. 6172351 0=-8.2-3.92(63)+5.68(64)-l.76(65)-l.6l(69)+Ll5(70)+0.-10(7l)+0.l5(74) +3.18(75) -3.33(76) APPLICATION OP LEAST SQUARES TO TKIANGULATION. Length equation 37 Symbol Angle Logarithm Tabular differ- ence Symbol Angle Logarithm Tabular differ- ence Tun -4+6 -2+3 -Dundas 113 41 86 18 59.6 43.3 -6 4.266771 9.9617359 9. 9990997 -0.92 +0.14 Ham-South Twin -10+11 40 00 30.0 -12+14 42 30 21.5 -1 3.898371 9.8255810 9.8297327 +2.34 +2.30 • -7+9 111 09 20.5 9.9696969 -0.81 / + 7- 9 \ +14-15 }21 40 38.8 9.5674745 +5.30 +15-16 +25-27 -23+25 -28+31 -21+22 } 36 40 26 26 08 27 20 22 04.3 12.5 06.8 55.0 9.7706188 9.8121311 9.6470132 9. 6477280 +2.88 +2.47 +4.25 +4.25 -25+27 -31+32 -42+45 -49+52 89 116 74 89 23 59 42 47 18.6 49.5 34.2 52.8 9.9999753 9.9498922 9.9843478 9.9999973 +0.02 -1.07 +0.58 +0.01 -39+40 -54+55 -63+65 -69+70 42 40 78 61 26 45 24 25 51.8 17.4 38.4 13.8 9.8292505 9.8147959 9.9910544 9. 9435708 +2.30 +2.44 +0.43 +1.15 -59+61 -71+72 -74+75 -77+79 49 36 60 85 58 34 40 04 12.0 55.5 06.2 23.4 9. 8840631 9.7752272 9.9404164 9. 9983924 +1.77 +2.84 +1.18 +0.18 2. 6534656 2. 6534708 0=-5.2-0.14(2)+0.14(3)+0.92(4)-0.92(6)-4.49(7)+4.49(9)+2.34(10)-2.34(ll) +2.30 (12)-7.60 (14)+8.18 (15)-2.88(16)-4.25(21)+4.25(22)-2.47(23)+5.37(25) -2.90 (27) -4.25 (28)+3.18 (31)+1.07(32)-2.30(39)+2.30(40)+0.58(42) -0.58(45) +0.01 (49)-0.01 (52)-2.44 (54)+2.44(55)+1.77(59)-1.77(61)-0.43(63)+0.43(65) -1.15 (69)+1.15 (70)+2.84 (71) -2.84(72)+1.18(74)-1.18(75)+0.18(77) -0.18(79) *» South Twin Lim — Rounds ten ^Beaver ^B Turn Nichols A "Dundas Tower Tow Hi// Fia. 5. 38 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Formation of azimuth equation Turn-Dundas 357 -1+2. 24 Turn-Tower 21 Azimuth difference* — 180 .Tower-Turn 201 +9-10 51 Tower-Lazaro 150 Azimuth difference — 180 Lazaro-Tower 330 f 101 -14+21 | 22 I 78 Lazaro-Roimd 173 Azimuth difference —180 Round-Lazaro 353 +40-42 63 Round-Cat 289 Azimuth difference +180 Cat-Round 109 33 07.7 17 25.4 50 33.1 07 07.6 43 25.5 11 09.1 32 16.4 14 01.2 18 15.2 38 55.1 32 57.3 57 17.7 27 25.3 01 37.8 25 47.5 49 17.6' 36 29.9 05 59.6 42 29.5 Cat-Round 109 -52+55 87 Cat-Beaver 197 Azimuth difference +180 Beaver-Cat 17 -59+63 102 Beaver-Lim » 119 Azimuth difference —180 Lim-Beaver 299 +70-71 40 Lim-South Twin 258 Azimuth difference +180 South T win-Lim 79 -75+76 33 South Twin-Ham 112 Fixed azimuth 112 Discrepancy 42 29.5 34 53.9 17 23.4 01 40.2 19 03.6 39 45.5 58 49.1 05 20.3 53 28.8 55 17.8 58 11.0 06 48.1 04 59.1 30 23.0 35 22.1 35 29.2 -7.1 Preliminary computation of triangles nation Of Symbol Observed angle Cor- rection Spher- ical angle Spher- ical excess Plane angle Logarithm 0.2 0.2 0.1 24 29.8 17 30.7 59.5 4. 266771 0. 1744195 9. 6142483 9. 9617359 0.5 4. 0554388 4. 4029264 0.6 0.7 0.6 51 20.9 42.6 10 56.5 4. 4029264 0. 1702686 9. 9990996 9. 8916183 1.9 4. 5722946 4. 4648133 • 2.2 2.2 2.1 21 40 36.6 05.0 18.4 4. 5722946 0. 4325371 9.8653119 9. 9696986 6.5 4. 8701436 4. 9745303 3.6 3.6 3.6 36 15.0 44.3 08 00.7 4. 9745303 0. 0000248 9. 9105723 9. 7706083 10.8 4; 8851274 4. 7451634 -10+11 - 4+ 6 -12+14 -2+3 -14+15 -7+9 -25+27 -15+16 Turn-Dundas Tower Turn Dundas Tower-Dundas Tower-Turn Turn-Tower Lazaro Turn Tower Lazaro-Tower Lazaro-Turn Lazaro-Tower Tow Hill Lazaro Tower 42 00 30.0 30.9 113 41 59.6 42 30 21.5 86 18 43.3 57.1 38.8 47 10 07.2 111 09 20.5 Tow Hill-Tower Tow Hill-Lazaro Lazaro-Tow Hill Nichols Lazaro Tow Hill Nichols-Tow Hill Nichols-Lazaro 23 18.6 54 28 47. 04.3 * See position computation, p. 40. APPLICATION OF LEAST SQUARES TO TKIANGULATION. 39 Preliminary computation of triangles — Continued Desig- nation of angle Symbol Station Observed angle Cor- rection Spher- ical angle Spher- ical excess Plane angle Logarithm B C A -31+32 -23+25 Lazaro-Nichols Ken Lazaro Nichols Ken-Nichols Ken-Lazaro o / It 116 59 49.5 60.2 40 27 12.5 0.8 0.7 0.7 22 48.7 32 59.5 11.8 4. 7451634 0. 0501070 9. 5837509 9. 8121294 4. 3790213 4. 6073998 2.2 B C A -42+45 -28+31 Lazaro- Ken Bound Lazaro Ken Round-Ken Round-Lazaro 74 42 34.2 20.9 26 20 06.8 0.6 0.7 0.6 78 33.6 57 20.2 06.2 4. 6073998 0. 0156525 9. 9918811 9. 6470108 4. 6149334 4. 2700631 1.9 B A C -49+52 -21+22 Lazaro-Round Cat Lazaro Round Cat-Round CatKLazaro 89 47 52.8 26 22 55.0 12.6 - 0.2 0.1 0.1 63 52.6 54.9 49 12.5 4. 2700631 0. 0000027 9. 6477276 9. 9529926 3. 9177934 4.2230584 0.4 B C A -59+61 -39+40 Cat-Round Beaver Cat Round Beaver-Round Beaver-Cat 49 58 12.0 56.4 42 26 51.8 0.1 0.0 0.1 87 11.9 34 56.4 51.7 3. .9177934 0. 1159371 9. 9996132 9. 8292503 4. 0333437 3. 8629808 0.2 B C A -71+72 -51+55 Beaver-Cat Lira Beaver Cat Lim-Cat Lim-Beaver 36 34 55.5 47.2 40 45 17.4 0.0 0.1 0.0 102 55.5 39 47.1 17.4 3. 8629808 0. 2247728 9. 9893057 9. 8147959 4.0770593 3. 9025495 0.1 B A C -74+75 -63+65 Beaver-Lim South Twin Beaver Lim South Twin-L South Twih-B 60 40 06.2 78 24 38.4 15.5 m saver • 0.0 0.1 0.0 40 06.2 38.3 55 15.5 3. 9025495 0. 0595836 9. 9910543 9. 8162528 3. 9531874 3. 7783859 0.1 40 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION TOWEB a Second angle a 4a a' $ A s COS a B h 1st term 2d term 3d and 4th \ terms J —Aj> Turn to Dundas Dundas and Tower 357 + 24 33 17 07.7 30.9 Turn to Tower Tower to Turn First angle of triangle 21 50 7 38.6 07.6 180 201 42 43 00 31.0 30.0 - 54 48 12 06. 742 39. 419 >85 128- 159 Turn X AX X' 5.7609 2. 3672 130 + 56 8 04. 052 43.993 54 4.4029 9. 9676 8.5097 264 416 251 35 sii 5 1 2 <* C 7.323 8.80 9.14 1.55 Tower Arg. s AX 131 -h « 2 sin E 2a 34 48. 045 2. 8803 7. 9471 6. 4574 2.8802931 +759.0896 + 0.3175 3d term 4th term s sin a A' sec^' 9. 50172 +0.0134 -0. 0019 -6 4. 4029264 9. 5706383 8. 5087480 0. 2370138 8. 1281 -11 + 5 AX sin £(0+tf') sec hU) 7. 2848 2. 7193259 9. 9117440 7 +759. 4071 + 0.0115 +759. 4186 54 41 47 IX 2. 7193259 +523. 9935 Corr. - 6 -Act 2. 6310706 // +427. 63 STATION LAZAEO Second angle * A) 48.045 2.2091 8. 0172 6. 4528 6. 6791 2. 7543502 9. 9112981 2. 6656483 // +463.07 STATION LAZARO Third angle 4 A COS a B 1st term 2d term 3d and 4th \ terms / -A iW+tf') Tower to Turn Lazaro and Turn Tower to Lazaro Lazaro to Tower o / 54 35 17 54 52 27.323 30.520 4. 5722946 9. 9398800 8. 5097404 -1051.7559 + 1.2O04 -1050.5555 0. 0358 -1050. 5197 54 44 12. 6 57. 843 sin 2 c C 3d term 4th term « sin a A' sec*' AX 9. 14459 9. 38353 1. 55122 Tower Lazaro 0.07934 +0. 0258 +0. 0100 4. 5722946 9. 6917656 8. 5087409 0. 2401420 3. 0129424 +1030.2498 Arg. 8 AX Corr. X AX 6.0428 2. 3683 8. 4111 -25 + 1S 201 - 51 180 131 -h 2 sin 2 < E 31.0 57.1 33.9 01.2 48.045 10.250 58.295 AX sin £(*+*') sec i(A4) 3. 0219 8.5281 6. 4516 8.0016 3. 0129424 9. 9119609 2. 9249033 it +841.20 42 COAST AND GEODETIC SXJEVEY SPECIAL PUBLICATION NO. 28., Preliminary 'position computation, STATION TOW HILL Second angle a 4> 4$ 4-' COS a B 1st term 2d term 3d and 4th \ terms / *«+*') Lazaro to Tower Tower and Tow Hill Lazaro to Tow Hill Tow Hill to Lazaro First angle of triangle 04 57. 843 31. 870 25.973 -1 Lazaro Tow Hill X 4. 9745303 9. 9794727 8. 5097191 3. 4637221 +2908.8550 + 2.8851 +2911. 7401 + 0. 1299 +2911.8700 54 28 41.9 sin 2 < C 3d term 4th term sin a A' sec$' J\ 9.94906 8. 95521 1. 55589 (W 6.9283 2. 3667 0. 46016 9.2950 +0. 1972 -0. 0673 -117 4.9745303 9. 4776065 8. 5087606 0. 2315532 Arg. s -159 + 42 3. 1924389 Corr. -117 +1557. 5388 330 + 47 180 197 21 131 131 -h 5 2 sin 2 a E 47 32.7 07.2 39.9 07.7 32.2 38.8 58.295 57. 539 J\ sinj(0+*') sec H4$) 3. 4637 8. 9043 6. 4597 9. 9105687 108 3. 1030184 +1267.70 STATION NICHOLS Second angle cos a B 1st term 2d term 3d and 4th \ terms / H+') Lazaro to Tow Hill Tow Hill and Nichols Lazaro to Nichols Nichols to Lazaro First angle of triangle 54 52 57. 843 27. 012 30.831 Lazaro Nichols 4. 7451634 9. 4909674 8. 5097191 2. 7458499 +556. 9933 + 10.0559 +567. 0492 - 0.0374 +567.0118 54 48 14. 3 « 2 sin 2 < C 3d term 4th term s sin a A' sec^' Ji J). 9. 49033 9. 95620 1. 55589 (W)« 5. 5072 2. 3667 1.00242 7. 8739 +0. 0075 -0.0449 +92 4. 7451634 9. 9781021 8.5087447 0. 2384494 Arg. 8 AX - 56 +148 3. 4704688 Corr. + 92 +2954.3966 17 + 54 71 180 251 89 131 132 -h « 2 sin 2 a E 11 39.9 47.9 27.8 14.3 13.5 18.6 58. 295 14. 397 A). sin J(*+^') sec t(J4>) 12. 692 +1 2. 7458 9. 4465 6. 4597 8. 6520 3. 4704688 9. 9123203 3. 3827891 +2414. 3 APPLICATION OF LEAST SQUARES TO TBIANGULATION. 43 •primary triangulation — Continued STATION TOW HILL Third angle a Ac # COS a B 1st term 2d term 3d and 4th \ terms / ««+«') Tower to Lazaro Tow Hill and Lazaro Tower to Tow Hill Tow Hill to Tower 54 54 4. 8701436 9. 8881103 8. 5097404 04 27.323 01. 351 25.972 Tower Tow Hill 3. 2679943 +1853. 5073 + 7. 8786 +1861.3859 - 0. 0351 + 1861.3508 54 19 56. 6 sin 2 a C 3d term 4th term sm a A' sec^' X AX 9. 74029 9. 60494 1. 55122 (*g) 2 6.5397 2.3683 0. 89645 8.9080 +0. 0809 -0. 1160 +16 4. 8701436 9. 8024699 8. 5087606 0. 2315532 Arg. < JX - 98 +114 3. 4129289 ' Corr. + 16 +2587. 7893 150 -111 180 218 131 48 33.9 20.5 13.4 02.4 11.0 48.045 07. 789 -h 2 sin 2 a E F A\ sin §(#+*') see iU4>) 7.972 3. 4129289 9. 9097770 44 3. 3227103 +2102. 38 STATION NICHOLS Third angle ' cos a B 1st term 2d term 3d and 4th ' terms -A A4> #' < COSH B h 1st term 2d term 3d and 4th \ terms / JW+tf') Lazaro to Nichols Nichols and Ken 71 + 22 57 33 27.8 00.2 Lazaro to Ken Ken to Lazaro First angle of triangle 94 30 30 28.0 53.7 180 273 116 59 59 34.3 49.5 54 + 52 1 57. 843 37. 056 480 731 589 lazaro X A\ X' 4.0250 2. 3667 131 + 21 37 58.295 45. 861 54 4. 607399 8. 895391 8. 509719 8 7 1 54 s 2 sin* C 3 a 4.899 9.21 9.99 1.55 Ken W£) 2 Arg. 3 J\ 131 -h * 2 sin 2 E a 59 44. 156 2.0125 9. 2121 6. 4597 2. 0125106 - 102.9226 + 5. 8614 3d term 4th term s sin a A' sec^' 0. 76800 +0.0002 +0.0048 +58 4. 6073998 9. 9986545 8. 5087403 0. 2404328 6. 3917 -29 +87 A\ sin JW+tf') sec JU0) 7. 6843 3.355234 9. 912798 - 97.0612 + 0.0050 - 97. 0562 54 53 46. 4 J> 3.3552332 +2265. 8607 Corr. +58 —Aa 3. 268032 n +1853.67 STATION ROUND Second angle a Aa 4 A COS a B 1st term 2d term 3d and 4th \ terms / iW+tf') Lazaro to Ken Ken and Round Lazaro to Round Round to Lazaro First angle of triangle 54 4. 2700631 9. 9971678 8. 5097191 -598. 3427 I- 0.0161 + 0.0084 57.843 58. 318 02 sin 2 e C 56. 161 Lazaro Round 3d term 4th term sin a A' sec$' 8. 54013 8. 11255 1. 55589 8. 20857 +0.0083 +0.0001 4. 2700631 9. 0562746 0. 2419389 2.0770129 // +119.4024 X AX (*fj) 2 5. 5538 2. 3667 7.9205 Arg. s A\ -6 Corr. -6 94 + 78 173 180 353 74 131 -h 2 sin 2 a E AX sin J(*+*') sec J(J0) -Aa 28.0 20.9 48.9 37.8 11.1 34.2 58.295 59. 402 57. 697 2. 7769 6. 6527 6. 4597 5. 8893 2. 077013 9.913183 +97.77 APPLICATION OF LEAST SQUARES TO TRIANGULATION. 45 primary triangulation — Continued STATION KEN Third angle J4, COS a B 1st term 2d term 3d and 4th \ terms / -H ito+*') Nichols to Lazaro Ken and Lazaro Nichols to Ken Ken to Nichols 54 54 4.3790213 30. 831 04. 068 8. 5097307 2. 8225729 -664. 6192 + 0. 5381 -664.0811 + 0.0132 -664. 0679 54 49 02. 9 s 2 sin 2 e C 34.899 8. 75804 9. 41947 1.55337 3d tem 4th term sin a A' sec$' J\ Nichols Ken 9. 73088 +0. 0103 +0. 0029 4. 3790213 9. 7097334 8. 5087403 0. 2404328 2. 8379276 -688. 5375 Arg. Corr. X JX 5.6450 2. 3676 8.0126 251 - 40 210 180 30 131 -h s 2 sin 2 a E 13.5 12.5 12. 693 28.537 59 44. 156 2. 8226 8. 1775 6.4553 JX sin Xt+tj)') sec iGW —ia 7. 4554 2. 837928 9.912392 2. 750320 -562. 76 STATION ROUND Third angle a ia COSa B 1st term 2d term 3d and 4th terms m+v) Ken to Lazaro Round and Lazaro Ken to Round Round to Ken 54 55 4. 6149334 9. 5799436 8. 5097172 02 34. 899 21. 261 56. 160 +1 2. 7045942 -506. 5172 + 5.2288 -501.2884 + 0.0270 -501. 2614 54 58 46.0 sin 2 c C 3d term 4th term sm a A' sec^' J\ 9. 22987 9. 93222 1. 55631 Ken Round MP 0. 71840 +0.0058 +0.0212 +49 4. 6149334 9. 9661083 8. 5087369 0. 2419389 3. 3317224 -2146. 458 Arg. s JX Corr. X 5.4091 2. 3666 7. 7757 -30 +79 +49 273 - 26 180 68 29 35 131 -h 6' 2 sin 2 a E 23 34.3 06.8 27.5 17.8 45.3 44. 156 46. 458 JX sin i(#+0') sec JC0) 57. 698 -1 2. 7046 9. 1621 6. 4604 8.3271 3. 331722 9. 913254 3. 244976 // -1757.83 46 COAST AND GEODETIC SURVE Y ' SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION CAT a Second angle a Aa. a' J0 8 COS a B h 1st term 2d term 3d and 4th \ terms / -Atj, JW+tf') Lazaro to Round Bound and Cat 173 + 26 27 22 48.9 55. Lazaro to Cat Cat to Lazaro First angle of triangle 199 + 50 4 43.9 ' 21.5 180 19 89 55 47 05.4 -.1 52.8 54 + t 52 8 57. 843 28.257 812 164 589 jazaro X A\ X' 5.4124 2. 3667 •131 21 5 58.295 19.363 55 4.2230 9.9734 8.5097 584 102 191 01 si] 2 s 2 l 2 a C 6.100 8.44 9.06 1.55 Cat Arg. s J\ 131 -h s 2 sin E a fi 38. 932 2.7062 7.5078 6. 4597 2. 7061877 -508.3791 + 0.1158 3d term 4th term s sin a A' sec$' 9. 06365 II +0.0060 +0.0005 -3 4. 2230584 9. 5308211 8. 5087375 0. 2416677 7. 7791 -5 +2 A\ sin J(#— tf ') sec i(.A$) 6.6737 2. 504284 9.913117 -508. 2633 + 0.0065 -508. 2568 54 57 4 2.0 i\ 2. 5042844 -319.3629 Corr. -3 -Aa 2. 417401 -261. 46 STATION BEAVER tx Second' angle a Aa a' A$ <*' s COS a B h 1st term 2d term 3d and 4th \ terms J -A$ Cat to Round Round and Beaver 109 + 87 t 42 34 58.1 56.4 Cat to Beaver Beaver to Gat First angle of triangle 197 + 17 1 54.5 40.2 180 17 49 19 58 34.7 12.0 - o 55 + i 01 3 It 26.100 45. 192 60 65 84 Cat X A\ y 4.705 2.366 131 16 2 38. 932 02. 273 55 3.8629 9.9798 8.5097 S08 982 090 05 si 1 s 2 l 2 a C 1.292 7.72 8.94 1.55 Beaver ,Arg. 8 A\ 131 -h « 2 sin E a 4 J e 36.659 .353 .672 .464 2. 3525880 -225. 2102 + 0.0170 3d term 4th term a sin a A' sec$' 8.2309 +0.0012 -1 3. 8629808 9. 4732671 8. 5087360 0. 2423463 7.071 -1 A\ sin i($+4>') sec iU4>) 5.489 2.087330 9.913657 -225. 1932 +0.0012 -225. 1920 55 03 1 8.7 d\ 2. 0873301 it -122. 2728 Corr. -1 -At X 2.000987 it -100.23 APPLICATION OP LEAST SQUARES TO TBIANGULATION. 47 ■primary triangulation — Continued STATION CAT Third angle a Act a' A# s COS a " B h 1st term 2d term 3d and 4th 1 terms J -A Kound to Lazaro Cat and Lazaro 353 - 63 26 49 11.1 12.6 Round to Cat Cat to Round 289 + 36 5 58.5 59.6 180 109 42 58.1 55 1 02 1 56. 161 30. 061 56 31 36 Round \ A\ X' 3.9069 2. 3658 131 23 7 57. 697 18. 765 55 3.917793 9. 525975 8. 509707 1 I 01 S 2 sin' C 2 or 6.100 7.83 9.94 1.55 Cat Arg. a A\ 131 -h s 2 sin' E ct 16 38. 932 1.9535 7.7837 6. 4643 1. 9534761 // +89. 8413 + 0. 2199 3d term 4th term s sin or A' sec$' 9.3423 // +0. 0002 -0. 0002 +2 3.9177934 9. 9740335 8. 5087375 0. 2416677 6. 2727 -1 +3 A\ sin i(4+A') seo t(A4>) -Act 6.2015 2. 642232 9. 913558 +90.0612 +90.0612 t it 55 02 11. 1 2. 6422323 -438. 7654 Corr. +2 2. 555790 // -359. 58 STATION ] 3EAVER or Third angle a Act c? A4, s COS a B h 1st term 2d term 3d and 4th 1 terms / — A4> Round to Cat Beaver and Cat Round to Beaver Beaver to Round O 289 - 42 36 26 58.5 51.8 247 + 10 7 06.7 40.0 180 67 17 46.7 55 + 02 2 11 56. 161 15. 131 67 91 86 Round A\ X' 4.264 2.366 131 23 9 57. 697 21. 038 55 4. 033343 9. 588856 8.509707 7 2 1 05 S 2 sin 2 C 1 a 1.292 8.06 9.92 1.55 Beaver Arg. 8 A\ 131 -h s 2 sin E or 14 t 36. 659 .132 .996 .464 2. 1319070 -135.4899 + 0.3584 3d term 4th term s sin a A' sec^' A\ 9. 5544 +0. 0004 +0.0004 +3 4. 0333437 9. 9645661 8.5087360 0. 2423463 6.630 -2 +5 . i\ - sin JW+0') seo i(^) -Act 6.592 2. 748992 9. 913723 -135. 1315 + 0.0008 -135. 1307 i . it 65 04 03. 7 - -3. 7489924 -561. 0382 Corr. +3 - -2. 662715 11 -459. 95 91865°- ■15 i 48 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION LIM Second angle Aa 4>' s COS a B 1st term 2d term 3d and 4th ' terms Beaver to Cat Cat and Lim Beaver to Linn Lim to Beaver 55 First angle of triangle 11.292 08.973 20.265 3.9025495 9. 6988310 8. 5097044 2. 1110849 -129. 1471 + 0.1735 -128. 9736 + 0.0006 55 06 15. 8 s 2 sin 2 a C 3d term 4th term sin a A' sec$' A\ 7. 8051 9. 8751 1. 5591 +0. 0004 +0.0002 +2 3.9025495 9. 9375769 8. 5087351 0. 2427356 2. 5915973 // +390. 4787 Beaver Lim Arg. s Corr. X A\ 4.222 6.588 -1 +3 +2 17 +102 119 180 299 131 + 131 -h s 2 sin 2 a E 34.7 47.2 21.9 20.3 01.6 55.5 36. 659 30. 479 21 07. 138 2.111 7.680 6.465 4X sin !(#+$') sec i(/l$) 6.256 2. 591597 9.913918 2. 505515 +320. 27 STATION SOUTH TWIN Second angle a Aa a! 4> A *' s COSa B 1st term 2d term 3d and 4th \ terms / -A) 2.549 7.354 6.464 2.'428468 9. 913752 2. 342220 +219.90 STATION SOUTH TWIN Third angle a Aa a' J' +2 3. 9531874 9. 9919164 8. 5087347 0. 2429024 Arg. s JX -2 +4 2. 6967411 Corr. +2 JX -497. 4404 299 - 40 180 79 58 05 131 -h s 2 sin 2 ot E 01.6 15.5 46.1 48.1 34.2 07. 138 17. 440 49. 698 -1 JX sin i(0+<4') sec J(J^) 1.744 7.890 6.467 6.101 2. 696741 9. 914053 2. 610794 -408. 12 50 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. B CM CO CM CM OS CM CM CM OS CM CO CO co- O CO 00 05 I— 1- CM CM 3 © CO 00 *©■ o CM CO CM CM CO CO O CO CM iO iO 5g a; +7 g LL J BO © "*ci CM t*. CO CO CO CO ^ CO ■* •©■ + 1 B OS iO i-H CO »< 1 a IO -©■ •^ © ~H CO Tfl iO CD in cm CM CO >-H IO CO Tt« CM CM CM IO € s >o Cm tH O +1 + ,_5 _J ,-* w CO "* CM CJ •e-l o + i 1 I 1 1 1 1 1 -< CM W ^ CM CO CO ^H CO ft 00 •!■§ 7 CO o iO o 5D iO OS 1 + + 1 + + + + + ^< CM iO CM i-l CM cq + + 1 1 + 4- + + + + + i-H 1 1 >-H ++ 1 i 1 1 1 1 I-' 1 ■^ CM iO - rH CO ■** c» i-H i-H r~ tH "*■ ^ +1 T to CO CM ,_; ■*!.£ 1 1 1 + 1 1 1 i 1 *©■ s ~' ^ ~* i-t ■* CO © CO CO CO CO ^ 8 r~ o 00 l- 1 © CO OS os o ,-* pj ,< + + 1 I 1 1 1 1 1 1 t> ,_, §3 CO CO CO 0s 1 8 LO iO iO iO *©■ + + + + + + + + + + o 3 Os CD OS Ol OS 3 s CM r-i s ©' s ,_! tH 8 ■* i-H i< i-H o O CO CO CO CO CO CO CO CO CO CO CO ,_, © CO to s <-* OS tf CO IO CM CM CM CM J ■©■ o o ° s iO s iO 3 »o iO IO »o £ iO IO m »o § | a g 1 o ca 1 fl 1 a ! o 02 H H h3 hJ vA >-) K O PQ 3 APPLICATION OF LEAST SQTJAEES TO TEIANGULATION. 51 Latitude equation 0= -2.5334 -0.14(2)+0.14(3)+0.39(4) -0.39(6) -0.69(7)+0.69(9)+0.67(10) -0.67(11) -0.66(12) - 1.36(14)+ 1.25(15) - 0.55(16) - 0.09(21) 4- 0.09(22) - 0.49(23) 4- 0.93(25) -0.44(27) - 0.76(28)4- 0.49(31) + 0.27(32) - 0.20(39) + 0.20(40) - 0.02(42) + 0.02(45) +0.14(49) - 0.14(52) - 0.10(54)+ 0.10(55)+ 0.08(59) - 0.08(61) + 0.10(63) - 0.10(65) +0.07(71) -0.07(72)+ 0.11(74) -0.11(75) Longitude equation •0= -20.0499+1.20(2) -1.20(3) -0.59(4)+0.59(6)+0.41(7) -0.41(9) -0.35(10)+0.35(11) +1.39(12) - 0.34(14) - 0.75(15) - 0.30(16)+ 0.67(21) - 0.67(22) - 0.34(23) + 0.07(25) +0.27(27) - 0.17(28) - 0.29(31)+ 0.46(32) - 0.16(39) + 0.16(40) - 0.62(42) + 0.62(45) +0.20(49) - 0.20(52) - 0.07(54) + 0.07(55) - 0.32(59) + 0.32(61) + 0.07(63) - 0.07(65) -0.16(71)+ 0.16(72) -0.06(74)+ 0.06(75) 52 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Correlate 1 2 3 4 5 6 7 8 9 10 11 12 13 11 15 16 17 IS 19 20 21 22 1 -1 +1 -i" +i -1 +i" 2 -1 +1 3 4 5 -l +i 7 8 -1 +1 9 -1 +1 10 -l +i 11 12 -l +i -1 13 14 + 1 -1 15 16 + 1 -1 + 1 -1 17 -1 -1 18 + 1 19 +1 -1 20 -1 21 +1 +1 -1 +1 22 +1 23 -1 -i' +i -1 +1 24 25 -1 +1 +1 26 27 28 -1 29 -1 30 -1 31 -1 + 1 + 1 + 1 32 -i" +i + 1 33 34 -1 + 1 35 36 -1 +1 + 1 37 38 -1 39 -1 + 1 40 -1 -1 -1 + 1 41 42 -1 -1 +1 43 +1 44 +1 45 + 1 46 -1 +1 47 -1 +1 -1 48 49 -1 50 -1 -1 51 + 1 -1 + 1 52 +1 + 1 -1 53 -1 54 -1 + 1 55 + 1 + 1 +1 56 -1 57 -1 + 1 -1 +1 58 + 1 -i" 59 -1 -1 90 61 +i 62 +1 +i" -1 + 1 63 -1 64 1 -1 + 1 65 1 1 1 1 1 + 1 ;M\ APPLICATION OP LEAST SQUARES TO TEIANGULATION. 53 equations 23 24 25 26 27 28 29 80 31 I 32 33 X 34 I -0.55 +0.47 +0.08 -1 + 1 - 3.55 + 2.39 + 1.16 - 0.28 - 1.00 + 1.28 - 9.26 + 0.88 + 8.39 + 1.90 - 1.91 + 1.92 + 2. 14 -16.31 +16.86 - 7.88 - 0.56 + 0.53 + 0.52 - 5.20 + 6.56 + 1.42 - 5.70 + 1.34 + 5.69 + 4.64 - 5. 97 - 7.38 + 0.92 - 2.41 + 4.91 + 3.96 - 1.00 0.00 + 1.00 0.00 + 1.00 - 1.00 - 3.07 + 2.73 - 0.66 -2.23 + 1.00 + 1.65 + 0.58 - 4.79 + 3.20 + 1.59 - 1.66 - 6.22 + 2.99 + 0.40 - 6.11 + 2.76 + 7.84 - 1.00 - 1.00 + 2.00 - 8.74 +10. 16 - 5.70 + 3.47 - 4.37 + 4.68 + 0.50 -0.14 +0.14 +0.92 -0.14 +0.14 +0.39 + 1.20 -1.20 -0.59 2 3 4 5 6 -0.92 -4.49 -0.39 -0.69 +0.59 +0.41 +0.01 +0.24 -0.25 -0.43 +1.14 -0.71 -4.49 +1.88 +2.61 + 1 -1 +4.49 +2.34 -2.34 +2.30 +0.69 +0.67 -0.67 +0.66 -0.41 -0.35 +0.35 +1.39 9 10 13 -5.30 +8.18 -2.88 -1 -7.60 +8.18 -2.88 -1.36 +1.25 -0.55 -0.34 -0.75 -0.30 -0.27 +0.74 -0.47 16 -0.30 17 IS +0.52 10 -4.20 +8.45 -4.25 ?0 -0.22 + 1 -4.25 +4.25 -2.47 -0.09 +0.09 -0.49 +0.-67 -0.67 -0.34 99 -0.40 +1.34 -0.94 ?4 -0.74 +3.64 -2.90 +5.37 +0.93 +0.07 ?1 26 -2.90 -4.25 -0.44 -0.76 +0.27 -0.17 ?7 -1.20 +1.92 9fl 99 -1.41 +1.25 +0.16 ?.n -0.72 +3.18 +1.07 +0.49 +0.27 -0.29 +0.46 31 3? 33 34 35 3fi 37 38 +0.59 +1.07 -1.66 -2.30 +2.30 -0.20 +0.20 -0.16 +0.16 31 +1 4(1 41 -0.17 -1 +0.58 -0.02 -0.62 4? 43 +0.65 -0.48 44 -0.58 +0.02 +0.62 45 -3. 79 +3.20 +0.59 -0.01 46 47 48 +0.01 +0.14 +0.20 4<) -4.22 50 +2.99 -2.98 51 +2.73 -1 -0.01 -0.14 -0.20 52 -5.11 +6.37 -1.26 53 -2.44 +2.44 -0.10 +0.10 -0.07 +0.07 54 +1.49 + 1 55 56 57 r>s -3.99 +9.16 -5.17 -1.28 -1 +1.77 +0.08 -0.32 59 6(1 -1.77 -0.08 +0.32 til +3.47 -2.19 m -3.92 +5.68 -1.76 + 1 -0.43 +0.10 +0.07 63 64 1 1 +6.43 -6.io -6.07 65 54 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Correlate 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 66 -1 + 1 67 -1 +1 68 69 -1 +1 70 -1 +1 71 -1 +1 -1 72 73 +1 74 -1 +1 -1 +1 -1 + 1 Hi' +i -l 75 76 77 78 79 APPLICATION OF LEAST SQUAEES TO TEIANGULATION. 55 equations — Continued 23 24 25 26 27 28 29 30 a 31 ;J2 33 i 34 ~ -1.00 0.00 + 1.00 - 3.76 + 3.30 - 0.03 + 3.25 - 2.76 - 0.62 + 0.95 - 0.33 - 1.82 + 1.00 + 0.82 fifi 67 68 -L-61 +1.15 +0.46 +1 -l -1.15 +1.15 +2.84 -2. 84 KM 70 -1.24 +5.00 -3.76 +0.07 -0.07 -0.16 +0.16 71 n 73 +0.15 +3.18 -3.33 -l +i +1.18 -1.18 +0.11 -0.11 -0.06 +0.06 74 75 7fi +0.18 77 78 -0.18 79 List of corrections v's* Adopted v's. V. u's.* Adopted v's. V. 1 +0.699 +0.7 0.49 41 -0. 789 -0.8 0.64 2 +2. 448 +2.4 5.76 42 -1.997 -2.0 4.00 3 -3. 146 -3.2 10.24 43 +0. 028 +0.0 0.00- 4 -1.369 -1.4 1.96 44 +1.554 +1.6 2.56 5 -0. 207 -0.2 0.04 45 +0. 581 +0.6 0.36 6 +1.576 +1.6 2.56 46 +0. 697 +0.7 0.49 7 +0.806 +0.7 0.49 47 -1.478 -1.5 2.25 8 -1.876 -1.9 3.61 48 +0. 781 +0.8 0.64 9 +0. 498 +0.5 0.25 49 +0. 735 +0.8 0.64 10 -0.117 -0.1 0.01 50 +1.145 +1.2 1.44 11 +0.688 +0.7 0.49 51. +0. 294 +0.3 0.09 12 +3.097 +3.1 9.61 52 -0.317 -0.3 0.09 13 +1.159 + 1.2 1.44 53 -1. 522 -1.5 2.25 14 -2.691 -2.7 7.29 54 -0. 138 -0.1 0.01 15 -1. 472 -1.4 1.96 55 -0. 197 -0.2 0.04 16 -0. 728 -0.7 0.49 56 +0. 741 +0.7 0.49 17 +0. 755 +0.8 0.64 57 +0.525 +0.5 0.25 18 -0. 168 -0.1 0.01 58 -1.266 -1.3 1.69 . 19 +0. 945 +1.0 1.00 59 -0.592 -0.6 0.36 20 -0.090 -0.1 0.01 60 -0. 262 -0.2 0.04 . 21 +0. 102 +0.1 0.01 61 +0. 524 +0.6 0.36 22 -0. 910 -0.9 0.81 62 +1. 193 + 1.2 1.44 23 -1.665 -1.6 2.56 63 +0. 065 +0.1 0.01 2-1 +0. 614 +0.7 0.49 64 +0. 364 +0.4 0.16 25 -1.570 -1.5 2.25 65 -1.294 -1.3 1,69 26 +2.090 +2.1 4.41 66 -0. 190 -0.2 0.04 27 +0.530 +0.5 0.25 67 -0. 898 -0.9 0.81 28 -0.966 -1.0 1.00 68 +1.088 + 1.1 1.21 29 +0.183 +0.2 0.04 69 -0. 748 -0.8 0.64 30 -1.164 -1.2 1.44 70 -0. 490 -0.5 0.25 31 +0. 311 +0.3 0.09 71 +0. 134 +0.1 0.01 32 +1.636 +1.6 2.56 72 +1.234 +1.2 1.44 33 -0. 457 -0.4 0.16 73 -0. 130 -0.2 0.04 34 +1.167 +1.2 1.44 74 +1.360 +1.3 1.69 35 -0. 710 -0.7 0.49 75 -0.203 -0.2 0.04 36 —0. 494 -0.5 0.25 76 -1.158 -1.2 1.44 37 +1.484 +1.5 2.25 77 +0. 445 +0.4 0.16 38 -0.990 -1.0 1.00 78 -1.482 —1.5 2.25 39 -0.319 -0.3 0.09 79 +1.037 +1.0 1.00 40 +0.941 +0.9 103. 76 * These values result from the computation on p. 69. 56 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. £ NNWOIO l> Q) Tf« iC Ol Sootoi »ooooe5oo «no>h Tii r»- ■* eo o» o'o'o'h'ih ddndo ++ I ++ + I + I + i-H i-H rH I + I++ +++++ tOOOOOJ ONW(N« id-WCflOON OCONWM I ++ I I I I I II ecooc dddHH +111+ ++ I ++ WO>-l©>-l rHOOl-HO I I I ++ + + + + I cdccr-icdod odeoeoci^ I M++ ++++ I WHVH + + I + + + 00 00 + + C>|iCCO ooc5o i-i00 I + + oo + I NOTOO I + + + I ++ + ONOO + + I + I I I + + + + + I + I + I + I + + + + hcscottio cor~ooo>o 1 Tf< NHT5 CO t»- t- -3* CO C ■-tOOOO O^HOOO I++ I I "I +++ I I ++++ +++ I + OONNW COHNHCO lOi-Jcdidc4 NiOi-HNCsJ + I++I +1 I ++ NOOOiH OOOOO I I I I + ++++ I OOOOO OOOOO I I I 1+ ++++I (O -* i-H i-H ^-i r-i>or--oi t^CSCOCOW fflOOMt-H ieicd cidedcoo + + I I + ++ I I + I I + ++++ I I I + O 000 N00 O + ++ I I + + ++ I +++ 7++ i + I I + ' + + C) CI CI CO + +++ + ++ + + + I I + COCO + + create + I + + + saas "2 «i>ooojs APPLICATION OP LEAST SQUARES TO TRIANGULATION. 57 ^SBS 8S3K8E: 33388 sssss i-It-JcJoiH ©'goo"©" I++++ +I++I OOQ^O HHNO NNNIO WONH i-io'de* 1 + I + ^ oseo ©COO "S3 +++++ W3 eo co go os Ohio C4cOo I I I I I NO*C(00 eoeoeSoco NO-V CN Tf* OHNNO ■o'oic'dd + + I + I + 11 + OONOO«i-i t- CM tP qho I + + + + + 1 + I + + + + + + + OHHOO) I ++ I I SB CSCC i-i I I 00 CO r- ++ i-< + + I + I -t- I + + + ++ ++ SS?338 8S8SS SSS5S 58 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Solution of 1 2 3 23 4 6 24 7 5 +6 ft 1 +2 -0.33333 -2 +0.33333 +0.53 -0.08833 +6 -0.6667 +2 +0.6667 +2.20 -0.1767 +5.3333 ft 1 2 +2:6667 -0.50001 +2.0233 -0.37937 +6 -0.6667 -1.3333 -0.44 +0. 1767 -1.0117 -2 - 7.91 +4 ft 1 2 3 -1.275 +0.31875 -2 +0.5 - 7.91 + 1.9775 +2.6386 -0.0468 -0. 7676 -0. 4064 +0.72 -0. 6375 + 3.7891 - 2.5213 +1.4178 Ob 3 23 +0.0825 -0.05819 + 1.2678 - 0.89420 +6 -1 -0. 0048 -2 + 7.53 - 3.955 - 0.0738 -2 +4.9952 ft 4 -2 +0.40038 + 3.5012 - 0. 70091 -2 +0.40038 +6 -0.8008 + 2.14 + 1.4018 -2 +2 -0.8008 +5. 1992 ft 3 23 4 6 + 3.5418 - 0.68122 -2 +0.384675 +1.1992 -0. 23065 +156. 0106 - 15.6420 - 1.1337 - 2.4540 - 2.4127 +1.3624 +2.14 +1.4018 -0.8169 +134.3682 fti +1.3624 -0.0101393 +2. 7249 -0.0202794 6 24 +6 -0.7693 -0.0138 +2 +0.4613 -0.0276 +5.2169 ft +2. 4337 -0.46650 APPLICATION OP LEAST SQUARES TO TKIANGULATION. 59 normals 25 31 32 33 34 5 2 +3 -0.5 - 6.66 + 1.11 - 2.26 + 0.37667 +3.08 -0.51333 - 5.5 + 0.91667 - 1.81 + 0.30167 +1 -1 - 3.08 + 2.22 - 0.86 + 0.16125 - 0.91 . + 0.7533 - 0.1567 + 0.02938 -2.00 -1.0267 -3.0267 __ +0.56751 + 4.0 + 1.8333 + 5.8333 - 1.09375 + 11.21 + 0.6033 + 11.8132 - 2.21499 -4 +1 -3 +0.75 - 11.77 - 2.22 + 0.43 - 13.56 + 3.39 - 1.76 -' 0. 7533 + 0.0783 - 2.435 + 0.60875 -4.07 + 1.0267 +1.5133 -1.53 +0.3825 +12.0 - 1.8333 - 2.9167 + 7.25 - 1.8125 - 13.95 - 0.6033 - 5.9067 - 20.4600 + 5.115 +1.50 -0.2650 -0.9562 +0.2788 -0. 19664 + 5.5439 + 0.5883 + 0.3263 - 4.3222 + 2.1363 - 1.50677 + 0.9624 + 0.1996 + 0.0594 - 0.7762 + 0.4452 - 0.31401 -0.0639 -0.2721 +1. 1482 -0. 4877 +0.3245 -0.22888 - 2.22 + 0.4858 - 2.2130 + 2.3109 - 1.6363 + 1.15411 + 15.1601 + 0.1599 - 4.4816 - 6.5216 + 4.3166 - 3.04458 +0.67 +0.67 -0.13413 +2 -1.5 -0.0162 +0.4838 -0.09685 + 3.84 - 6.78 - 0.1243 - 3.0643 + 0.61345 + 0.57 - 1.2175 - 0. 0259 - 0.6734 + 0. 13481 -0.44 -0. 7650 -0.0189 -1.2239 +0.24502 - 8.0 + 3.6250 + 0.0952 - 4.2798 + 0.85678 + 6.89 - 10.23 - 0.2512 - 3.5912 + 0.71893 -2.48 +0.2683 -2.2117 +0. 42539 +0. 1937 +0. 1937 -0.03726 + 8.25 - 1.2269 + 7.0231 - 1.35080 + 1.48 - 0.2697 + 1.2103 — 0. 23279 +0.37 -0.4900 -0.12 +0.02308 - 0.0 - 1.7135 - 1.7135 + 0.32957 + 13.76 - 1.4378 + 12.3221 - 2.37000 +1.4732 -0. 4696 +1.5067 +2.5103 -0.0186822 +7.91 -5.9325 -0.2493 -0.3391 -0.1320 +1.2571 -0.0093556 +151. 8020 - 26.8149 - 1.9103 + 2.1478 - 4.7843 +120.4403 - 0.8963453 +24.5038 - 4. 8152 - 0.3981 + 0.4720 - 0.8245 +18.9380' - 0.1409411 -7.2148 -3. 0256 -0.2902 +0.8578 +0.0817 -9.5911 +0.0713792 - 5.7 +14.3369 + 1.4632 + 2.9998 + 1.1673 +14.2672 - 0.1061799 +336. 4739 - 40.4597 - 3.8599 + 2.5171 - 8.3941 +286.2773 - 2.1305435 +3.31 -0. 8508 -0.0255 +2.4337 -0.46650 +0.0745 -0.0127 +0.0618 -0.01185 + 3.54 + 2:7016 - 1.2212 + 5.0204 - 0.96233 + 0.76 + 0. 4656 - 0. 1920 + 1.0336 - 0. 19813 +0.80 -0. 0462 +0.0972 +0.8510 -0.16312 g2 - o!6591 - 0. 1447 - 4.0038 + 0.76747 + 11.21 + 4.7400 - 2.9027 + 13.0473 - 2.50097 60 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Solution of 5 25 8 9 26 10 12 27 4 6 24 7 +6 -0.8008 -0. 2766 -0.0553 -1.1353 —0.62 +0.2683 +0.5101 -0.0509 -1. 1353 -2 -2 +0.42 +3. 7320 ft 4 6 24 7 5 -1.0278 +0.27540 -2 +0.53591 -2 +0.53591 +0.42 -0. 11254 +7.2568 -0. 0899 -0. 9408 -0.0469 -1.1353 -0.2831 +0.51 -0.5508 +0.51 -0.5508 -1.1220 +0.1157 +4. 760S 5 25 -0.0408 +0.00857 -0. 0408 +0.00857 -1.0063 +0.21137 +6 -1.0718 -0.0003 +2 -1.0718 -0.0003 -1.82 +0.2251 -0.0086 2 +4.9279 ft 5 25 8 +0.9279 -0. 18830 -1.6035 +0.32539 -2 +0.40585 +6 -1.0718 -0.0003 -0. 1747 +0.25 +0.2251 -0.0086 +0.3019 +2 +0.3766 +8.45 +4. 7532 C, 5 25 8 9 +0.7684 -0. 16166 +2.3766 -0.5 +8.45 -1.77775 +6. 7354 -0. 0473 -0. 2127 -0. 5218 -0. 1242 +0.08 -0. 6508 -0.3842 -1.8590 -1.3660 • +5.8294 Cm 8 9 26 -0.9550 +0. 16382 -3.2250 +0.55323 +6 -0.8117 -1.1883 -0. 1564 +8.45 -4.2250 -0.5283 +3. 8436 do +3.6967 -0.96178 +6 Cm +0.34 -0.05667 APPLICATION OF LEAST SQUAEES TO TBIANGULATION. 61 normals — Continued 11 13 31 32 33 34 n 2 +0. 1937 -0.0447 -0.0255 -0.0288 +0.0947 -0. 02538 +8.61 -1.2269 -1.6199 -2. 4425 -2.3420 +0.9787 -0. 26225 +1.75 -0.2697 -0. 2792 -0. 3841 -0. 4822 +0.3348 -0.08971 +1.46 -0. 4900 +0. 0277 +0. 1945 -0.3970 +0. 7952 -0. 21308 -2.9 -1.7135 +0.3952 -0. 2893 +1.8678 -2. 6398 +0. 70734 +14.86 - 1.4378 - 2. 8421 - 5. 8055 - 6.0866 - 1.3122 + 0.35161 -0.0649 +0.0824 -0.0235 -0.0288 +0.0261 -0.0087 +0. 00183 +0. 8640 +0.4110 +2. 9876 -2.2501 -2.3420 +0.2695 -0.0600 +0.01260 +0. 1260 +0. 0903 +0.5148 -0.3538 -0. 4822 +0.0922 -0.0127 +0.00267 -0. 1377 +0. 1642 -0. 0510 +0. 1792 -0. 3970 +0.2190 -0.0233 +0.00489 -6.09 +0. 5740 -0. 7289 -0.2665 +1.8678 -0. 7270 -5.3706 +1.12809 + 4.2703 + 0.4817 + 5.2417 - 5.3483 - 6.0866 - 0.3614 - 1.8024 + 0.37859 ■ +0.0508 -0.0001 +0.0507 -0.01029 +3.18 +0.5245 -0.0005 +3. 7040 -0. 75164 +0.49 +0. 1794 -0.0001 +0. 6693 -0. 13582 -0.29 +0. 4262 -0.0002 +0. 1360 -0.02760 -2.3 -1.4147 -0.0460 -3. 7607 +0. 76314 + 3.77 - 0. 7032 - 0.0154 + 3.0516 - 0.61925 -2 -2 +0.42077 +2 +0. 0508 -0.0001 -0.0095 +2.0412 -0.42944 +2.02 +0.5245 -0.0005 -0.6975 +1.8465 -0. 38848 +1.20 +0. 1794 -0. 0001 -0. 1260 +1.2533 -0.263675 +1.79 + 0.4262 -0.0O02 -0.0256 +2. 1904 -0. 46083 -3.2 -1. 4147 -0.0460 +0. 7081 -3.9526 +0.83157 +19.02 - 0.7032 - 0.0154 - 0.5746 + 17.7270 - 3.72949 +0.05 +0.3233 +0.3733 -0.06404 -0.05 -0. 0107 -0. 0018 +0.0165 -0.3300 -0.3760 +0.06450 +3. 9252 -0. 1101 -0.0127 +1.2052 -0.2985 +4.7091 -0. 80782 +0.5728 -0. 0377 -0. 0027 +0.2178 -0.2026 +0.5476 -0.09394 +0.0732 -0. 0895 -0.O049 +0. 0443 -0.3541 -0.3310 +0.05678 -2.60 +0.2971 -1. 1352 -1.2237 +0. 6390 -4.0228 +0.69009 + 4.6556 + 0.1477 - 0.3810 + 0.9930 - 2.8657 + 2.5496 - 0.43737 —2 + 1 +0.0612 -0.9388 +0.24425 +2 +0.0206 -1.0206 -0.0616 +0.9384 -0.24415 -4.83 +1.5033 -0. 9233 +0.7714 -3.4786 +0.90504 -0.07 +0.2716 -0. 6266 +0. 0897 -0.3353 +0.08724 +1.29 +0.0552 -1.0952 -0.0542 +0. 1956 -0.05089 -3.2 -1.5263 +1.9763 -0. 6590 -3.4090 +0.88693 + 7.72 + 1.2385 - 8.8635 + 0.4177 + 0.5126 - 0.13336 +2 -0.33333 -2 +0.33333 +4.24 -0. 70667 -0.05 +0. 00833 -0.87 +0. 145 -1.0 +0.16667 + 8.66 - 1.44333 62 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Solution of 9 26 10 12 27 11 13 14 15 28 16 17 +149. 8778 - 15.0220 - 1. 7842 - 3.5554 - 0.0193 -15.67 + 3. 5555 + 0.2065 + 0.9029 - 0.1133 +1.02 +0. 1133 -2.98 +2.98 - 8.1354 +129.4969 C27 9 26 10 12 27 -11.1184 + 0. 0858584 +1. 1333 -0. 0087516 -2.98 +0.0230121 +2.98 -0.0230121 - 8.1354 + _ 0.0628231 + 6 - 0.8415 - 0.0239 - 0. 2293 - 0.6667 - 0.9546 +2 +0. 6667 +0. 0973 +2 -0,2559 -2 +0. 2559 + 1.66 - 0.6985 + 3.2840 Cii 12 27 11 +2.7640 -0.84166 +1. 7441 -0. 53109 -1.7441 +0.53109 + 0.9615 - 0.29278 +6 -0.6667 -0. 0099 -2.3263 +2 +0.0261 -1.4679 -2 -0.0261 +1.4679 + 1.66 + 0.07*12 - 0.8093 +2.9971 Ca 27 11 13 +0.5582 -0. 18625 -0.5582 +0.18625 - + 0. 9219 „-- 0.30760 +6 -0.0686 -0. 9263 -0. 1040 -2 +0.0686 +0. 9263 +0. 1040 + 4.22 - 0.1872 - 0.5106 - 0.1717 + 2 , +4. 9011 Ch 27 11 13 14 -0. 9011 +0. 18386 + 3.3505 - 0. 68362 + 2 - 0.40807 +6 -0. 0686 -0,9263 -0. 1040 -0. 1657 - 1.94 + 0.1872 + 0.5106 + 0.1717 + 0.6160 + 2 + 0.3677 +2 \ +4. 7354 Ca 27 11 13 14 15 - 0. 4545 + 0.09598 + 2. 3677 - - 0.5- - +2 -0.42235 ;\ +158. 2846 - 0.5111 -- 0. 2815 - 0.2836 - 2. 2905 - 0.0436 +18.86 - 1.3672 + 0.2273 - +5.48 J +0.1920 + 154.8743 14 15 28 +17. 7201 - 0. 114416 +5. 6720 -0. 036623 + 6 - 0.8161 - 1. 1839 - 2.0275 +2 -1 -0. 6490 + 1.9725 CI. +0.3510 -0. 17795 APPLICATION OF LEAST SQUARES TO TRIANGULATION. 63 normals — Continued 18 29 31 32 33 34 1 I + 11.43 — 3.6287 - 0.2080 - 0.9025 + 6.6908 — 0.0516676 -53. 9453 - 3. 2826 + 2.6052 + 3. 3456 - 0. 2403 -51.5174 + 0.3978273 -0.7272 -2. 2281 +0. 3029 +0. 3225 +0. 0028 -2.3271 +0. 0178703 +9. 1030 -3. 8940 -0. 1831 -0. 1881 +0. 0493 +4. 8871 -0. 0377391 +0.0 +7.0267 -2. 2255 +3. 2787 +0. 0567 +8.1366 -0. 0628324 +108.3339 - 31. 5142 + 1. 4105 - 4930 - 0. 4908 + 77.2464 - 0. 5965116 - 4 + 0.8589 + 0. 0241 + 0.229? + 0.5745 - 2.3133 + 0.70442 + 6.76 + 0. 7770 - 0. 3016 - 0.8496 - 1.4133 - 4.4232 + 0.5493 - 0. 16727 -0.32 +0. 5274 -0. 0351 -0. 0819 +0. 0167 -0. 1998 -0.0927 +0. 02823 -2.52 +0. 9217 +0.0212 +0. 0478 +0. 2900 +0. 4196 -0.8197 +0. 24960 +5.0 -1.6631 +0. 2576 -0. 8326 +0. 3333 +0. 6986 +3. 7938 '-1. 15524 - 3.04 + 7. 4590 - 0. 1633 + 0.1252 - 2.8866 + 6.6323 + 8.1269 - 2.47470 - 2 - 0.0586 + 1.9470 - 0.1116 + 0.03724 - 2.31 + 1.4133 + 0.4509 - 0.4623 - 0.9081 + 0.30299 -0.34 -0. 0167 +0. 0204 +0.0780 -0.2583 +0.08618 -0.36 -0. 2900 -0.0428 +0. 6899 -0.0029 +0.00097 +3.7 -0. 3333 -0.0712 -3. 1931 +0. 1024 -0.03417 + 7.37 + 2. 8866 - 0.6760 - 6.8401 + 2.7405 - 0. 91438 i - 2 + 0.1540 + 1.2286 + 0.0208 - 0.5966 + 0. 12173 - 2.31 - 1.1855 - 0.2917 + 0. 1691 - 3.6181 + 0.73822 -0.34 -0. 0536 +0. 0492 +0.0481 -0. 2963 +0. 06046 -0.36 +0. 1125 +0. 4353 +0. 0005 +0.1883 -0.03842 +5.2 +0. 1872 -2.0148 -0. 0191 +3. 3533 -0. 68419 + 11.43 + 1. 7776 - 4. 3161 - 0.5104 + 8.3811 - 1. 71004 +2 +2 -0. 42235 +0.02 +0.02 -0. 00422 + 4 - 0. 1540 - 1.2286 - 0.0208 - 0. 1097 + 2.4869 - 0.52517 + 3.51 + 1.1855 + 0. 2917 - 0. 1691 - 0.6652 + 4.1529 - 0.87699 +0.48 +0. 0536 -0. 0492 -0. 0481 -0.0545 +0. 3818 -0. 08063 + 1.23 -0. 1125 -0. 4353 -0. 0005 +0. 0346 +0. 7163 -0. 151265 -2.5 -0. 1872 +2. 0148 +0. 0191 +0. 6165 -0.0368 +0. 00777 + 13.78 - 1. 7776 + 4.3161 + 0. 5104 + 1.5409 + 18.3697 - 3.87923 ' +5.4S +0. 1920 +5.6720 -0.036623 +3. 2298 +0.0019 +3. 2317 -0. 020867 + 3.82 + 0. 4203 + 0. 6773 + 0. 0343 + 0.4078 + 0. 2387 + 5. 5984 - 0.036148 + 6.8009 - 3.2365 - 0. 1608 + 0.2793 + 2.4734 + 0:3986 + 6.5549 - 0.042324 -0.0428 -0. 1462 +0. 0271 +0. 0795 +0. 2026 +0.0366 +0. 1568 -0. 001012 -0. 7425 +0.3070 +0.2400 +0. 0009 -0. 1287 +0.0688 -0. 2545 +0. 001643 +6.6 +0. 5112 -1. 1107 -0. 0315 -2. 2924 -0.0035 +3. 6731 -0. 023717 +205. 2346 + 4. 8529 - 2.3794 - 0.8430 - 5. 7295 + 1.7631 +202. 8988 - 1.310087 +2 -1 -0. 6490 +0.3510 -0. 17795 +0.02 -0.01 -0.3698 -0.3598 +0. 18241 + 2 + 0.2435 - 1.2435 - 0.6405 + 0.3595 - 0.18226 + 0.67 + 1.4764 - 2.0764 - 0. 7500 - 0.6800 + 0.34474 +0.02 +0. 1209 -0. 1909 -0.0179 -0.0679 +0.03442 +0.39 -0. 0768 -0. 3582 +0. 0291 -0. 0159 +0. 00806 +2.8 -1.3684 +0. 0184 -0. 4203 +1.0297 -0. 52203 + 38. 76 - 3.4201 - 9.1849 - 23. 2149 + 2.9401 - 1. 49054 91865°— 15- 64 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Solution of 17 18 29 19 / 20 21 22 15 2S 16 +6 -0. 8447 -0. 2077 -0. 0625 +2 -0. 8447 -0.2077 -0.0625 + 8.60 - 0.0084 - 0.1184 + 0. 0640 -2 +4.8851 On 15 28 16 17 +0. 8851 -0. 18118 + 8.5372 - 1. 74760 -2 +0. 40941 +6 -0. 8447 -0. 2077 -0. 0625 -0. 1604 - 2.30 - 0.0084 - 0.1184 + 0.0640 - 1. 5468 +2 +0. 3624 -2 +4. 7247 Cm 15 28 16 17 18 - 3. 9096 + 0. 82748 +2. 3624 -0. 50001 -2 +0. 42331 +127.4272 - 0.0001 - 0.0674 - 0.0656 - 14.9196 - 3. 2351 -8.18 +3. 4952 +1.9548 +0.95 -1.6550 +109. 1394 17 18 29 2 73 +o! 025014 -0. 7050 +0. 006460 +6 -0. 8188 -1. 1812 -0.0683 -2 +1 -0. 0176 +3. 9317 Cm 18 29 19 -1. 0176 +0. 25882 +6 -0.8466 -0. 0046 -0.2634 +2 -2 +4.8854 C20 20 +2 -0.40938 -2 +0. 40938 +6 -0. 8188 +2 +0. 8188 +5. 1812 Csi 20 21 +2. 8188 -0. 54404 +6 -0. 8188 -1. 5335 +3. 6477 APPLICATION OF LEAST SQUAKES TO TEIANGULATION. normals — Continued 65 30 31 32 33 34 1 V +2 -1.0503 -0. 2050 -0. 0640 +0. 6807 -0. 13934 + 0.67 - 1. 7540 - 0.2401 + 0.1210 - 1.2031 + 0. 24628 +0.02 -0. 1613 -0. 0057 +0. 0121 -0. 1349 +0. 02761 +0.39 -0. 3025 +0. 0093 +0. 0028 +0. 0996 -0. 02039 - 5.1 + 0. 0155 - 0.1345 - 0.1832 - 5.4022 + 1. 10585 +22. 06 - 7.7584 - 7.4308 - 0. 5232 + 6.3475 - 1.29936 - 4.38 - 4.38 + 0. 92704 +4 -1.0503 -0. 2050 -0. 0640 -0. 1233 +2. 5574 -0. 5412S - 3.00 - 1.7540 - 0.2401 + 0. 1210 + 0.2180 - 4.6551 + 0.98527 +0.08 -0. 1613 -0. 0057 +0. 0121 +0. 0244 -0.0505 +0. 01069 +0.85 -0. 3025 +0. 0093 +0. 0028 -0. 0180 +0. 5416 -0. 11463 - 1.7 + 0. 0155 - 0. 1345 - 0. 1832 + 0.9788 - 1.0234 + 0. 21661 +11.03 - 7.7584 - 7.4308 - 0. 5232 - 1. 1500 - 5.8325 + 1.23447 + 8. 0144 - 3.6244 + 4.3900 - 0.040224 -0.93 -0. 0105 -0. 1168 +0. 0656 -1. 1896 +2. 1162 -0. 0651 +0. 000596 -37. 6627 - 0.0175 - 0.1368 - 0.1240 + 2. 1025 - 3.8520 -39. 6905 + 0. 363668 — 1. 5212 -0. 0016 -0.0033 -0. 0124 +0. 2358 -0. 0418 -1.3445 +0. 012319 +0. 7206 -0. 0030 +0.0053 -0. 0029 -0. 1741 +0. 4482 +0.9941 -0. 009109 -18.4 + 0. 0002 - 0. 0766 + 0.1878 + 9.4409 - 0.8468 - 9.6945 + 0.0SS827 +79. 9881 - 0. 0776 - 4. 2339 + 0. 5363 -11. 0929 - 4.8263 +60. 2939 - 0.552449 - 4.38 + 2.19 + 0.1098 - 2.0S02 + 0.52908 +2 +0. 2787 -1.2787 -0.0016 +0. 9984 -0. 25394 - 3.27 - 0.4926 + 2.3276 - 0.9928 - 2.4278 + 0.61749 +0.03 -0. 0552 +0. 0252 -0. 0336 -0. 0336 +0. 00855 +0.23 +0. 0408 -0. 2708 +0. 0249 +0.0249 -0. 00633 + 2.1 - 2. 2117 + 0. 5117 - 0.2425 + 0.1575 - 0.04006 - 7.47 + 2. 5987 + 2. 9163 + 1.5082 - 0. 4467 + 0. 11361 + 4.50 - 1.8541 + 0.0284 - 0.6384 + 2. 1359 - 0. 43720 -4 +1. 0826 -0. 0004 +0.2584 -2.6594 +0.54436 + 0.19 - 1.9706 - 0.2564 - 0.6284 - 2.6654 + 0.54558 -0.35 -0. 0214 -0. 0087 -0. 0087 -0. 3888 +0. 07958 -0.18 +0. 2293 +0. 0064 +0. 0064 +0. 0621 -0. 01271 + 2.3 - 0.4332 - 0.0626 + 0. 0408 + 1.8450 - 0.37766 + 5.41 - 2. 4690 + 0. 3895 - 0.1157 + 3. 2148 - 0.65804 -10. 92 - 0.8744 -11. 7944 + 2.27638 +1 +1.0887 +2. 0887 -0. 40313 - 0.93 + 1.0912 + 0.1612 - 0. 03111 -0.21 +0. 1592 -0. 0508 +0. 00980 -0.01 -0.0254 -0. 0354 +0. 00683 + 6.1 - 0.7553 + 5.3447 - 1.03156 + 5.03 - 1.3161 + 3.7140 - 0.71682 - 3.75 + 0.8744 + 6.4166 + 3.5410 - 0.97075 +3 -1. 08S7 — 1. 1363 +0. 7750 -0. 21246 + 3.12 - 1.0912 - 0.0877 + 1.9411 - 0.53214 +0.11 -0. 1592 +0. 0276 -0. 0216 +0. 00592 -0.06 +0. 0254 +0. 0193 -0.0153 +0.00419 + 0.1 + 0. 7553 - 2.9077 - 2.0524 + 0.56266 + 8.52 4, 1.3161 - 2. 0206 + 7.8155 - 2.14258 66 COAST AND GEODETIC SXJEVEY SPECIAL PUBLICATION NO. 28. Solution of normals — Continued 30 31 32 33 34 V • J +76.0764 - 9.74 + 1.8338 - 0.5171 - 0.0430 - 8.2 + 48.4945 18 - 4.0604 + 2.3708 — 4.3155 - 0.0468 + 0.5021 - 0.9487 — 5.4070 29 - 0.1766 + 0.0026 + 1. 5965 + 0.0541 - 0.0400 + 0.3900 — 2.4253 19 - 1.1006 + 0.5282 — 1. 2845 - 0.0178 + 0.0132 + 0.0833 — 0.2363 20 - 0.9338 + 1.1627 + 1. 1653 + 0. 1700 - 0.0272 - 0.8066 — 1.4055 21 -26.8485 + 4.7547 + 0. 3670 - 0.1156 - 0.0806 +12. 1666 + 8.4545 22 - 3.4374 - 0.7523 — 1.8843 + 0.0210 + 0.0149 + 1.9924 - 7.5869 +39.5191 - 1.6733 _ 2.5217 - 0.4522 + 0.3394 + 4.6770 + 39.8883 C M + 0.042342 + 0.063810 + 0.011443 - 0.008588 - 0.-118348 — 1.009342 +16 + 6.82 + 1.67 + 3.69 - 7.3 + 46.82 1 -1.5 + 3.33 + 1.13 - 1.54 + 2.75 + 0.9050 3 - 2.2S — 10.17 - 1.8262 - 1.1475 + 5.4375 — 15.3450 23 - 0.0548 — 0. 4201 - 0.0875 - 0.0638 + 0.3218 — 0.8488 4 - 0.0469 + 0.2968 + 0.0652 + 0.1185 + 0.4145 + 0.3478 6 - 0.0072 — 0.2617 - 0.0461 + 0.0045 + 0.0038 0.4591 24 - 0.0118 — 1.1268 - 0.1772 + 0.0897 - 0.1335 — 2.6783 7 - 0.0007 — 0.0595 - 0.0122 - 0.0101 + 0.0474 — 0. 1546 5 - 0.0024 — 0.0248 - 0.0085 - 0.0202 + 0.0670 + 0.0333 25 — — 0.0001 — — - 0.0098 — 0.0033 8 - 0.0005 — 0. 0381 - 0.0069 - 0.0014 + 0.0387 _ 0.C314 9 - 0.8766 — 0. 7930 - 0.5382 - 0.9406 + 1.6974 — 7.6127 26 - 0.0243 + 0. 3037 + 0.0353 - 0.0213 - 0.2595 + 0. 1644 10 - 0.2291 + 0.8493 + 0.0819 - 0.0478 + 0.8323 0.1252 27 - 0.3457 + 2. 6618 + 0.1202 - 0.2525 - 0.4204 — 3.9911 11 - 1.6295 + 0. 3869 - 0.0053 - 0.5774 + 2.6724 + 5.7248 13 - 0.0042 — 0.0338 - 0.0096 - 0.0001 + 0.0038 + 0. 1021 14 - 0.0726 — 0. 4404 - 0.0301 + 0.0229 + 0.4082 + 1.0202 15 - 1.3060 — 2. 1810 - 0.2005 - 0.3762 + 0.0193 — 9.6472 28 - 0.2024 — 0.2369 - 0.0057 + 0.0092 - 0.1328 _ 7.3344 10 - 0.0055 + 0. 1239 + 0.0124 + 0.0029 - 0.1877 — 0.559 17 - 0.0948 + 0. 1676 + 0.0188 - 0.0139 + 0.7527 — 0.8845 18 - 1.3843 + 2.5197 + 0.0273 - 0.2932 + 0.5539 + 3. 1570 29 — — 0.0237 - 0.0008 + 0.0006 - 0.0058 + 0.0359 19 - 0.2535 + 0.6165 + 0.0085 - 0.0063 - 0.0400 + 0. 1134 20 - 1.4477 — 1. 4509 - 0.2116 + 0.0338 + 1.0043 + 1.7500 21 - 0.8420 — 0.0650 + 0.0205 + 0.0143 - 2.1546 1. 4972 22 - 0.1647 — 0.4124 + 0.0046 + 0.0033 + 0.4301 — 1.6605 30 - 0.07C9 — 0.1068 - 0.0191 + 0.0144 + 0.1980 + 1.6890 + 3.1119 + 0.2312 - 0.0558 - 1.3082 + 7.0750 + 9.0541 Cm — 0.07430 + 0.01793 + 0.42039 - 2.27353 2. 90951 +351. 4744 +49.3161 -13.0822 - 5.2 +478.9301 1 — 7.3926 - 2.5086 + 3.4188 - 6.1050 — 2.0091 2 — 0.1387 - 0.0253 - 0.4881 + 0.9406 + 1.9049 3 — 45.9084 - 8.2546 - 5.1867 +24. 5775 69. 3594 23 — 3.2189 - 0.6708 - 0.4889 + 2.4655 _ 6. 5041 4 — 1.8798 - 0.4131 - 0.7508 - 2.6254 — 2.2030 6 — 9.4868 - 1.0349 + 0.1621 + 2.3146 — 16. 6447 24 -107.9561 -16.9750 + 8.5969 -12. 7883 -256.6033 7 — 4.8313 - 0.9947 - 0.8189 + 3.8530 — 12. 5558 5 — 0.2567 - 0.0878 - 0.2085 + 0.6923 + 0.3441 25 — 0.0008 - 0.0002 - 0.0003 - 0.0677 0.0227 8 — 2.7841 - 0.5031 - 0. 1022 + 2.8267 _ 2.2937 9 — 0.7173 - 0.4869 - 0.8509 + 1.5355 _ 6.8866 26 — 3.8041 - 0.4424 + 0.2674 + 3.2497 — 2. 0596 10 — 3. 1483 - 0.3035 + 0. 1770 - 3.0853 + 0.4639 12 — 2. 9963 + 0.0353 + 0.6148 + 0.7067 6. 1198 27 — 20. 4950 - 0.9258 + 1.9442 + 3.2370 + 30.7307 11 — 0.0919 + 0.0155 + 0. 1371 - 0.6346 — 1. 3594 13 — 0. 2751 - 0.0783 - 0.0009 + 0.0310 + 0. 8303 14 — 2. 6710 - 0.2187 + 0. 1390 + 2.4755 + 6. 1871 15 — 3.6421 - 0.3348 - 0.6282 + 0.0323 16. 1100 28 — 0.2774 - 0.0066 + 0.0108 - 0. 1555 _ 8.5875 16 — 0.2344 - 0.0234 - 0.0055 + 0.3550 + 1. 0136 17 — 0.2963 - 0.0332 + 0.0245 - 1.3305 + 1.5633 18 — 4. 5865 - 0.0498 + 0.5336 - 1.0083 5. 7466 29 — 14. 4342 - 0.4890 + 0.361b - 3.5256 + 21 .'> V 19 — 1.4991 - 0.0207 + 0.0154 + 0.0973 0.2758 20 — 1.4542 - 0.2121 + 0.0339 + 1.0066 + 1.7539 21 — 0. 0050 + 0.0016 + 0.0011 - 0. 1663 0. 1155 22 — 1.0329 + 0.0115 + 0.0081 + 1.0922 4.1589 30 — 0. 1009 - 0.0289 + 0.0217 + 0.2984 + 2. 5453 31 — 0.0172 + 0.0041 + 0.0972 - 0.5257 0.6727 +105.7210 + 13.6619 - 6.0470 + 14.5692 +127.9051 G32 - 0.129226 + 0.057198 - 0.137808 1. 209836 APPLICATION OP LEAST SQUARES TO TRIANGTJLATION. Solution of normals — Continued 67 33 34 1 - +8.7558 -1.0341 - 2.5334 +79.9111 1 -0.8513 +1.1601 - 2.0717 - 0.6818 2 -0.0046 -0.0889 + 0.1714 + 0.3471 3 -1.4823 -0.9314 + 4.4134 -12. 4550 23 -0. 1398 -0. 1019 + 0.5138 - 1.3555 4 -0.0908 -0. 1650 - 0.5770 ' - 0. 4841 6 -0. 2817 +0.0279 + 0.3989 - 2.8685 24 -2. 6691 +1.3518 - 2.0108 -40. 3482 7 -0.2048 -0. 1686 + 0.7933 - 2.5851 5 -0.0300 -0.0713 + 0.2368 + 0. 1177 25 — -0.0001 - 0.0143 - 0.0048 8 -0.0909 -0.0185 + 0.5108 - 0.4145 9 -0.3305 -0.5776 + 1.0422 - 4.6742 26 -0.0514 +0.0311 + 0.3779 - 0.2395 10 -0.0293 +0.0171 - 0.2974 + 0.0447 12 -0.0004 -0.0072 - 0.0083 + 0.0721 27 -0.0416 +0.0873 + 0.1454 + 1.3804 11 -0.0026 -0.0231 + 0.1071 + 0.2294 13 -0.0223. -0.0002 + 0.0088 + 0.2302 14 -0.0179 +0.0114 + 0.2027 + 0.5067 15 -0.0308 -0.0578 + 0.0030 - 1.4811 28 -0.0002 +0.0003 - 0.0037 - 0.2053 16 -0.0023 -0.0005 + 0.0354 + 0.1012 17 -0.0037 +0.0027 - 0. 1492 + 0. 1753 18 -0.0005 +0.0058 - 0. 0109 - 0.0623 29 -0.0166 +0. 0122 - 0.1194 + 0.7428 19 -0.0003 +0. 0002 + 0.0013 - 0.0038 20 -0.0309 • +0.0049 + 0. 1468 + 0.2558 21 -0.0005 -0.0003 + 0.0524 + 0.0364 22 -0.0001 -0.0001 - 0.0122 + 0.0463 30 -0.0052 +0.0039 + 0.0535 + 0. 4564 31 -0.0011 -0.0261 + 0.1410 + 0.1804 32 -1.7655 +0.7814 - 1.8827 -16.5287 +0. 5568 +0.2254 - 0.3351 + 0.4471 Css -0. 40481 + 0.60183 - 0.80298 +9. 4562 -20.0499 -18.6051 1 -1.5811 + 2.8233 + 0.9291 2 -1.7177 + 3.3105 + 6.7041 3 -0.5825 + 2.7731 - 7.8260 23 -0.0743 + 0.3745 - 0.9880 4 -0. 2999 - 1.0486 - 0.8799 6 -0.0028 - 0.0395 + 0.2844 24 -0.6846 + 1.0184 +20. 4342 7 -0.1388 + 0.6531 -' 2. 1283 5 -0. 1694 + 0.5625 + 0.2796 25 -0.0001 - 0.0263 - 0. 0088 8 -0.0038 + 0. 1038 - 0. 0842 9 -1.0094 + 1.8215 - 8. 1691 26 -0. 0188 - 0.2284 + 0. 1448 10 -0. 0100 + 0. 1735 - 0.0261 12 -0. 1262 - 0. 1450 + 1.2557 27 -0. 1844 - 0. 3071 - 2.9152 11 -0.2046 + 0. 9469 + 2.0285 13 — + 0.0001 + 0.0027 14 -0. 0072 - 0.1288 - 0.3220 15 ' -0.1084 + 0. 0056 - 2.7787 28 -0.0004 + 0. 0060 + 0.3334 16 -0.0001 + 0.0083 + 0.0237 17 -0.0020 + 0. 1102 - 0. 1594 18 -0.0621 + 0. 1173 + 0.6686 29 -0. 0091 + 0. 0883 - 0. 5492 19 -0. 0002 - 0.0010 + 0.0028 20 -0.0008 - 0.0234 - 0.0409 21 -0. 0002 + 0.0365 + 0.0254 22 -0.0001 - 0.0086 + 0.0327 30 -0. 0029 - 0.0402 - 0. 3426 31 -0.5500 + 2. 9743 + 3.8063 32 -0. 3459 + p. 8333 + 7.3159 33 -0.0912 + 0. 1357 - 0. 1810 +1.4672 - 3. 1701 - 1.7029 Cm + 2. 16065 + 1.10065 68 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Bad solution 34 33 32 31 30 22 21 20 +2. 16065 +0.60183 -0.874C5 -0. 13781 +0. 12358 +0. 03526 -2.27353 +0.90832 -0.00489 -0.00156 -0.11835 -0.01856 -0. 00312 +0.0C134 -0. 058C8 +0.5627 +0.0091 -0. 0016 -0. 0112 +0.2914 +0. 1910 -1.0316 +0.0148 -0.0027 -0.0007 +0.5530 -0.4479 -0.5666 -0.3777 -0.0275 -0.0217 +0.0115 -0.7467 +0.0860 +0.42S3 +0.6065 +2. 16065 -0. 27282 +0.02103 -1.37166 -0. 10C77 + 1.0414 -1.4817 -0. 0433 19 29 18 17 16 28 15 14 -0.0401 -0.0137 -0.0023 +0.0130 +0.3483 -O.1041 -0.0112 +0.088S3 -0.01968 -0.00336 +0.00765 -0.00082 +0.00791 -0.00028 +0.00475 +0.2166 -0.2477 -0.0029 +0.0207 +0.7425 -0. 1824 -0.0183 -0. 0950 +0. 0703 +1.1058 -0.0441 -0.0075 +0.0052 +0.1911 +0.0777 -0. 1485 -0. 0913 -0.5220 +0.0174 -0.0094 +0. 0072 +0. 2500 +0.0155 -0. 0897 -0. 1937 -0.02372 +0.00355 +0.00028 -0.00089 +0. 04958 -0. 00177 -0. 01845 -0.03986 +0.06003 +0.0078 -0.3268 +0.0220 -0.0184 +0.7204 -0. 0004 -0.2128 -0.4597 +0.2624 +0.0028 -0.6842 -0.0830 -0.0165 +0.0155 -0.1670 +0.2141 1 -0.0197 -0.0005 +0. 1SC9 +0.08500 +1.0S84 -0.5247 -0.7413 +0.5038 +0. 02875 -0.0027 13 11 27 12 10 26 9 8 +1 1 l+l+l oooooooo OOQCOMOWIOJi WO!00l-"*»-Oil-'W -1.1552 +0.5393 -0.0077 -0.0C35 -0.9602 -0. 0084 -0.0014 +0.3037 -0. 0240 -0. 06283 -0.08154 -0. 00488 +0. 00837 +0. 07CS7 +0.00181 +0. 000C6 -0.01706 -0. 00025 -0. 10590 +0. 1667 +0.3133 -0.0023 -0.0149 +0.0095 +0.4111 +0. 0108 +0. 8869 -0.1100 -0.0238 +0.0190 +0.3349 -0.3013 +0. 1840 +0. 69009 +0. 12268 +0. 02563 -0. 01C99 -0. 0SS47 +0. 07899 -0. 10586 +0. 16213 +0.8316 -0.9957 +0. 0719 -0.0C82 +0.5890 -0.5190 +0.3402 -0.4948 ' —0. 1404 +0.7631 -0. 0596 +0. 0371 -0.0158 +0.0141 +0.4017 +0.2825 +0.0613 +0. 8942 +0.9897 +0.0285 +0. 8CS20 +1.4844 -1.2334 -0.3254 -0. 19135 25 5 7 24 * 6 4 23 3 +1.12809 +0.01057 -0.00073 +0.00026 -0.00251 +0. 18351 -0. 00279 +0. 01272 +0. 7073 -0.4604 +0.0245 -0.0055 +0. 0348 -0.0977 -0. 1744 +0. 7955 +0.3660 +0.7675 -0. 3524 +0. 0541 -0. 0202 +0. 0163 -0. 6200 -0.5552 -0. 10618 +0. 15423 +0.03845 -0. 01885 +0.01283 -0.02483 -0. 02413 +0. 00720 +0.3296 +0.0499 +0.0635 -0. 0284 +0.0511 +0.5654 -0. 2745 -0. 2731 -0. 0204 +0. 85C8 +0.5294 -0.0368 +0. 0129 +0. 1328 -0. 1783 +0.4765 -0.0271 +0. 1830 +1. 15411 -0. 49453 +0. 08567 -0. 03169 +0.26972 -0.03462 -0. 11342 -1.8125 +018264 -0. 1661 +0. 0713 -1.0287 +0. 0766 +0. 9746 +0. 2662 -0. 7099 +0. 83524 +1.32912 +0. 03872 -0. 7922 +1.1901 +0.4571 + 1.9492 2 1 Probable error of an ol jserved direc tion=±0.67 ""V 34 - ±1".2 -1.0938 +1.2262 -0.0080 +0. 0034 -0.3169 +0.3961 +0.9107 -1. 1091 -0. 1028 +0.0233 +0. 6858 -0. 0738 -0. 2641 -0. 0690 +0.2070 +0. 0070 APPLICATION OP LEAST SQUARES TO TEIANGULATION. 69 Computation of corrections 1 2 3 4 5 6 7 8 -0. 007 -0.207 -0.459 +1.372 +0.007 +0. 792 +0.393 -1.372 -0. 003 +0. 038 +2.593 +0. 207 -0. 792 +0. 067 +0. 003 -0. 038 -2.593 -0. 007 +0. 019 -0.106 -1.275 -0.207 +0.007 +0.207 -0.019 +0. 106 + 1.275 -0. 174 -0. 094 +0. 188 +0. 886 -1.949 +0. 073 -0.207 -0.2 -1.876 -1.9 +0. 699 +0.7 -1.369 -1.4 +0. 806 +0.7 +1.576 +1.6 -3. 146 -3.2 +2.448 +2.4 9 10 11 12 13 14 15 16 +0.792 + 1.949 +0. 008 +0. 101 -1.372 +0.094 -0.188 -0.886 -0. 007 -0. 792 +0. 200 +1.372 +0.049 -0.183 -0.756 +0.007 -0.209 -0. 049 +0.183 +0.756 -0.207 +0. 792 -0.359 +0.048 -0. 180 +3. 003 +0.207 +0.952 -0. 792 -1.949 -0.593 -0.205 +1.372 -0. 160 +0.371 -0. 735 +0.317 +0.172 -0.341 -1.620 +1.949 -1.190 -0.457 -0.112 -0.359 -0.061 +0. 150 -0.648 +1.159 +1.2 -1. 472 -1.4 +0.688 +0.J +3.097 +3.1 -0. 117 -0.1 +0.498 +0.5 -2. 691 -2.7 -0. 728 -0.7 17 18 19 20 21 22 23 24 +1.190 -1.484 +0.325 +0. 984 -0.260 +0.457 -0.625 +1.484 -0.990 +0.451 -0.894 +0.804 -0.325 +0.990 +1.233 -0. 191 -1.617 -1.372 -0. 089 +0. 025 +1.448 -1.233 +0.894 +0.813 +0.089 -0.025 -1.448 -1.190 +0.710 -0.532 -0. 052 +0. 134 -0.735 -0.457 -0.710 + 1.781 -0.168 -0.1 -0.090 -0.1 +0.945 +1.0 +0. 614 +0.7 +0.755 +0.8 -0. 910 -0.9 -1.665 -1.6 +0. 102 +0.1 25 26 27 ~28 29 30 31 32 -1.949 +1. 190 +0.457 -0.029 -1.249 +0. 113 -0.254 +0. 151 +1.949 +0. 141 -0. 112 -0. 061 +0. 120 +0.583 +0.325 -1.042 -0. 089 +0. 207 -0. 367 -1.484 + 1.667 +0.710 -1.874 -1.190 +1.484 -0.325 +1.661 -0.625 +0. 067 -0. 134 -0. 627 +1. 190 -0. 710 +0. 213 +0.023 -0. 074 +0. 994 +2.090 +2.1 +0.183 +0.2 -1.164 -1.2 +0.530 +0.5 -0. 966 -1.0 +1.636 +1.6 -1.570 -1.5 +0.311 +0.3 33 34 35 36 37 38 39 40 -0.457 +0.457 +0. 710 -0.710 -1.484 +0.990 + 1.484 -0.990 +0.003 +0. 017 -0. 048 +0. 065 -0.346 +1.233 -0.028 +0.741 -0. 003 +0.031 -1.372 +0.048 -0.055 +0.346 -0.457 -0.4 -0.710 -0.7 + 1.484 + 1.5 -0. 990 -1.0 +1. 167 +1.2 -0. 494 -0.5 -0.319 -0.3 +0.941 +0.9 *70 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Computation of corrections — Continued 41 42 43 44 45 46 47 48 ; -0. 741 -0. 048 +0.325 -0.990 -1.233 -0. 148 +1.372 +0. 012 +0.005 -1.340 +0.028 +0.990 +0. 564 -0. 325 -0.417 -0. 012 -0. 005 + 1.340 -0.028 +0.725 -0.894 +0. 028 -0. 612 +0.894 -0. 113 +0.028 +0.0 -0. 789 -0.8 +1. 554 +1.6 +0. 697 +0.7 +0.781 +0.8 -1. 478 -1.5 +0.581 +0.6 -1.997 -2.0 49 50 51 52 53 54 55 56 +1.233 -0.894 +0.002 -0.038 +0.432 +0.741 +0.525 -0. 121 +0. 894 -0. 028 -0.572 -1.233 +0.028 -0.741 +0.003 +0.570 +0.078 +1.372 +0. 038 -0.432 -1. 088 -0.434 -0.504 +0. 541 -0.051 +0. 027 -0. 151 -0.003 -0.525 +1.088 +0.504 +0.043 -0. 107 -1.372 +0.051 -0.027 +0. 151 +0. 741 +0.741 +0.7 -1.522 -1.5 +1.145 +1.2 +0.294 +0.3 +0. 735 +0.8 -0. 138 -0.1 -0.317 -0.3 -0. 197 -0.2 57 58 59 60 61 62 63 64 +0.525 -0. 741 -0.525 +0. 003 +0.525 -1. 088 -0. 504 -0. 115 -0. 109 +1.372 +0.037 -0. 022 -0.691 -0.525 +0.263 -0.003 -0. 149 -0.037 +0. 022 +0. 691 +1.088 -0. 190 +0.295 +0.504 +0. 190 +0.043 -0. 186 +0. 771 -1.372 -0.009 -0.027 +0. 151 +1.482 -1.118 +0. 525 +0.5 -1.266 -1.3 -0. 262 -0.2 +0.364 +0.4 +1.193 +1.2 +0.524 +0.6 +0.065 +0.1 -0.592 -0.6 65 66 67 68 69 70 71 72 -0.043 -1.482 +0. 346 +0. 009 +0. 027 -0.151 -0. 190 -1.088 +0. 190 +1.088 ' -1.041 +0.317 -0. 024 +0.043 +1.041 -0.226 -1.372 +0.024. -0.504 -0. 190 -0.043 -0. 105 -0.091 +1.372 +0. 060 -0. 019 -0.346 +0.504 +0.425 -0. 060 +0. 019 +0.346 -0. 190 -0.2 +1.088 +1.1 -0. 898 -0.9 -0. 748 -0.8 -0. 490 -0.5 +1.234 +1.2 -1.294 -1.3 +0. 134 +0.1 73 74 75 76 77 78 79 +0. 190 -0.320 +0.043 +1.482 -0.030 +0. 025 -0.030 -0. 130 ' -0.043 -1.041 -0. 626 +1.372 -0.025 +0. 030 +0. 130 -1.482 +1.041 +0. 655 -1.372 + 1.482 -1.041 +0. 004 -1. 482 +1.041 -0. 004 -1.482 -1.5 -0. 130 -0.2 +1.037 +1.0 +0.445 +0.4 -1.158 -1.2 + 1.360 + 1.3 -0. 203 -0.2 APPLICATION OF LEAST SQUARES TO TKIANGULATION. Final solution of triangles 71 Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane an- gle Logarithm -10+11 -1+2 -4+6 Tum-Dundas Tower Turn Dundas Tower-Dundas Tower-Turn 42 00 24 17 113 41 30.0 25.4 59.6 + 0.8 + 1.7 + 3.0 30.8 27.1 62.6 0.1 0.2 0.2 30.7 26.9 42 02.4 4. 266771 0. 174417 9. 614231 9. 961733 4. 055419 4. 402921 + 5.5 0.5 -12+13 -1+3 -5+6 Turn-Dundas Lazaro Turn Dundas Lazaro-Dundas Lazaro-Turn 25 52 110 36 43 31 38.1 08.7 18.5 - 1.9 - 3.9 + 1.8 36.2 048 20.3 0.4 0.5 0.4 35.8 04.3 19.9 4. 266771 0. 360081 9. 971300 9. 837989 4. 598152 4.464841 - 4.0 1.3 -12+14 -2+3 - 9+10 / Turn-Tower Lazaro Turn Tower Lazaro-Tower Lazaro-Turn 42 30 86 18 51 11 21.5 43.3 09.1 - 5.8 - 5.6 - 0.6 15.7 37.7 08.5 0.6 0.7 0.6 15.1 37.0 07.9 4. 402921 0. 170282 9.999099 9. 891638 4. 572302 4.464841 -12.0 1.9 -13+14 -4+5 - 9+11 Dundas-Tower Lazaro Dundas Tower Lazaro-Tower Lazaro-Dundas 16 37 70 10 93 11 43.4 41.1 39.1 - 3.9 + 1.2 + 0.2 39.5 42.3 39.3 0.4 0.4 0.3 39.1 41.9 39.0 4. 055419 0. 543408 9. 973475 9. 999325 4. 572302 4. 598152 - 2.5 1.1 -14+15 -7+9 Lazaro-Tower Tow Hill Lazaro Tower Tow Hill-Tower Tow Hill-Lazaro 47 10 111 09 38.8 07.2 20.5 ( + 1.3 - 0.2 37.7 08.5 20.3 2.2 2.2 2.1 21 40 35. 5 06.3 18.2 4. 572302 0. 432543 9. 865314 9.969699 4. 870159 4 974544 6.5 -25+26 -14+16 -8+9 Lazaro-Tower Nichols Lazaro Tower Nichols-Tower Nichols-Lazaro 30 04 101 38 48 16 51.8 55.1 10.2 + 3.6 + 2.0 + 2.4 55.4 57.1 12.6 1.7 1.7 1.7 53.7 55.4 10.9 4.572302 0.299961 9. 990962 9.872905 4863225 4. 745168+ 1 + 8.0 5.1 -25+27 -15+16 Lazaro-Tow Hill Nichols Lazaro Tow Hill Nichols-Tow Hill Nichols-Lazaro 89 23 54 28 18.6 47.9 64.3 + 2.0 + 0.7 20.6 48.6 61.6 3.6 3.6 3.6 17.0 45.0 36 07 58. 4.974544 0.000025 9. 910573 9.770601 4 885142 4 745170-1 10.8 \ 72 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. Final solution of triangles — Continued Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane an- gle Logarithm -26+27 -7+8 Tower-Tow Hill Nichols Tower Tow Hill Nichols-Tow Hill Nichols-Tower 59 18 62 53 26.8 10.3 35.1 - 1.6 - 2.6 ti 25.2 07.7 39.3 4.1 4.0 4.1 21.1 03.7 57 48 35.2 4.870159 0. 065550 9. 949433 9. 927516 4. 885142 4. 863225 12.2 -31+32 -16+17 -23+25 Lazaro-Nichols Ken Lazaro Nichols Ken-Nichols Ken-Lazaro 116 59 22 32 40 27 49.5 57.3 12.5 + 1.3 + 1.5 + 0.1 50.8 58.8 12.6 0.8 0.7 0.7 50.0 58.1 11.9 4. 745169 0. 050108 9. 583744 9. 812130 4. 379021 4. 607407 + 2.9 2.2 -33+34 -16+18 -24+25 Lazaro-Nichols Sepl Lazaro Nichols Seal-Nichols Seal-Lazaro 128 55 38 29 12 35 09.3 18.8 33.3 + 1.6 + 0.6 - 2.2 10.9 19.4 31.1 0.4 0.5 0.5 10.5 18.9 30.6 4. 745169 0. 109005 9. 794041 9. 338465 4. 648215 4. 192639 +-U0 1.4 —33+35 -17+18 -30+31 Lazaro-Ken Seal Lazaro Ken Seal-Ken Seal-Lazaro 154 32 15 56 9 31 21.8 21.5 16.8 - 0.3 - 0.9 + 1.5 21.5 20.6 18.3 0.2 0.1 0.1 21.3 20.5 18.2 4.607407 0. 366640 9. 438723 9. 218592 4. 412770+ 1 4. 192639 + 0.3 0.4 -34+35 -23+24 -30+32 Nichols-Ken Seal . Nichols Ken Seal-Ken Seal-Nichols 25 37 27 51 126 31 12.5 39.2 06.3 - 1.9 + 2.3 + 2.8 10.6 41.5 09.1 0.4 0.4 0.4 10.2 41.1 08.7 4'. 379021 0. 364122 9. 669628 9. 905072 4. 412771 4. 648215 + 3.2 1.2 -36+37 -17+19 -29+31 Lazaro-Ken Mid Lazaro Ken Mid-Ken Mid-Lazaro 128 20 35 17 16 21 51.4 43.3 23.9 + 2.0 + 0.2 + 0.1 53.4 43.5 24.0 0.3 0.3 0.3 53.1 43.2 23.7 4. 607407 0. 105542 9. 761771 9. 449655 4. 474720 4. 162604 + 2.3 0.9 -42+44 -19+21 +36-38 Lazaro-Mid Round Lazaro Mid Round-Mid Round-Lazaro 51 05 43 39 85 15 11.2 34.4 11.7 + 3.6 - 0.9 + 0.5 14.8 33.5 12.2 0.2 0.2 0.1 14.6 33.3 12.1 4. 162604 0. 108962 9. 839081 9. 998508 4. 110647 4. 270074 + 3.2 0.5 APPLICATION OF LEAST SQUARES TO TEIANGULATION. Final solution of triangles — Continued 73 Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane an- gle Logarithm -42+45 -17+21 -28+31 Lazaro-Ken Round Lazaro Ken Round-Ken Round-Lazaro 74 42 34.2 78 57 17.7 26 20 06.8 + 2.6 - 0.7 + 1.3 36.8 17.0 08.1 0.6 0.7 0.6 36.2 16.3 07.5 4. 607407 0. 015651 9. 991879 9. 647016 4. 614937 4. 270074 + 3.2 1.9 -44+45 -37+38 -28+29 Mid-Ken Round Mid Ken Round-Ken Round-Mid 23 37 23.0 146 23 56.9 9 58 42.9 - 1.0 -2.5 + 1.2 22.0 54.4 44.1 0.2 0.1 0.2 21.8 54.3 43.9 4. 474720 0. 397167 9. 743050 9. 238760 4. 614937 4. 110647 - 2.3 0.5 -49+52 -21+22 -40+42 Lazaro-Round Cat Lazaro Round Cat-Round Cat-Lazaro 89 47 52.8 26 22 55.0 63 49 17.6 - 1.1 - 1.0 -2.9 51.7 54.0 14.7 0.2 0.1 0.1 51.5 53.9 14.6 4. 270074 0. 000003 9. 647723 9. 952995 3. 917800 4. 223072 - 5.0 0.4 -46+/17 -40+43 -51+52 Round-Cat Spur Round Cat Spur-Cat Spur-Round 33 20 40.5 111 30 09.4 35 09 14.0 - 2.2 - 0.9 - 0.6 38.3 08.5 13.4 0.1 0.0 0.1 38.2 08.5 13.3 3. 917800 0. 259903 9. 968671 9. 760250 4. 146374 3. 937953 -3.7 0.2 • -46+48 -42+43 -20+21 Round-Lazaro Spur Round Lazaro Spur-Lazaro Spur-Round 105 41 48.0 47 40 51.8 26 37 18.2 + 0.1 + 2.0 + 0.2 48.1 53.8 18.4 0.1 0.1 0.1 48.0 53.7 18.3 4. 270074 0. 016506 9. 868888 9. 651373 4. 155468 3. 937953 + 2.3 0.3 -47+48 -49+51 -20+22 Cat-Lazaro Spur Cat Lazaro Spur-Lazaro Spur-Cat 72 21 07.5 54 38 38. 8 53 00 13.2 + 2.3 - 0.5 - 0.8 09.8 38.3 .12.4 0.1 0.2 0.2 09.7 38.1 12.2 4. 223072 0. 020934 9. 911462 9. 902368 4. 155468 4. 146374 + 1.0 0.5 -59+61 -52+55 -39+40 Cat-Round Beaver Cat Round Beaver-Round Beaver-Cat 49 58 12.0 87 34 53.9 42 26 51.8 + 1.2 + 0.1 + 1.2 13.2 54.0 53.0 0.1 0.0 0.1 13.1 54.0 £2.9 3.917800 0.115935 9. 999613 9. 829253 4. 033348 3. 862988 + 2,5 0.2 74 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. Final solution of triangles — Continued Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane an- gle Logarithm -56+57 -39+41 -60+61 Round-Beaver Snipe Round Beaver Snipe-Beaver Snipe-Round 52 14 105 37 22 08 18.1 20.7 21.3 - 0.2 - 0.5 + 0.8 17.9 20.2 22.1 0.1 0.0 0.1 17.8 20.2 22.0 4.033348 0. 102063 9. 983652 9. 576182 4. 119063 3. 711593 + 0.1 0.2 -56+58 -40+41 -50+52 Round-Cat Snipe Round Cat Snipe-Cat Snipe-Round 79 10 63 10 37 39 11.9 28.9 24.5 - 2.0 - 1.7 - 1.5 09.9 27.2 23.0 0.0 0.0 0.1 09.9 27.2 22.9 3.917800 0. 007806 9. 950551 9. 785987 3.876157 3.711593 - 5.2 0.1 -57+58 -59+60 -50+55 Beaver-Cat Snipe Beaver Cat Snipe-Cat Snipfc-Beaver 26 55 27 49 125 14 53.8 50.7 18.4 - 1.8 + 0.4 - 1.4 52.0 51.1 17.0 0.0 0.0 0.1 52.0 51.1 16.9 3. 862988 0. 343980 9. 669189 9. 912095 3. 876157 4. 119063 - 2.8 0.1 -67+68 -59+62 -53+55 Beaver-Cat Khwain Beaver Cat Khwain-Cat Khw ain-B ea ver 62 13 58 43 59 03 29.4 17.2 08.4 + 2.0 + 1.8 + 1.3 31.4 19.0 09.7 0.1 0.0 0.0 31.3 19.0 09.7 3. 862988 0. 053161 9. 931792 9. 933305 3. 847941 3.849454 , + 5.1 0.1 -71+72 -59+63 -54+55 Beaver-Cat Lira Beaver Cat 36 34 102 39 40 45 55.5 45.5 17.4 + 1..1 + 0.7 - 0.1 56.6 46.2 17.3 0.0 0.0 0.1 56.6 46.2 17.2 3. 862988 0. 224770 9. 989306 9. 814795 + 1.7 0.1 Iiim-Cat Lim-Beaver 4.077064 3.902553 -71+73 -62+63 -66+67 B eaver-Khwain Lim Beaver Khwain Lim-Khwain Lim-Beaver 59 25 43 56 76 38 24.7 28.3 09.2 - 0.3 - 1.1 - 0.7 24.4 27.2 08^5 0.0 0.0 0.1 24.4 27.2 08.4 3. 849454 0. 065022 9. 841307 9.988077 3. 755783 3.902553 - 2.1 0.1 -72+73 -53+54 -66+68 Cat-Khwain Lim Cat Khwain Lim-Khwain Lim-Cat 22 50 18 17 138 51 29.2 51.0 38.6 - 1.4 + 1.4 + 1.3 27.8 52.4 39.9 0.1 0.0 0.0 27.7 52.4 39.9 3. 847941 0. 410972 9. 496870 9. 818151 3.755783 4.077064 + 1.3 0.1 APPLICATION OF LEAST SQUAEES TO TKIANGTJLATION. Final solution of triangles — Continued 75 Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane an- gle Logarithm -74+75 -53+65 -70+71 Beaver-Lim South Twin Beaver Lira South Twin-Lim South Twin-Beaver 60 78 40 40 24 55 06.2 38.4 17.8 - 1.5 - 1.4 + 0.6 04.7 37.0 18.4 0.0 0.1 0.0 04.7 36.9 18.4 3. 902553 0. 059585 9. 991054 9. 816260 3. 953192 3. 778398 - 2.3 0.1 -77+78 -74+76 -64+65 South Twin-Beaver Ham South Twin Beaver Ham-Beaver Ham-South Twin 35 94 50 38 10 11 30.7 29.2 06.3 - 1.9 - 2.5 - 1.7 28.8 26.7 04.6 0.0 0.1 0.0 28.8 26.6 04.6 { 3. 778398 0. 234548 9. 998847 9. 885424 4.011793 3. 898370+1 3. S98371 - 6.1 0.1 -77+79 -75+76 -69+70 South Twin-Lim Ham South Twin Lim Ham-Lim Ham-South Twin 85 33 61 04 30 25 23.4 23.0 13.8 + 0.6 - 1.0 + 0.3 24.0 22.0 14.1 0.1 0.0 0.0 23.9 22.* 14.1 3. 953192 0. 001608 9. 741959 9. 943571 3. 696759 3. 898371 - 0.1 0.1 -78+79 -63+64 -69+71 Beaver-Lim Ham Beaver Lim Ham-Lim Ham-Beaver 49 28 102 25 13 20 52.7 32.1 31.6 + 2.5 + 0.3 + 0.9 55.2 32.4 32.5 0.0 0.0 0.1 55.2 32.4 32.4 3. 902553 a 119395 9. 674811 9. 989845 3. 696759 4. 011793 + 3.7 0.1 76 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION TOWER Second a Aa J$ COS a B 1st term 2d term 3d and 4th \ terms / i»+<4') Turn to Dundas Dundas and Tower Turn to Tower Tower to Turn First angle of triangle 06. 742 39. 415 27. 327 -1 4. 402921 9. 9676448 8. 5097251 2. 8802909 +759.0859 + 0. 3175 +759. 4034 + 0.0115 +759. 4149 a r u 54 41 47 s 2 in 2 o 3d term 4th term sin a A' sec^' 8. 80585 9. 14128 1. 55459 Turn Tower +0. 0134 -0. 0019 4. 402921 9. 5706188 8. 5087480 0. 2370138 2. 7193010 +523.9635 Arg. s JX Corr. X 5. 7609 2. 3672 -11 + 5 - 6 357 + 24 180 201 42 130 131 -h s 2 sin 2 a E 04 sin iU>+^') sec iU) 07.7 27.1 34.8 07.6 27.2 30.8 04. 052 43. 9C3 48.015 2.8803 7. 9471 6. 4574 7. 2848 2. 7193010 9. 9117440 7 2. 0310157 +427. CI STATION LAZARO a Second a A COS a B 1st term 2d term 3d and 4th terms i(«+« Turn to Tower Tower and Lazaro Turn to Lazaro Lazaro to Turn o / 54 48 4 54 52 06. 742 51. 079 57. 821 -1 First angle of triangle Turn 4. 464841 9. 4935464 8. 5097251 2. 4681125 -293. 8411 + 2. 7534 -291. 0877 + 0.0085 -291. 0792 a I it 54 50 32. 3 s 2 sin 2 £ C 3d term 4th term sine* A' sec^' 8. 92963 9. 95564 1. 55459 Lazaro W +0. 0020 +0.0065 +27 4. 464841 9. 9778267 8. 5087409 0. 2401420 3. 1915533 + 1554.3661 Arg. s Corr. JX 4. 9281 2. 3672 -15 +42 +27. 21 108 180 257 42 131 -h s 2 Sin 2 a E 3i.8 37.7 12.5 10.8 01.7 15.7 04. 052 54. 366 sin iW+cS') sec iUw 58. 418 -1 2. 4681 8. 8853 6. 4574 7. 8108 3. 1915533 9. 9125251 3. 1040784 +1270.8 APPLICATION OF LEAST SQUARES TO TBIANGULATION. 77 primary triangulation statio:t tower Third angle a Act a' A COS or B 1st term 2d term 3d and 4th terms !«+*') Dundas to Turn Tower and Turn Dundas to Tower Tower to Dundas 35 09. 559 42. 233 27. 326 Dundas Tower 4.055419 9. 6439895 8. 5097372 2. 2091457 +161. 8623 + 0.3708 +162. 2331 +0.0001 + 162.2332 54 36 48. 4 sm 2 a C 3d term 4th term sm a A' sec<£' X A\ 8. 11087 9. 90630 1. 55194 D 4.4203 2. 3681 9. 56911 6. 7884 f 0. 0006 -0. 0005 +3 4. 055419 9. 9531462 8. 5087480 0. 2370138 Arg. A\ -2 +5 2. 7543273 Corr. +3 +567. 9725 177 -113 63 1X0 243 43.6 02.6 41.0 43.0 58. 131 -h s 2 sin 2 a E 04 20.042 27. 973 A\ sin 4W+0') sec i(j) 48. 015 2.2091 8. 0172 6. 4528 6. 6791 2. 7543283 9. 9112981 2. 6656264 + 463.05 STATION LAZAEO Third angle a a' Atf, ' COS a B 1st term 2d term 3d and 4th 1 terms / -A4> *(#+#') Tower to Turn Lazaro and Turn Tower to Lazaro Lazaro to Tower o / 54 35 17 54 52 27. 326 3d. 494 57.820 4. 572302 9. 9398618 8. 5097404 3. 0219042 -1051. 7298 1.2004 1050. 5294 0. 0358 -1050. 4936 54 44 12.6 sin 2 « C 3d term 4th term sin a A' sec.0' A\ Tower Lazaro AX 9. 14459 9. 38353 1. 55122 (W) J 6. 0428 2.3683 0. 0793 1 8.4111 +0. 0258 +0. 0100 -7 4. 572302 9. 6918222 8. 5087409 0. 2401420 Arg. s A\ -25 +18 3. 0130064 Corr. - 7 +1030. 4012 201 - 51 180 330 131 -h ! sin 2 ot E 27.2 08.5 18.7 01.3 48. 015 10. 401 A\ sin H+4>') sec i(A) -Aa 58. 416 +1 3. 0219 8. 5281 6. 4516 3. 0130064 9. 9119609 2. 9249673 +841.3 78 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position, computation, STATION TOW HILL Second angle a d 4>' COS a B 1st term 2d term 3d and 4th \ terms / *(#+*') Lazaro to Tower Tower and Tow Hill Lazaro to Tow Hill Tow Hill to Lazaro 54 04 57.820 32. 022 25. 798 First angle of triangle Lazaro Tow Hill A\ 4. 974544 9. 9794818 8. 5097191 3. 4637449 +2909. 0080 + 2. 8842 +2911. 8922 + 0.1299 +2912. 0221 O / It 54 28 41.9 S 2 sin 2 a C 3d term 4th term sin a A' J\ 9.94909 8:95504 1.55589 G5g) 2 6.9283 2.3667 0. 46002 9.2950 +0. 1972 -0.0673 -117 4.974544 9. 4775129 8. 5087606 0. 2315526 Arg. s -159 + 42 3. 1923584 Corr. -117 +1557.2502 330 + 47 180 197 21 131 -h s 2 sin 2 a E 17.4 08.5 25.9 07.5 18.4 37.7 58. 417 57.250 47 55. 667 -2 3. 4637 8.9043 6.4597 sin i(<£+$') sec JGW 8. 8277 3. 1923584 9.9105687 108 3. 1029379 +1267. 5 STATION NICHOLS Second angle 4> •/>' S COS a B 1st term 2d term 3d and 4th terms iW+tf') Lazaro to Tow Hill Tow Hill and Nichols Lazaro to Nichols Nichols to Lazaro 54 52 54 4.745169 9. 4910534 8. 5097191 57.820 27.129 30. 691 First angle of triangle Lazaro Nichols X 2. 7459415 +557. 1107 + 10.0559 +567. 1666 - 0.0374 +567. 1292 sin 2 c C 3d term 4th term sma A' sec $' JX 9. 49034 9. 95619 1. 55589 (ag) 2 5. 5072 2. 3667 1. 00242 7.8739 +0. 0075 -0. 0449 +92 4. 745169 9. 9780929 8. 5087447 0. 2384490 Arg. 8 - 56 +148 3. 4704648 Corr. + 92 +2954.3694 17 + 54 71 180 251 131 25.9 14.5 14.3 00.2 20.6 58. 417 14. 369 12. 786 +1 -h s 2 sin 2 a E sin JW+f) sec tU4>) -da 2.7458 9. 4465 6. 4597 3. 4704648 9. 9123203 3. 3827851 +2414. 27 APPLICATION OF LEAST SQUARES TO TKIANGULATION. 79 primary triangulation — Continued STATION TOW HILL Third angle a Act. a! 8 COSa B 1st term 2d term 3d and 4th 1 terms / -A4> i(W) Tower to Lazaro Tow Hill and Lazaro Tower to Tow Hill Tow Hill to Tower 54 35 31 54 04 27. 326 01. 527 25. 799 -1 Tower Tow Hill 4. 870159 9. 8881363 8. 5097404 3. 2680357 +1853. 6842 + 7. 8777 +1861.5619 - 0. 0351 +1861.5268 sin 2 £ C 3d term 4th term sin a A' sec#' JX X jx 9. 74032 9. 60486 1. 55122 (+$') sec i(J) 3. 3706 8. 7959 6. 4373 3. 1452347 9. 9101427 70 3. 0553844 +1136. 02 91865°— 15- 80 COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION KEN Second angle cos a B 1st term 2d term 3d an1 4th \ terms / -A4> «*+*') Lazaro to Nichols Nichols and Ken Lazaro to Ken Ken to Lazaro 1 54 52 1 54 54 57.820 36. 965 First angle of triangle Lazaro Ken X 4. 607407 8. 8949989 8. 5097191 -102. 8312 + 5. 8614 - 96.9698 + 0.0050 sin ! < C 3d term 4th term sin a A' d\ 9. 21480 9. 99731 1. 55589 +0.0002 +0. 0048 +58 4. 607407 9. 9986569 8. 5087403 0. 2404325 3. 3552425 +2265. 9094 (M) 2 Arg. s /(X Corr. 4.0250 2. 3667 -29 +87 +58 71 + 22 180 273 116 131 14.5 58.8 13.3 53.7 19.6 50.8 58. 417 45.909 -h « 2 sin 2 a E sin £(*+*') sec iU4) 44. 326 2.0125 9. 2121 6. 4597 7. 6843 3. 355242 9. 912798 +1853.70 STATION ROUND Second angle il+') sec $(J0) 13.3 17.0 30.3 37.8 52.5 36.8 58. 417 59.499 57. 916 +1 2. 7769 6. 6527 6. 4597 2.077365 9. 913183 1. 990548 +97. 85 APPLICATION OP LEAST SQUARES TO TEIANGULATION. 81 ■primary triangulation — Continued STATION KEN Third angle a J(6 COS a B 1st term 2d term 3d and 4th V terms / JW+tf') Nichols to Lazaro Ken and Lazaro Nichols to Ken Ken to Nichols o / 54 43 11 54 54 30. 691 04. 093 4. 379021 9. 9338378 8. 5097307 2. 8225895 -664. 6446 + 0.5380 -664. 1066 + 0.0132 -664. 0934 54 49 02. 9 « 2 sin 2 i C 34. 784 + 1 8. 75804 9. 41937 1. 55337 3d term 4th term sin a A' sectj,' d\ Nichols Ken :w)« 9. 73078 +0.0103 +0. 0029 -2 4. 379021 9. 7096861 8. 5087403 0. 2404325 2. 8378797 Arg. s JX Corr. X 5. 6450 2. 3676 8. 0126 -10 + 8 251 - 40 180 30 132 -h s 2 sin 2 a E 59 00.2 12.6 47.6 22.7 10.3 + .1 12. 787 28.462 sin i(tf+<*') sec J(J0) 44.325 + 1 2.8226 8. 1775 6.4553 7. 4554 2. 837880 9. 912392 2. 750272 STATION ROUND Third angle COS a B 1st term 2d term 3d and 4th \ terms / -A$ U4>+4>') Ken to Lazaro Round and Lazaro Ken to Round Round to Ken 54 54 34.785 21. 362 4. 614937 9.5800256 8. 5097172 2. 7046798 -506.6171 + 5.2285 -501. 3886 + 0.0270 -501. 3616 54 58 46. s 2 sin 2 a C 3d term 4th term sin<* A' sec$' J\ Ken Round X 9. 22987 9. 93219 1.55631 W # 2 5.4091 2. 3666 0. 71837 7. 7757 +0. 0058 +0. 0212 +49 4. 614937 9. 9660945 8. 5087369 0. 2419389 Arg. s -30 + 79 3. 3317122 Corr. +49 -2146. 4074 247 131 19.6 08.1 11.5 17.8 44. 326 46. 407 -h 2 sin 2 a E JX sin J(0+tf') sec J0W 57. 919 -2 2. 7046 9. 1621 6. 4604 8. 3271 3. 331712 9. 913254 3. 244966 -1757.79 82 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION CAT Second angle a COS a B 1st term 2d term 3d and 4th \ terms / m+) 6.6737 2. 504184 9.913117 STATION BEAVER Second angle Ja H # I COSa B 1st term 2d term 3d and 4th \ terms / «*+*') Cat to Round Round and Beaver Cat to Beaver Beaver to Cat 55 01 55 3.862988 9. 9799133 8.5097090 05 26.110 45.204 11.314 First angle of triangle Cat Beaver 109 + 87 197 180 17 49 131 16 37.4 54.0 31.4 40.2 11.6 13.2 39.129 02.231 2.3526103 -225. 2218 + 0.0170 -225. 2048 + 0.0012 -225. 2036 e r n 55 03 18. 7 s 2 sin 2 a C 3d term 4th term sm a A' sec^' J\ 7.7260 8. 9465 1.5584 +0.0012 -1 9. 4731109 8. 5087360 0. 2423463 2. 0871811 -122. 2309 (**)» 4.705 2.366 7.071 Arg. * -1 Corr. -1 -h s 2 sin 2 a E sin J(*+#') 2.353 6.672 6.464 2. 087181 9.913657 -100. 19 APPLICATION OF LEAST SQUARES TO TRIANGULATION. 83 primary triangulation — Continued STATION CAT a Third angle a Act ot' j$ s COS a B h 1st term 2d term 3d and 4th \ terms / -A$ i«+d') Round to Lazaro Cat and Lazaro 353 - 63 25 49 It 52.5 14.7 Round to Cat Cat to Round 289 + 36 5 37.8 59.6 180 109 42 37.4 o 55 t 02 1 56. 147 30. 037 56 81 36 Round X A\ X' 3.9069 2.3658 131 23 7 57. 917 18.788 55 3. 91781 9.5258 8.5097 30 533 }71 01 sii 2 S 2 C fi.110 7.83 9.94 1.55 Cat Arg. s A\ 131 -h s 2 sin ! E a 16 39. 129 1.9535 7.7837 6. 4643 1. 9533604 ii +89. 8174 + 0.2199 3d term 4th term J sin a A' seed' 9.3423 +0.0002 -0.0002 +2 3.917800 9.9740491 8.5087375 0. 2416677 6.2727 -1 +3 A\ sin J(d+d') sec l(i) 6.2015 2. 642254 9. 913558 +90.0373 +90. 0373 Q 1 II 55 02 1 1.1 i X 2. 6422545 II -438. 7877 Corr. +2 -Act 2.555812 II -359. 59 STATION BEAVER Third angle a Aa a' * id d' COS or B 1st term 2d term 3d and 4th \ terms ) i(d+d') Round to Cat Beaver and Cat Round to Beaver Beaver to Round o / 55 02 2 55 05 56.147 15. 166 11. 313 +1 4.033348 9. 5889657 8. 5097071 2. 1320208 -135.5254 + 0.3584 -135. 1670 + 0.0008 -135. 1662 O I II 55 04 03. 7 S 2 sin 2 a C 3d term 4th term sin a A' seed' A\ 8.0667 9.9291 1. 5586 Round Beaver +0.0004 +0.0004 +3 4. 033348 9. 9645467 8.5087360 0. 2423463 2. 7489773 -561.0186 Arg. s A\ Corr. X A\ 4.264 2.366 247 180 67 131 17 6.630 -2 +5 +3 131 -h s 2 sin 2 a E 37.8 53.0 44.8 24.7 +.1 57. 917 21. 019 36. 898 A\ sin j(d+d') sec 4(Jd) -Aol 2.132 7.996 6.464 6.592 2.748977 9.913723 2.662700 -459. 94 84 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION LIM a Second angle a Act a' * s COS a B h 1st term 2d term 3d and 4th \ terms / -A Beaver to Cat Cat and Lim 17 +102 19 39 ir 11.6 46.2 Beaver to Lim Lim to Beaver First angle of triangle 119 58 5 57.8 20.3 180 299 36 53 34 37.5 56.6 55 + 05 2 11. 314 08. 948 51 51 91 ieaver X A\ X' 4.222 2.366 131 + 14 6 36.898 30. 508 55 3.9025 9. 6987 8.5097 53 431 044 07 si] 2 S 2 1 2 <* C 0.262 7.80 9.87 1.55 Lim Arg. s A\ 131 -h s 2 sin E a 21 07. 406 +1 2.111 7.680 6.465 2. 1110005 n -129. 1221 0. 1735 3d term 4th term s sin a A' sec$' 9.2393 // +0.0004 +0.0002 +2 3.902553 9.9376062 8. 5087351 0. 2427356 6. 588 -1 +3 i\ sin iW+tf') see JGW 6.256 2. 591630 9. 913918 -128.9486 + 0.0006 -128.9480 55 06 1 5.8 t IX 2. 5916301 +390. 5082 Corr. +2 -Aa 2. 505548 +320.3 STATION SOUTH TWIN Second angle 4>' COS a B 1st term 2d term 3d and 4th ' terms -A iW+0') Beaver to Lim Lim and South Twin Beaver to South Twin South Twin to Beaver First angle of triangle 55 3. 778398 9. 9772271 8. 5097044 11.314 04.203 2. 2653295 */ -184. 2169 + 0.0130 -184. 2039 + 0.0008 -184. 2031 55 06 43. 4 sin* a C 15. 517 15.517 7. 5568 8. 9984 1. 5591 Beaver South Twin 3d term 4th term sm a A' sec ' AX 8. 1143 +0.0008 3. 778398 9. 4990449 8. 5087347 0. 2429024 2. 0290799 n -106.9252 Arg. » A\ Corr. X 119 + 78' 198 180 18 60 131 131 -h s 2 sin 2 a E 57.8 37.0 34.8 27.7 02.5 04.7 36.898 46.925 49. 973 +1 49. 974 JX sin J(*+*') sec JCW 2.265 6.555 6.465 5.285 2. 029080 9. 913958 1. 943038 -87.7 APPLICATION OP LEAST SQUARES TO TEIANGULATIOK. 85 primary triangulation — Continued STATION LIM a Third angle a Act a' 1 A6 4>' 8 COS a B h 1st term 2d term 3d and 4th 1 terms / — A6 if.6+6') Cat to Beaver Lira and Beaver 197 - 40 17 45 31.4 17.3 Cat to Lira Lira to Cat 156 32 3 14.1 40.0 180 336 28 34.1 55 + / 01 5 26.110 54. 152 41 99 34 Cat X JX V 5.099 2.366 131 + 16 4 39.129 28. 278 55 4.0770 9.9625 8.5097 54 205 090 07 sii 2 s 2 C 0.262 8.15 9.19 1.55 Lira (W) 2 Arg. 5 A\ 131 I -h s 2 sin 2 a E JX sin i(6+6') sec l(A6) 21 07. 407 2.549 7.354 6.464 2. 5492935 -354. 2367 + 0.0817 3d term 4th term 8 sin a A' sec 6' 8. 9124 r " +0.0029 +0.0002 4.077064 9. 6000497 8. 5087351 0. 2427356 7.465 -2 +2 6.367 2. 428584 9. 913752 -354. 1550 + 0.0031 -354. 1519 t ft 55 04 2 3.2 * IX 2. 4285844 +268. 2776 Corr. -Aa 2. 342336 +219. 96 STATION SOUTH TWIN Third angle a da 4> A4, COS a B 1st terra 2d term 3d and 4th \ terms / M+4') Lim to Beaver South Twin and Beaver Lim to South Twin South Twin to Lim 55 07 08 20.262 55.255 Lim South Twin 3. 953192 9. 2816903 8. 5097018 1. 7445841 -55. 5372 + 0. 2818 -55. 2554 + 0.0002 -55. 2552 55 07 47. 9 sin 2 a C 3d term 4th term srn<* A' sec^' JX 7.9064 9.9838 1. 5597 9. 4499 +0. 0001 +0.0001 +2 3. 953192 9. 9919052 8. 5087347 0. 2429024 2. 6967345 -497. 4329 Arg. 5 Corr. X 3.488 2.365 ISO 79 131 37.5 18.4 19.1 48.1 07.2 07. 407 17. 433 49. 974 -2 +4 +2 -h 2 sin 2 a E 4X sin 1(6+6') sec tU6) 1.744 7.890 6.467 6.101 2. 696734 9. 914053 2. 610787 -408. 12 86 COAST AND GEODETIC STJKVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION SEAL Second angle COS a B 1st term 2d term 3d and 4th terms iW+«i') Nichols to Ken Ken and Seal Nichols to Seal Seal to Nichols 30. 691 22. 365 53. 056 -1 First angle of triangle Nichols Seal X A\ 4. 648215 9. 7157086 8. 5097307 -747. 5741 + 5.1654 -742.4087 + 0.0438 -742. 3649 54 49 41. 9 sin 2 a C 3d term 4th term sin a A' sec $' AX 9. 2964 9. 8633 1. 5534 (*g) 2 5.747 2.368 0. 7131 8.115 +0.0130 +0.0308 +4 4. 648215 9. 931652 8. 508740 0.240667 Arg. s AX -4 +8 3.329278 Corr. +4 -2134. 4108 210 + 27 238 180 59 131 -h *2 sin 2 a E 35 47.6 41.5 29.1 04.7 33.8 + .1 10.6 12.787 34. 411 38. 376 + 1 AX sin i(#+*') sec i(J) 2.874 9.160 6.455 3. 329278 9.912450 3. 241728 -1744. 72 STATION MID Second angle 4> J.!, 4>' COS a B 1st term 2d term 3d and 4th terms i(#+tf') Round to Lazaro Lazaro and Mid Bound to Mid Mid to Round o / 55 02 4 54 57 56. 147 57. 777 58. 370 1 i. 110647 9. 853ft)3 8. 509707 2. 473457 +297. 4795 + 0.2961 +297. 7756 + 0.0014 +297. 7770 o / // 55 00 27.3 sin 2 a C 3d term 4th term s sin a A' sec^' AX First angle of triangle Round Mid 8.2213 9. 6916 1.5586 9. 4715 +0. 0021 -0. 0007 4. 110647 9. 845806 8. 508740 0. 241043 2. 706236 +508. 4356 A) 2 Arg. s AX Corr. X AX 4.947 2.366 7.313 353 + 51 180 224 85 131 23 52.5 14.8 07.3 56.5 10.8 12.2 57.917 28. 436 26. 353 -1 -h ' sin 2 a E AX sin J(d+#') „sec t(.A) > -Ac, 2.473 7.913 6.464 6.850 2.706236 9.913405 APPLICATION OP LEAST SQUARES. TO TBIANGULATION. 87 primary triangulation — Continued STATION SEAL Third angle s cos a B 1st term 2d term 3d and 4th \ terms / «*+#') Ken to Nichols Seal and Nichols Ken to Seal Seal to Ken 34. 785 18. 270 4. 412771 8.984161 8. 509717 1.906649 2. 3867 -78.2716 + 0.0017 -78.2699 54 55 13. 9 Sitfa C 3d term 4th term sma A' sec' 4X 8.8255 9. 9960 1. 5563 Ken Seal 0. 3778 +0. 0002 +0. 0015 +3 4.412771 9. 997972 8. 508740 0. 240667 3. 160153 Arg. * JX Corr. X JX 3.813 2.367 30 -126 264 180 84 131 47 10.4 09.1 01.3 43.3 44.6 -.1 44. 326 05. 949 6.180 -1 + 4 +3 -h s 2 sin 2 a E sin JW+0') sec i(4$) 1.907 8.821 6.460 7.188 3. 160153 9. 912942 3. 073095 -1183.30 STATION MID Third angle a COS a B 1st term 2d term 3d and 4th 1 terms / -40 «*+*')' Lazaro to Round Mid and Round Lazaro to Mid Mid to Lazaro 54 4. 162604 9.806246 8. 509719 57 57.820 00. 550 2. 478569 -301. 0017 + 0.4488 -300. 5529 + 0.0032 -300. 5497 54 55 28. 1 sin 2 a C 3d term 4th term sma A' sec^' 8. 3252 9. 7710 1. 5559 Lazaro Mid +0. 0021 +0. 0011 4. 162604 9. 885527 8. 508740 0. 241043 2. 797914 +627. 9340 Arg. s JX Corr. X JX 4.957 2.367 173 - 43 180 309 47 32 30.3 33.5 56. 22.9 + .1 58. 417 27. 934 26. 351 + 1 -h s 2 sin 2 a E sin i(+ •/>'.) sec JU0) 2.479 8.096 6.460 7.035 2. 797914 9. 912963 2. 710877 // +513. 90 88 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 2S. Final position computation, STATION SPUR a Second angle a Aa a' A 4>' t COS a B h 1st term 2d term 3d and 4th \ terms / -Aif, Hound to Cat Cat and Spur Round to Spur Spur to Bound First angle of triangle 289 +111 36 30 it 37.8 08,5 41 06 4 46.3 22.7 180 221 33 02 20 23.6 38.3 55 02 3 56.147 31. 319 Round X A\ X' 4.649 2.366 131 + 23 5 57.917 20.567 54 3.937 9.877 8.509 353 035 707 59 sii 2 C 4.828 -1 7. 8759 9.6358 1.5586 Spur Arg. 8 A\ 131 -h s^sin' E a 29 18.484 2.325 7.512 6.464 2. 324695 // +211.2006 + 0.1176 3d term 4th term s sin a A' soe^' 9. 0703 +0.0010 -0.0002 3. 937953 9. 817925 8. 508738 0. 241303 7.015 sin ito+0') sec J(J^) 6.301 2.505919 9.913468 +211.3182 + 0.0008 +211. 3190 o / // 55 01 1 0.5 A 2. 505919 // +320. 5672 Corr. -Aa 2. 419387 // +262.66 STATION SNIPE a Second angle a Aa a' * A$ 4>' a cos a B h 1st term 2d term 3d and 4th \ terms / -Aij, Round to Cat Cat and Snipe 289 + 63 36 10 37.8 27.2 Round to Snipe Snipe to Round First angle of triangle 352 + 47 05.0 29.8 180 172 79 47 10 34.8 09.9 55 02 2 56.147 45. 140 232 980 586 Round X A\ X' 4.436 2.366 131 23 57. 917 36. 371 55 3.711 9.996 8.509 593 547 707 00 sii 1 C 1.007 7.4 8.1 1.5 Snipe Arg. * A\ 131 -h « 2 sin E a 23 5 J e 21. 546 .218 .621 .464 2. 217847 +165. 1380 + 0.0015 3d term 4th term a sin a A' sec#' 7. 1798 +0. 0006 3. 711593 9. 098982 8. 508738 0. 241442 6.802 JX sin JU+0') sec J(J^) 4.303 1.560755 9. 913502 +165. 1395 + 0.0006 +165. 1401 'HI II 55 01 3 3.6 A 1. 560755 -36. 3710 Corr. -Ac i 1. 474257 it -29.80 APPLICATION OF LEAST SQUARES TO TKIANGTJLATION. 89 primary triangulation — Continued STATION SPUR Third angle a COS a B 1st term 2d term 3d and 4th \ terms / -J$ JW+tf') Cat to Round Spur and Bound Cat to Spur Spur to Cat 55 26.110 01.283 24. 827 Cat Spur 4. 146374 9. 425347 8. 509709 2. 081430 + 120.6230 + . 0. 6592 +121.2832 -0.0003 +121.2829 55 00 25. 5 s 2 sin 2 a C 3d term 4th term sin a A' sec (ft 1 JX 8.2927 9. 9681 1. 5582 9. 8190 +0.0003 -0.0006 +1 4. 146374 9. 984029 8. 508738 0. 241303 2. 880445 +759.3552 (»£)' Arg. s JX Corr. X JX 4.163 2.366 6.529 + 1 + 1 109 - 35 180 254 23 131 -h 2 sin* a 'E 37.4 13.4 24.0 22.1 39.129 39. 355 18. 484 JX sin J(*+^') sec i(J) 2.081 8.261 6.464 6.806 2.880445 9. 913402 2. 793847 +622. 07 STATION SNIPE a Third angle a AOL a' A' s COS a B h 1st term 2d term 3d and 4th \ terms / -A$ M+4>') Cat to Round Snipe and Round Cat to Snipe Snipe to Cat 109 - 37 42 39 37.4 23.0 72 03 5 14.4 29.7 180 251 57 44.7 55 01 1 26.110 15. 103 523 567 582 Cat X JX X' 3.949 2.366 131 + 16 6 39.129 42. 417 55 3.876 9.488 8.509 157 721 709 00 si 1 S 2 Q2<* C 1.007 7.7 9.9 1.5 Snipe Arg. 8 JX 131 -h 8 2 sin E a 23 i ( 21.546 .975 .709 .464 1. 874587 +74.9181 + 0.1850 V 3d term 4th term s sin a A' sec $' 9.2672 +0. 0002 -0.0001 3. 876157 9. 978339 8. 508738 0. 241442 6.315 JX sin U4,+') sec J(J^) 6.148 2.604676 9. 913436 +75. 1031 + 0.0001 +75. 1032 55 00 4 8.6 41 2. 604676 +402. 4167 Corr. -J< t 2. 518112 +329. 69 90 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, primary triangulation — Continued STATION 'KHWAIN Second angle J4> COS a B 1st term 2d term 3d and .4th terms «*+*') Beaver to Cat Cat and Khwain Beaver to Khwain Khwain to Beaver 55 55 3. 849454 9.3824014 8. 5097044 04 11.314 55.322 15. 992 -1 First angle of triangle Beaver Khwain 1. 7415598 +55. 1518 + 0. 1706 +55.3224 0.0000 +55.3224 o / // 55 04 43. 7 sin 2 a C 3d term 4th term sin a A' sec 4>' A\ 7. 6989 9.9740 1. 5591 +0.0001 -0. 0001 3. 849454 9. 986983 8. 508736 0. 242180 2. 587353 +386. 6812 (M)> Arg. Corr. X A\ 3.483 2.366 17 +58 ISO 255 62 131 11.6 19.0 30.6 17.1 13.5 31.4 26.681 03.579 -h s 2 sin 2 a E A\ sin £(#+*') sec i(A) 1.742 7.673 6.465 2. 587353 9.913782 2. 501135 +317.06 STATION KHWAIN* Third angle a Aa COS a B 1st term 2d term 3d and 4th 1 terms J !(#+*') Cat to Beaver Khwain and Beaver Cat to Khwain Khwain to Cat 55 26.110 49.881 3. 847941 9 872700 8. 509709 -169.9613 + 0.0797 -169.8816 + 0.0008 55 02 51. 1 sin 2 a C 3d term 4th term sin a A' sec^' A\ 7. 6959 9. 6470 1. 5584 Cat Khwain +0.0007 +0.0001 3. 847941 9. 823487 8. 508736 0. 242180 2. 422344 II . +264. 4502 Arg. s A\ Corr. X A\ 4.461 2.366 6.82*7 197 - 59 180 318 131 -h 2 sin 2 c E 10 21 31.4. 09.7 21.7 44.9 39.129 24. 450 03.579 A\ sin i(tf+f ) sec j(j0) 2.230 7.343 6.464 2. 422344 9.913616 2.335960 // +216.75 * This is right-hand portion ol computation above. APPLICATION OF LEAST SQUARES TO TRIANGULATION. 91 List of geographic positions— Felice Strait, Alaska, southeast Alaska datum Station Latitude and longitude Sec- onds in meters Azimuth - Back azimuth To station Distance Loga- rithm Tower 1907 54 131 35 04 n 27.326 48.015 845.0 862.3 201 243 43 43 27.2- 58.0 21 63 50 51 34.8 41.0 Turn Dundas Meters 25288.4 11361. 1 4.402921 4.055419 Laza.ro 1907 54 131 52 21 57. 820 58. 417 1788.0 1041.5 287 313 330 48 40 18 01.7 37.9 17.4 108 134 150 09 02 32 12.5 23.3 18.7 Turn Dundas Tower 29163.6 39641.7 37351.0 4. 464841 4.598152 4.572302 Tow Hill 1908 54 131 04 47 25.798 55.665 797.6 1012.2 197 218 07 47 18.4 56.1 17 39 28 22 25.9 58.4 Lazaro Tower 94307. 74158. 2 4. 974544 4.870159 Nichols 1907 54 132 43 11 30.691 12. 787 949.0 228.9 251 281 340 17 21 40 00.2 55.6 20.8 71 102 160 57 16 59 14.5 06.1 16.8 Lazaro Tower Tow Hill 55612. 1 72983. 6 76761.2 4. 745169 4. 863225 4. 885142 Ken , 1907 54 131 54 59 34.785 44.326 1075. 7 789.7 273 30 59 59 19.6 10.4 94 210 30 49 13.3 47.6 Lazaro Nichols 40495. 5 23934.3 4. 607407 4. 379021 Seal 1907 54 131 55 35 53.055 38.377 1640. 6 683.4 290 59 84 15 10 47 23.0 33.9 44.5 110 238 261 26 41 28 33.9 29.1 01.3 Lazaro Nichols Ken 15582. 6 44485. 1 25868. 5 4.192639 4.648215 4.412771 Mid 1914 54 131 57 32 58.370 26.352 1805.0 468.8 309 78 39 00 23.0 16.4 129 257 47 37 56. a 55.6 Lazaro Ken 14541. 3 29834. 6 4. 162604 4.474720 Round 1914 55 131 02 23 56. 147 57.917 1736. 3 1028. 3 353 44 68 25 31 08 52.5 07.3 29.3 173 224 247 27 24 39 30.3 10.8 11.5 Lazaro Mid Ken 18624. 12901. 7 41203. 8 ' 4.270074 4. 110647 4. 614937 Spur 1914 54 131 59 29 24.827 18. 484 767.7 328.7 221 326 02 44 23.6 11.7 41 146 06 50 46.3 11.9 Round Lazaro 8668. 7 14304. 3 3. 937953 4. 155468 Cat 1914 55 131 01 16 26.110 39.129 807.4 695.2 19 74 109 54 33 42 45.7 24.0 37.4 199 254 289 50 23 36 24.3 01.9 37.8 Lazaro Spur Round 16713.7 14007. 9 8275.6 4.223072 4. 146374 3.917800 Snipe 1914 55 131 00 23 11.007 21.546 340.4 383.0 172 251 47 57 34.8 44.7 352 72 47 03 05.0 14.4 Round Cat 5147.5 7518. 9 3. 711593 3.876157 Beaver 1914 55 131 05 14 11.314 36.898 349.9 654.5 17 45 67 148 19S 19 09 17 12 23 11.6 02.7 24.8 30.2 34.8 197 225 247 328 18 17 01 09 08 25 31.4 52.7 44.8 19.3 02.5 Cat Snipe Round Ham South Twin 7294. 4 13154. 2 10798. 1 10275.3 6003. 4 '3. 862988 4. 119063 4.033348 4.011793 3.778398 Khwain 1914 55 131 04 21 15.991 03.579 494.5 63.5 255 318 57 10 13.5 44.9 76 138 02 14 30.6 21.7 Beaver Cat 7070. 6 7046.0 3. 849454 3. 847941 Lim 1914 55 131 07 21 20.262 07. 407 626.6 131.3 197 258 299 336 359 33 58 53 28 19 05.0 19.1 37.5 34.1 01.9 17 79 119 156 179 34 05 -,s 32 19 14.5 07.2 57.8 14.1 05.0 Ham South Twin Beaver Cat Khwain 4974.6 8978. 3 7990. 1 11941. 6 5698. 8 3. 696759 3. 953192 3.902553 4.077064 3.755783 ADJUSTMENT OF TRIANGULATION BY THE METHOD OF VARIATION OF GEOGRAPHIC COORDINATES DEVELOPMENT OP FORMULAS A scheme of triangulation may be adjusted not only by means of equations of condition * but also by means of observation equations in which the number of independent unknowns is just sufficient to * There is some confusion in usage as to the term equation of condition, or condition equation. In this publication the meaning is restricted to that of an equation expressing some condition which is imposed a priori and independently of anything arising from the observations themselves, and which must be rigorously satisfied by the adopted results. An equation which expresses the results of an observation, and which will, in general, be satisfied only approximately by the adopted results, is not herein termed an equation of condition, but an observation equation. 92 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. determine the entire triangulation. These independent unknowns may very conveniently be taken as the small corrections to the assumed approximate geographic coordinates (that is, the latitudes and longitudes) of the points in the triangulation. To form the observation equations the relation must be found that connects the small change in the direction of a line with the small arbitrary changes in the geographic coordinates of its ends. The following derivation of the formulas is based on the formulas for the compu- tation of geographic positions given in U. S. Coast and Geodetic Survey Special Publication No. 8 and on the notation there used. A "d" before the symbol of a quantity denotes a small arbitrary change in that quantity. and X are, respectively, the latitude and longitude of A v the initial point of the position computation, which may also be thought of as the occupied point, while ' and X' are the latitude and longitude of B u the terminal point in the position com- putation, which may also be thought of as the point sighted on. By definition also, J = $'-$ JX = X'-X Ti=sB cos a a is the azimuth at A ± of the line A 1 B 1 reckoned from the south toward the west. JX = sA' sec ' sin a h cos« =is AX sm cx-- sA' sec ' A' sec 6' Ji The meaning of A' and B is explained in Special Publication No. 8. By differentiating the preceding equation and neglecting the effects of changes in A', B, and sec ' there results: , , A' sec T J Xdh-hd (J X)~] -cosec* a d« = ^-^ (JXJ° J (JX) 2 Multiplying by -sin' «= - ^ ^ ^ APPLICATION OF LEAST SQUARES TO TEIANGULATION. 93 and dividing by arc 1" in order to express da in seconds instead of in radians gives, da in seconds- ^, ^ arc r , [M(Ji)-M] sB cos a */j)\ _ s A' sin a sec ' sx s'BA' sec ' arc \" d ^ A > S 2 BA' sec <£' arc l" dfl sin or cos or sin a cos a „-, jT70(^)- S.A' sec ' sin a arc \" u ^ af -> S B cos a arc 1"°'* _ sin a- cos « r )= -dh = d(f>'-dcf> Evidently, also, d(AX)=dX'-dX It thus appears that, to the degree of approximation here adopted, it is the difference in the changes of coordinates at the ends of a line that turns the line in azimuth. The formulas for computing da become, da * sec - = *BA' secV arc 1" [J ^' ~ '*> + ^ §k ' ~ ^ sin a cos a f d^' — d dX' + - arc 1" |_ h "*" AX — 1 In practice — A may be used for Ti, but if a position computation has been made over the line, log h will be immediately available- The change in the azimuth a' at B t of the line B 1 A x for given changes in the coordinates of A x and B ± may usually be taken the same as the change in a, the azimuth at A x of the line A x B v * If the point A x is fixed d and dX are zero, and if B t is fixed d' and dX' are zero. This formula will now be applied to three examples, first, the adjustment of a quadrilateral, next the adjustment of three new points connected with a number of fixed points, and, lastly, to a figure involving a closure in geographic position. The steps to be taken and the precautions to be observed will be explained as they arise in the course of the examples. * For more exact formula to be used with longer lines, see Dr. F. R. Helmert's Hbhere Geodasie, vol. 1, pp. 495 and 496. For such lines some of the approximations made in the derivation here given are no longer permissible. 94 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. In all cases treated by this method, however complicated they may be, a start is made with the assumed positions of the points to be determined and the assumed azimuths and lengths of the lines sighted over. These positions, azimuths, and lengths must be consistent with each other and not too far from the final result so that the corrections to the assumed quantities are in fact small, as is implied in the development of the formulas. Otherwise it is not important how these preliminary quantities are found. ADJUSTMENT OF A QUADRILATERAL WITH TWO POINTS FIXED As a simple example a quadrilateral, A u A 2 , A s , A tl with two points, A t and A 2 , fixed is adjusted. The coordinates of A 1 and A 2 and the length and direction of the line A x — A 2 are fixed as shown in the first lines of the position computation that follows. The angles of the preliminary computation of the triangles are obtained from the list of directions. To obtain the preliminary positions, directions and lengths, the triangles A u A 2 , A 3) andA 2) A 3 , J. 4 were made to close by correcting each angle by approximately one-third of the error of closure as indicated in the triangle computation. This determined the entire quadrilateral. In each of the other triangles two sides and an included angle became known and thus their remaining parts were computed. List of observed directions * AT A 2 AT A 3 Station Direction f Station Direction t Initial ill At At 00 00.0+zi 00 00.0+Di 101 44 45. 1+us 133 53 46.3+!>a Initial il4 A, At 00 00.0+zs 00 00.0+!)7 31 03 42.5+»s 61 47 35.0+!* AT Ai AT Ai Initial A, At At 00 00.0+Zj 00 00.0+fj 26 40 23.5+Bb 47 31 20.2+i>6 Initial At ill A, 00 00.0+Z4 00 00.0+Cio 25 15 16.2+Sn 116 47 20.0+t)ia *.Seefig. Ion p. 16. t Bach observed value has its symbolic correction affixed. APPLICATION OF LEAST SQUARES TO TRIAJSTGTJLATION. Preliminary computation of triangles 95 Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane angle Loga- rithm A.1-A1 • ' " " 7/ " " 3.772745 A, 30 43 S2.6 +0.8 53.3 0.291566 A 2 101 44 45.1 +0.7 45.8 0.1 45.7 9.990809 Ai 47 31 20.2 +0.8 21.0 9.867787 +2.3 0.1 A2-A1 4.055120 Aff-As 3.932098 A2-A1* 3.772745 A 4 25 15 16.2 17.3 0.369934 At 133 53 46.3 45.8 0.1 45.7 9.857694 Ai 20 50 56.7 57.0 9.551339 +0.9 0.1 A r Ai 4.000373 At-Ai 3.694018 Az-Az 3.932098 Ai 116 47 20.0 -1.2 IS. 8 0.1 18.7 0.049306 At 32 09 01.2 -1.2 00.0 9.726024 A 3 31 03 42.5 -1.2 41.3 9.712614 -3.6 0.1 Ai-Az 3. 707428 At-A-t 3.694018 Ax-Az* 4.055120 A, 91 32 03.8 01.5 0.1 01.4 0.000156 Ai 26 40 23,5 24.0 9.652153 A z 61 47 35.0 34.6 9.945097 -2.2 0.1 Ai-Az 3. 707429-1 Ai-Ai 4.000373 * This triangle is computed from two sides and the included angle. 91865°— 15 7 96 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION A 3 Second angle !(*+*') 2d and 3d terms As to A\ A\ and A3 Aito A 3 A3 to ^2 60 56 56 01.089 56.720 57.809 X 60 56 29 -57. 0388 + 0.3184 -56. 7204 sin a A' COS a B First angle of triangle As A, S 2 sin 2 a C 156 +101 258 180 78 30 34 45.8 12.4 05.9 00.00 18.3 53.3 19.237 15.877 JX 3. 932098 9. 990544 8. 508600 0.313737 2. 744979 II -535. 8774 1. 756170 sin i(0+0') 7. 86420 9. 98109 1. 65750 9. 05279 +0. 3183 +0.0001 h 2 D 2. 744979 9. 941572 2. 686551 -485.90 . 3.512 2.322 5.834 STATION A 4 Second angle a «#+#') 1st term 2d and 3d terms -H As to As As and As Aito At A t to As First angle of triangle 01. 089 55. 340 As At X JX 180 110 116 149 +55. 2418 + 0.0979 +55. 3397 sin a A' sec $' cos a B 3. 694018 9. 538951 8.509299 sin 2 c C 7. 38804 9. 94466 1. 65750 +0. 0978 +0.0001 12.4 OO.O 12.4 29.0 00.00 41.4 18.8 19.237 07. 794 h 2 D 3.484 2.322 J\ 3. 694018 9. 972328 8. 508601 0. 313313 2. 488260 -307. 7939 JX sin i(.+') 2. 488260 9. 941507 2. 429767 -269.0 APPLICATION OF LEAST SQUARES TO TRIANGULATION. secondary triangulation. STATION A„ 97 a A\ to A° Third angle A3 and At A\ to A) Aa J4> A3 to Ai 1st term 2d,3d,and\ 4th terms / 60 1 58 1 it 56. 416 58. 607 60 56 57. 809 +118.0810 + 0.5258 +118.60-8 sin a A' sec^' cos a B 4.055120 9. 507765 8. 509295 2. 072180 Ai A 3 sin z c C X A\ 8. 11024 9. 95248 1.65837 9. 72109 +0. 5261 +0.0003 -0.0006 4.055120 9.976241 8. 508600 0.313737 2. 853698 A\ sin i(+>t>') 9. 941676 2. 795374 // -624. 27 336 -47 180 108 149 08.4 21.0 47.4 24.3 00.00 11.7 -.1 57. 360 54.000 25 03. 360 h* D 4.144 2.322 6.466 -h s 2 sins a E 2.072 8.063 6.640 6.775 STATION A t Third angle a Aa i«+«') 1st term 2d and 3d terms -A A3 to A2 A^ and A2 Aa to At At to A 3 60 60 60 56 02 +111.9962 + 0.0639 55 57.809 52.060 05. 749 A, A, +112.0601 sin a A' sec$' COS a B 3. 707428 9. 832477 sin 2 a C A\ 3. 707428 9. 865257 8. 508601 0. 313313 2.394599 +248.0841 JX sin i(tf+0<) -Aa. X JX 78 31 180 227 149 7. 41486 9. 73052 1.65808 +0. 0636 +0.0003 18.3 41.3 37.0 36.8 00.00 00.2 03.360 08.084 h.2 D 11. 444 -1 4.098 2.322 6.420 2. 394599 9. 941540 2. 336139 tt +216.8 98 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. FORMATION OF OBSERVATION EQUATIONS The observation equations used in making the adjustments are formed on the assumptions of the direction- method* Each point- ing of the telescope is treated as an independent observation and the sum of the squares of the corrections to the separate pointings is to be made a minimum. A single pointing, however, taken by itself determines nothing, for if each of the pointings at a station be changed by the same amount the set of pointings has the same significance as before. The effect is simply a change in the zero direction, which is a purely arbitrary matter. If a set of corrections to directions at a point has been determined by any method and the mean of these corrections is not zero, the sum of the squares of these corrections can always be diminished by subtracting from each correction the mean of all of the corrections so that the algebraic sum of the reduced corrections is zero. Hence in any set of directions adjusted by the method of least squares the algebraic sum of the corrections at a point is zero.f To allow for this change of zero direction, or for the constant correction to all directions at a point, an unknown constant correction, "z," is introduced into all equations expressing the results of observations at a point, a different "z" for each point where observations are taken. The observation equation may be written, Assumed azimuth + doc — observed azimuth + z — v = 0. The coefficients of the d's and dX's come from the last equation on page 93. As a sample, take those in the expression for v a . A ± cor- responds to the A t and A 3 to the B x of the explanation of the for- mulas. Sin a, cos «, Ji, and AX come from the position computation on page 97. log sin a 9.9762n 4. 7984ra 4. 7984n log cos <* 9.5078 log ft 2.0722 log AX 2. 8537n colog arc 1" 5. 3144 2. 7262n 1.9447 4. 7984n Number -532 Number +88 The observed angles in the following formation of equations come from the list of directions on page 94. Azimuth A 2 to A x (initial direction) 156 20 26. 6 Observed angle initial direction to A x 00 00. 0— z^+v t Observed azimuth A 2 to A^ 156 20 26. 6-z 1 +v 1 Assumed azimuth A 2 to A x 156 20 26. 6+rf« Assumed azimuth— observed azimuth 0=0. O+da+z^— v x l * See Wright and Hayford, Adjustment of Observations, Chap. VII. t This does not necessarily hold good when a line whose direction has already been fixed enters into the set. APPLICATION OF LEAST SQUARES TO TRIANGTJLATION. 99 Azimuth A 2 to J. t is fixed. Therefore da = and v t = z t Azimuth A 2 to A x (initial direction) 156 20 26. 6 Observed angle initial direction to A 3 101 44 45. 1— z 1 -\-v 2 Observed azimuth A 2 to A 3 258 05 11. 7 — z x +v 2 Assumed azimuth A 2 to A 3 258 05 12.4+da Assumed azimuth— observed azimuth 0=+0. 7-\-da-\-z l —v 2 da=- 730<^ 3 - 75 s - 75 3 ) +414 («W 4 - n =z 4 -447 304+2213 X t +1. 1 D 12 =2 4 -918 303-414 3^+918 30 4 +414 3,l 4 -l. 2 These equations contain s's which are of no particular interest in themselves. The normal equations might be formed and the z's eliminated in the regular way, but this work is made easier by the following mechanical rule, the effect of which is to form at once the 100 COAST AND GEODETIC STJBVEY SPECIAL PUBLICATION NO. 28. reduced normal equations with the s's eliminated. For the proof of the rule and further particulars see Jordan's Handbuch der Ver- messungskunde, Vol. I, pages 151-171, of the third edition. Each direction is assumed to have equal weight. Write the observation equations dropping the s's and giving each unit weight. Add together as they stand all observation equations containing the z for any particular point. Drop the z term out and treat the result- ing equation as a new observation equation with a negative weight equal to — 1/r, where r is the total number of directions, both fixed and to be determined, that have been observed at the point in ques- tion. To reduce the new fictitious observation to unit weight it must be multiplied through by *J — = -^[r% where i = -J — 1. Table 1 below shows the coefficients of the unknowns, the coeffi- cients formed by adding the equations containing any partic- ular z, and the weights. Table 2 shows these equations divided through by 100 for convenience. This has no effect on the relative weights. The table also contains the fictitious observation equa- tions obtained by multiplying the sum equation by y— • From Table 2 the normal equations which do not contain the s's are formed in the ordinary way for observation equations of equal weight, using the i's strictly according to algebraic laws. Thus in the first line of Table 2 ( — 4.23i) 2 contributes to the first diagonal coefficient not +17.8929 but + 17.8929i 2 , or -17.8929, and (-4.23i) X + (1.26i) contributes toward the side coefficient not —5.3298, but -5.3298i 2 or +5.3298. Table for formation of normals, No.l 8fa *X 8 l$i SU I P VF Snm Zl 1 1 1 3 - 730 - 730 - 75 - 75 -1212 -1212 +218 +218 +0.7 -0.5 +0.2 1 1 1 -i l l i 0.58S Sum Za 1 1 1 3 - 532 - 532 + 88 + 88 - 447 - 447 +221 +221 -0.8 -0.3 -1.1 1 1 l -i 1 1 1 0.58! Sum Zs 1 1 1 3 - 918 - 730 - 532 -2180 -414 - 75 + 88 -401 + 918 + 918 +414 +414 +0.0 -1.2 -0.4 -1.6 1 1 1 -i 1 1 1 0.58! Sum Z( 1 1 1 3 - 918 - 918 -414 -414 -1211 - 447 + 918 - 740 +218 +221 +414 +853 +0.0 +1.1 -1.2 -0.1 l l l -i 1 1 1 0.58i APPLICATION OP LEAST SQUARES TO TEIANGULATION. Table for formation of normals, No. 2 101 tf& *Xs 34* S\ t 1 -T 2 3 Zl - 7.30 - 4.238 -0.75 -0.44s -12.12 - 7.03i +2.18 +1. 26i +0. 007 -0.005 +0.00116! - 8.043 - 9.945 -10. 438841 4 5 - 5.32 - 3.09J +0.88 +0.5H - 4.47 - 2.59! +2.21 +1. 28! -0.008 -0. 003 -0. 00638! - 4.448 - 2.263 - 3.89638! 7 8 9 Z3 - 9.18 - 7.30 - 5.32 -12.64J ' -4.14 -0.75 +0.88 -2.33J + 9.18 + 5.32! +4.14 +2. 40! +0.0 -0.012 -0.004 -0. 00928! 0.0 - 8.062 - 4.444 - 7.25928i 10 11 12 - 9.18 - 5.32i -4.14 -2. 40i -12. 11 - 4.47 + 9.18 - 4.29i +2.18 +2.21 +4.14 +4.951 +0.0 +0. Oil -0. 012 -0. 000582 - 9.93 - 2. 249 - 0.012 - 7.060581" Normal equations ^3 Sh 8 Duck-Lubec Chan- 3. 045619 nel Lighthouse -1+ 5 Gunner 115 34 42.9 + 7.1 50.0 0. 041803 +24 Duck 33 11 41.9 - 1.9 40.0 9. 738370 Lubec Channel 35.2 - 5.2 30.0 31 13 30.0 9. 714665 Lighthouse Gunner-Lubec 2. 828792+1 Channel Light- house Gunner-Duck 2 805087 Indian Point-Lar- rabee Gunner 3.236668 -2+ 3 31 57 41.4 + 7.0 48.4 0. 276234 +23 Indian Point 114 00 36.0 + 2.5 38.5 9. 960694 -22 Larrabee 34 01 32.0 + 1.1 33.1 9. 747852 +10.6 Gunner-Larrabee 3. 473596 Gunner-Indian 3. 260754 Point Indian Point-Mam 3. 470097 -2+ 4 Gunner 94 56 05.9 + 5.8 11.7 0. 001614 +23 Indian Point 47 05 36.3 + 2.5 38.8 9. 864792 -21 Mam 37 58 10.6 - 1.1 03.5 9. 789044 + 7.2 Gunner-Mam 3.336503" 1 Gunner-Indian 3.260755-' Point Indian Point^Lu- 3.315762 boc Channel -2+ 5 Lighthouse Gunner 101 48 54.9 + 9.3 64.2 0. 009304 +23 Indian Point 18 35 56.3 + 2.5 58.8 9.503728 Lubec Channel 68.8 57.0 59 34 57.0 9. 935688 Lighthouse G u n n e r-L u b e c 2. 828794-1 Channel Light- house Gunner-Indian 3.260754 Point Larrabee-Mam 3. 443126 -3+ 4 Gunner 62 58 24.5 - 1.2 23.3 0. 050223 +22 Larrabee 44 10 37.0 - 1.1 35.9 9. 843153 -21 Mam 72 51 01.9 - 1.1 00.8 9. 980247 - 3.4 Gunner-Mam 3.336502 Gunner-Larrabee 3. 473596 108 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary computation of triangles — Continued Symbol Station Observed angle Correc- tion Spheri- cal angle Spheri- cal excess Plane ingle Loga- rithm Larrabee-Lubec • / // " " • , " 3. 449573 Channel Light- house -3+ 5 Gunner 69 51 13.5 + 2.3 15.8 0.027417 +22 Larrabee 12 59 29.9 - 1.1 28.8 9. 351803 Lubec Channel 16.6 15.4 97 09 15.4 9.996606 Lighthouse Gunner-Lubec 2. 828793 Channel Light- bouse Gunner-Larrabee 3.473596 Larrabee-Lubec 3. 702878 church spire -3+7 Gunner 154 32 48.4 +20.7 69.1 33 09.1 0. 366851 +22 Larrabee 10 45 60.0 - 1.1 58.9 9.271388 Lubec church spire Gunner-Lubec 71.6 52.0 14 40 52.0 9. 403873 3.341111 church spire Gunner-Larrabee 3. 473596 Mam-Lubec Chan- 3. 176968 nel Lighthouse -4+5 Gunner 6 52 49.0 + 3.5 52.5 0. 921500 +21 Mam 3 04 49.6 + 1.1 50.7 8. 730324 Lubec Channel 21.4 16.8 170 02 16.8 9. 238033 Lighthouse Gunner-Lubec 2. 828792+1 Channel Light- house Gunner-Mam 3.336501+' Lubec church 3. 603183 spire-Indian Point + 2-7 Gunner 173 29 30.2 -27.7 02.5 0.945080 Lubec church 56 55.4 85.6 2 57 25.6 8. 712552 -23 spire Indian Point Gunner-Indian Point Gunner-Lubec church spire Gunner-Lubec Channel Light- house 3 33 34.4 - 2.5 31.9 8. 792909' 3.260754 3.341111 2.828793 - 9+10 Cranberry Point 75 45 12.8 0.0 12.8 0.013566 - 5+ 8 Gunner 90 05 58.7 0.0 58.7 9.999999 Lubec Channel 48.5 48.5 14 08 48.5 9.388114 Lighthouse Cranberry Point- 2.842358 Lubec Channel Lighthouse Cranberry Point- 2.230473 Gunner Gunner-Mam 3.336502 - 9+11 Cranberry Point 78 36 65.9 -13.1 52.8 0.008631 -4+8 Gunner 90 58 47.7 + 3.5 51.2 9.996768 -20+21 Mam 4 24 21.3 - 5.3 16.0 8.885341 -14.9 Cranberry Point- 3.341901 Mam Cranberry Point- 2.230474-1 Gunner Gunner-Lubec 3.341111 church spire - 9+13 Cranberry Point 174 08 32.3 +11.5 43.8 0.991389 -7+8 Gunner 5 24 23.8 -18.4 05.4 8.973748 Lubec church spire 03.9 10.8 27 10.8 7. 897971 Cranberry Pomt- 3. 306248 L u b ec church spire Cranberry Point- 2.230471+* Gunner APPLICATION OF LEAST SQUARES TO TBIANGTJLATION. Preliminary computation of triangles — Continued 109 Symbol Station Observed angle Correc- tion Spheri- cal angle Spheri- cal excess Plane angle Loga- rithm Lubec Channel ° , „ " " . , „ 3.176968 Lighthouse-Mam Cranberry Point -10+11 2 51 53.1 -13.1 40.0 1.301768 Lubec Channel 35.2 54.7 175 48 54.7 8. 863166 Lighthouse -20 Mam Cranberry Point- Mam Cranberry Point- 1 19 31.7 - 6.4 25.3 8.363626 3.341902-1 2. 842362-« Lubec Channel Lighthouse Lubec church 3.309982 spire-Treats -13+14 Cranberry Point 17 43 62.9 - 4.7 58.2 0.516300 Lubec church spire 19.0 28.0 144 41 28.0 9. 761916 -19 Treats Cranberry Points Treats Cranberry Point- Lubec church spire Lubec church s p i r e-Cranberry Point 17 34 38.1 - 4.3 33.8 9. 479966 3.588198 3.306248 3.306248 -15+16 Telegraph 128 38 59.9 -52.6 07.3 0. 107274 Lubec church spire Cranberry Point 51 47.5 85.9 30 52 25.9 9. 710244 -12+13 20 29 12.6 +14.2 26.8 9.544138 T e 1 e g r aph-Cran- 3. 123766 berry Point Telegraph-L u b e c church, spire 2.957660 Lubec church 3.341111 spire-Gunner Telegraph — -15+17 131 33 53.1 -54.8 58.3 32 58.3 0. 125876 Lubec church spire 24 36.1 75.1 30 25 15.1 9. 704449 -6+7 Gunner Telegraph-Gunner Telegraph- Lubec church spire Cranberry Point- Gunner IS 01 30.8 +15.8 46.6 9. 490673 3. 171436 2.957660 2.230473 -16+17 Telegraph Cranberry Point 2 54 53.2 - 2.2 51.0 1.293796 - 9+12 153 39 19. 7 - 2.7 17.0 9.647167 -6+8 Gunner 23 25 54.6 - 2.6 52.0 9.599497 - 7.5 Telegraph-Gunner 3. 171436 T e 1 e g r aph-Cran- 3. 123766 berry Point Cranberry Point- 2.842358 Lubec Channel i -16+18 Lighthouse Telegraph Cranberry Point 29 52 44.5 - 9.6 34.9 0.302657 -10+12 77 54 06.9 - 2.7 04.2 9. 990244 Lubec Channel 08.6 20.9 72 13 20.9 9.978751 Lighthouse Telegraph-L u b e c 3. 135259+1 Channel Light- house T e 1 e g r aph-Cran- 3.123766 berry Point Gunner-Lubec 2.828793 Channel Light- house -17+18 Telegraph 26 57 51.3 - 7.4 43.9 0.343516 -5+6 Gunner 66 40 04.1 + 2.6 06.7 9. 962951 Lubec Channel 01.6 09.4 86 22 09.4 9.999127 Lighthouse Telegraph-L u b e c 3.135260 Channel Light- house Telegraph-Gunner 3.171436 110 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION GUNNER Second angle J 1st term 2d term Duck to Lubec Channel Lighthouse Lu bee Channel Lighthouse and Gunner Duck to Gunner Gunner to Duck 44 50 44 44 SO -10.2275 + 0.0008 50 33.886 10. 227 44. 113 First angle of triangle Duck Gunner -10.2267 sma A' COS a B 2. 805087 9. 694202 8. 510480 1.009769 sin^c C JX 2. 805087 9. 939097 8. 508994 0. 149348 1. 402526 +25. 2654 JX sin i(#+*') X JX + 33 119 180 299 115 5. 6102 9.8782 1. 4016 1.402526 9. 848301 1.250827 +17.82 38 57 42.1 40.0 22.1 17.8 00.00 04.3 50.0 47. 822 25.265 13.087 STATION CRANBERRY POINT Second angle a Ja i«+tf') 1st term 2d term -A$ Gunner to Lubec Channel Lighthouse Lubec Channel Lighthouse and Cranberry Point Gunner to Cranberry Point Cranberry Point to Gunner First angle of triangle 44 + 50 50 44. 113 04.529 48. 642 Gunner Cranberry Point X JX 55 + 90 145 180 325 75 -4. 5288 0. 0000 COS a B 2. 230473 9. 915025 8. 510480 0. 655978 sin* a C 4.4609 9. 5103 1. 4016 5.3728 sin a A' JX 2. 230473 9. 755164 8. 508994 0.149357 0. 643988 +4. 4054 JX sin i(+$') 0. 643988 9. 848315 0. 492303 +3.11 18 58 54.3 58.7 53.0 03.1 00.00 49.9 12.8 13. 087 04.405 17. 492 APPLICATION OP LEAST SQUARES TO TEIANGULATION. Ill secondary triangulation STATION GUNNER Third angle a Aa. <*' 4> 1st term 2d term -40 Lubec Channel Lighthouse to Duck Gunner and Duck Lubec Channel Lighthouse to Gunner Gunner to Lubec Channel Lighthouse 44 50 44 O I It 44 50 36 -12. 4618 + 0.0008 50 31. 652 12. 461 44.113 Lubec Channel Lighthouse Gunner -12. 4610 sine* - A' sec0' cos a B 2. 828793 9. 756308 8. 510480 1. 095581 sin 2 a C 4X 9.914474 8. 508994 0. 149348 1. 401609 -25. 2121 A\ sin H0+0') JX X' 5. 6576 9.8289 1. 4016 - 31 235 ISO 55 1. 401609 9. 848294 -17. 78 58 06.5 30.0 36.5 17.8 00.00 54.3 38.299 25.212 13.087 STATION CRANBERRY POINT a. Third angle a a' Lubec Channel Lighthouse to Gunner Cranberry Point and Gunner 235 - 14 12 08 36.5 48.5 Lubec Channel Lighthouse to Cranberry Poin Cranberry Point to Lubec Channel Lighthouse 221 + 03 48.0 14.7 180 41 00 04 00.00 02.7 * 44 + 50 31.652 16. 990 Lubec Channel Lighthouse Cranberry Point X A\ X' 66 58 38.299 20. 807 ■ 44 50 48. 642 66 58 17. 492 4(^+0') 1st term 2d term -40 44 50 40 -16.9903 + 0.0005 s COS a B h 2. 842358 9. 877362 8. 510480 S 2 sin 2 a C 5. 6847 9.6350 1. 4016 1. 230200 6.7213 -16.9898 8 sin a; A' sec0' * 2. 842358 9. 817494 8.508994 0. 149357 AX sin K0+0') 1 9 318203 848303 1. 318203 1 166506 AX -20. 8067 -Aa -14.67 91865°— 15 8 112 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary •position computation, STATION TELEGRAPH Second angle 4' ito+tf') 1st term 2d term Cranberry Point to Lubec Channel Lighthouse Lubeo Channel Lighthouse and Telegraph Cranberry Point to Telegraph Telegraph to Cranberry Point First angle of triangle 48.642 20. 860 09.502 Cranberry Point Telegraph X 41 + 77 -20. 8635 + 0.0034 -20. 8601 sin a A' sec$' COS a 3 3. 123766 9. 685141 8.510480 sin 2 a C 6.2475 9.8839 1. 4016 3. 123766 9.941951 8. 508994 0. 149401 1. 724112 tt +52.9800 sin i(0+tf') 1.724112 9. 848343 1. 572455 +37. 36 180 298 66 58 02.7 04.2 06.9 37.4 00.00 29.5 34.9 17.492 52.980 10.472 APPLICATION OF LEAST SQUARES TO TEIANGULATION. 113 secondary triangulation — Continued . STATION TELEGRAPH a 1 n a Third angle a ■ Aa. a' Lubec Channel Lighthouse to Cranberry Point Telegraph and Cranberry Point Lubec Channel Lighthouse to Telegraph Telegraph to Lubec Channel Lighthouse 221 - 72 03 13 48.0 20.9 148 50 27.1 22.7 180 328 00 50 00.00 04.4 <*> 4? 44 + 50 31. 652 37.850 Lubec Channel Lighthouse Telegraph X JX X' '66 + 58 38.299 32. 173 44 51 09. 502 66 59 10. 472 1st term 2d term -Aj> 1 II 44 50 50 -37. 8511 + 0.0012 * COS a B h 3. 135260 9. 932338 8. 510480 «2 sin 2 a C 6.270 9.427 1.401 i 1.578078 7.099 -37. 8499 s sin a A' seo^' 3. 135260 9. 713841 8. 50S994 0. 149401 sin l(4>+4>') 1. 507496 9. 848324 1. 507496 1. 355820 JX +32. 1733 -Ja +22. 69 114 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. FORMATION OF OBSERVATION EQUATIONS In the first column of the following table is given the assumed azimuths of the various lines. Under station Gunner the azimuth to Duck, 299° 38' 04".3, comes from the position computation on page 110, and the other azimuths are obtained by adding to this the corrected angles from the preliminary computation of triangles. Thus in the first triangle in that list, Gunner-Duck-Larrabee, the angle at Gunner is 45° 43' 34."2, which, added to the azimuth of Duck, gives 345° 21' 38".5, as shown in the table. For any one station the assumed and observed azimuths of some one station may be taken as identical. At Gunner they are identical on station Duck. The observed azimuths in the second column of the table have their symbolic corrections affixed. These azimuths are ob- tained by adding the observed angles as derived from the list of observed directions on page 106 to the azimuth of A the line Gunner- Duck. At the fixed stations given in the lower part of the table the method of computing the assumed and observed azimuths is somewhat different. The assumed azimuths of the fixed lines come from the table of fixed positions on page 106. The assumed azimuths of the new lines are found by adding to one of these fixed azimuths the appropriate corrected angle from the computation of triangles on pages 107-109. In the second column of the table the observed azimuth of one fixed line used as an initial line is taken identical with its assumed azimuth, and the- other observed azimuths, whether of fixed lines or of new ones, are found by adding to this azimuth the observed angles between the initial line and each of the others as derived from the list of directions. The coefficients of the d's and Ws are found from the formulas on page 93. APPLICATION OF LEAST SQUARES TO TEIANGULATION. 115 o g H S P O ! <4 « *Jfa3IJ€ i i ON + L+ I I *S CO "+ + + +,< oeocsioos . -C M rl » H MMrtH J. ' L Lt+i h©«Oh os "O -^ r- eo rH l^tDOOrH I I I I I U | ' ttt N N N N "5" N N r s? » » ll II 11 [[ [I N H N M N N N N I I I I I I I I 6» rH hiQ »0 1-1 COHUJOOWOtOO -*cjodi-HTpi-Jr^co O'OCOOiOO'*!!) NWM rHi-Hi-H M o !* M Ph- IS M f! H « p. II i-» ps ® 3 c t ^ 1 oa © ". oo • £+ I ++2 *o 'o *a *© (< ,-h th w Sooo eq os-* cor~r-~ co *h ~h oo 3; t-- i-i OOhqn LLLLL ' N N N N N M II IK II II II II Si Si S- Si Si Si Si ~ » ^ » SJ tttttt M f* N N N N I I I I I I os cj us aS cj uj 00 ■* « CO t> rH in rH co oo os t- w •*!« -«i r-t co »ra OS t*« t"~ OS I s * O M Ph H « O w A H Eh i L IS coco 77 +++ 1 9338 OSOeOOQ -&-&■«.■& iSq '-O 'O «0 rH OS CO CO 00 00-*" rH rHVCOd 111 + S 1 £T £T 5 ++++ M M M N I I I I WHCOtD OQOiHOQ .-«6»COCO Eh H W « En So iff II J! i I oi r^ CO i-H rH "■* co»o coco cot-^ 116 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 23. tots S g ■>-* C^ 1-1 ^ _4 CO IffflJJ s"s *> Vs & K> S> Ss £> S> :a »& r? £ £ ++++++ tf «S tf <$ *s & I I I I t I ND5OMW00 ci eo ■* i-4 ci e 8 d,y o3 !•§ ■§§§'■3 + sf±\ a si as as » s «a a » a ■S8SS88 ++++++ n ^? n <•? b? iS I i I i I I rinSONN t^OOt^CD-*^ E-i "A l-H O £4 I— t A !z; raft J- * ** T l • + + + + + I I I I I COOOOOTtH t-. cS rt -^ « io ,_! MCOOO II + ++ APPLICATION OF LEAST SQUARES TO TKIANGULATION. 117 Figure 6 taken with the following two tables shows that z t is for directions taken at Gunner, z 2 for directions at Cranberry Point, z 3 for directions at Telegraph, and s 5 for directions at Mam. The scheme for eliminating these z's by the use of the sum equations, as fictitious additional observation equations with negative weights, is used here in the same manner as in the previous example. Each weight is the negative reciprocal of the total number of observed lines in the adjustment that radiates from the point in question. The weights are, respectively, —1/8, —1/6, —1/4, and —1/6, as shown in Table 1. At Treat 2 , Larrabee, Indian, and Duck where only one new line is to be determined the same process might be used, but the following method is identical in results and slightly shorter. Use is made of the fact that the directions taken at a point may each be changed by the same amount, a change equivalent to using merely a different zero point. Correct each of the directions by the averages of all the corrections necessary to reduce the observed results to the accepted results on the lines that have already been fixed. Then drop the z from the observation equation of the new line and assign the equation a positive weight equal to -, where s is the number of lines already fixed and therefore s + 1 is the total number of lines. Thus at Larrabee the constant terms of the observation equations repre- senting pointings on lines already fixed are + 2.6, —0.4, +9.0, —0.1, and 0.0, the mean of which is +2.8. Subtracting this from each of the preceding numbers we have —0.2, —3.2, +6.2, —0.1, and —2.8 as the new constant terms, also —0.8 instead of +2.0 on Gunner, the new point. These new values are inclosed in parentheses and are used in forming the normal equations. There are five fixed lines, so the weight of the new equation without z that is used to replace the six equations containing z is 5/6 and the equation itself is v 22 = + 541^ - 1473^ - 0.8 which in Table 2 corresponds to the line No. 22, + 0.49^-1.34^-0.07. The z's are computed from the sum equations as in the previous example, the result of substitution in the right-hand side being di- vided by-l/r, r being the total number of lines through the point to which z applies. For fixed points where only one new line occurs, substitute 8$ and d/i in the right-hand side of the observation equation on the new line omitting the z, and divide the result by - 1/r. Thus at Duck (see p. 116), z s = (8669*^ -3609^- 1.9) + (- 3) as shown in the computation below. 118 COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 28. When the z's are known the v's or corrections are computed from the equations on pages 115 and 116. The details are shown in the table on page 121. For convenience of solution in the normals it is best to divide the constant terms by 10 and the coefficients by 1000. The solution will then give 100^ and 100& 0>0 CNCN SO CM CMOS •oto CO OO oio COO HO wco-sh cor-o TPCMOS l-t**H O CN OS CO OCO OS CM tHO connco cooseocN rn-giNeo cooon rH l-H QCM iQCM O0H CM iQ CO >-lO0 lOCMr-i ION COOONCO g CNN noo TflOS Tfl CO NCNtPOOCMCS COQCMOSOOCO CiOOMWN cmSS QOO «CO 00 00 COO tot^ i-HO Tj< NrH cm'o MVi-(d NO COOOOi-H 3° 1 + cSdao Tfl s 00 rH Oi-H •OCM no iHOSlO noon ■*wo COi-fiO CM 00 nco Oi-H IQ00 OStH CN CMOS OS rH OS CO 00 i-i nioos N CM CM 1 OS CM CM i-H HOJCBNN *5fl rH CO 00 OS TH OCO*cfi CO OOirHON N rUN OCO CN N COCO CO i— ' Tjl CO Tt* OS OJOCOCDN^i INONiHtHM NOONQ0V CM + OO 1 + COO* + 1 coo' + 1 co'oco + 1 1 VO + 1 icowf 7+++ NO 1 + codes tjii-I CO i++++ 8° 1 + hoo'n'om + 1 1 1 1 1 CO 00 88 OS gSS COO s COOS lOO SJS? OS 3S3 WtNCOTjiio IO Tfl CO IS 2&§ ccnos 00 CO CM O NCOOOtH OCDCMrHO NOCMON o iO OS co M + ceo + 1 coco 7 + too 1 + oseo»o CO 1 ++ CM° I.+ CO CD IO OS ■CO + 111 too + 1 COOOOS CO + 1 1 1 1 n OS Tt< OS con »o MOO OB o CD NCO rHCO ICCO • t-HCMCO-W coco CO to nco OCO CM □ONOS COi-H COCO TJ1 Tf* OS OS OSi-H IOCS •»COi-trH I. 00*5? thc5 00 n + 1 l>Tj! 1 + 3° 7+ i-H weN* £0*0 "S OOcOi-H 1 ++ 1 + CM* CO O OS CO CO i-l + 111 + col SS8 00 CO Tt«rH CN »0 OCO COOS rHN« COOS ■Wl-f rPTjl TftQO OSrHCN CO II CO lOTjiCO CD o S3° CO 1 + coo cor- + 1 + 1 OSCOi-t + 1 1 + • n i-i CN CM Tfl CD »oao cm'o 1 i + Q-tf CM CO OSN + 1 o^ 1 *£* rH + CO CM iO CO - 1 1-1 N -. APPLICATION OP LEAST SQUARES TO TEIANGTJLATION. Bach solution 121 Ma -0. 34,3 +0. 69872 -0. 75924 -0.06052 -0. 12121 -0.09758 +0.01446 -0.20433 +0.02662 +0. 15787 -0.02003 -0. 00832 +0.09614 SXl +0. 00658 +0.01840 -0.00210 -0.18815 -0.08990 -0.25523 +0.00902 -0.00020 +0.00049 +0. 16170 +0. 08016 -0. 22038 +0.03079 Computation of corrections 1 2 3 4 5 6 7 8 + 2.669 + 8.956 -11.412 + 0.781 + 4.362 -11.412 - 2.2 + 0.167 + 3.760 -11.412 + 4.8 - 0.675 + 3.543 -11.412 + 3.6 - 2.388 + 9.783 - 11.412 + 7.1 - 1.122 - 4.114 - 2.205 + 14.007 -11.412 + 9.7 - 0.576 - 4.033 -11.412 +25.5 - 6.562 -55.918 +20. 490 +44. 767 -11.412 + 7.1 + 0. 213 + 0.2 - 8.469 - 8.5 - 2.685 - 2.7 - 4.944 - 4.9 + 3.083 + 3.1 + 9.479 + 9.5 + 4.854 + 4.9 - 1.535 - 1.5 2l 9 10 11 12 13 14 22 - 7. 705 -33. 662 +20.490 +44. 767 - 2.205 +14.007 +55.6 - 6.562 -55.918 +20.490 +44. 767 - 1.129 - 5.781 + 10.033 - 1.129 - 1.932 + 3.035 - 1.129 -13.1 - 4.027 - 3.371 - 2.535 + 14.337 - 1. 129 - 2.7 - 1.966 - 3.476 - 1.129 + 11.5 - 0. 612 - 2.203 - 1.129 + 6.8 - 6.562 -55. 918 + 6.171 +48.783 - 2.535 + 14.337 + 2.5 + 3.123 + 3.1 -13.126 -13.1 + 4.929 + 5.0 19 + 2.856 ■f 2.9 + 1.648 + 1.7 + 0.575 + 0.6 +91.292 -11.412 + 6.776 - 1.129 15 16 17 18 Zs Zi 19' + 0.715 -42. 768 +44.369 - 4.027 - 3.371 - 2.535 +14.337 +44.369 -52.6 - 1.122 - 4.114 - 2.205 +14.007 +44. 369 -54.8 - 1.460 +24.669 +44.369 -62.2 - 1. 122 - 4.114 - 4.027' - 3.371 - 5.486 + 10.245 -169.6 - 0.612 - 2.203 - 0.742 + 4.3 - 0.612 - 2.203 + 4.3 - 0.742 - 0.742 - 0.7 + 2.316 + 2.3 + 1.485 - 0.742 + 5.378 + 5.4 + 0.743 + 0.8 — 3.827 - 3.8 - 3.865 - 3.8 -177.475 + 44.369 20 21 25 20' 20" 20'" 20"" 22 - 1.932 + 3.035 - 1.912 + 6.4 - 0.675 + 3.543 - 1.912 + 1.1 - 0.675 + 3.543 - 1.932 + 3.035 + 7.5 - 1.912 + 1.1 - 1.912 + 0.4 - 1.912 - 4.6 - 1.912 + 3.1 + 0.167 + 3.760 - 0.521 - 0.8 - 0.812 - 0.8 - 1.512 - 1.5 - 6.512 - 6.5 + 1.188 + 1.2 + 5.591 + 5.6 + 2.056 + 2.1 + 2.606 + 2.6 +11.471 - 1.912 2c 22' 22" 22'" 22"" 22" 23 27 + 0. 167 + 3.760 - 0.8 - 0.521 - 0.2 - 0.521 - 3.2 - 0.521 + 6.2 - 0.521 - 0.1 - 0.521 - 2.8 + 0.781 + 4.362 - 0.469 - 2.8 + 0.781 + 4.362 - 2.8 - 0.721 - 0.7 - 3.721 - 3.7 + 5.679 + 5.7 - 0.621 - 0.6 - 3.321 - 3.3 + 3.127 - 0.521 + 2.343 - 0.469 + 1.874 + 1.9 23' 23" 23'" 23"" 24 Z8 24' '24" - 0.469 - 3.8 - 0.469 - 5.0 - 0.469 - 2.4 - 0.469 + 11.2 + 2.669 + 8.956 - 3.242 - 1.9 + 2.669 + 8.956 - 1.9 - 3.242 + 0.2 - 3.242 - 0.1 - 4.269 - 4.2 - 5.469 - 5.5 - 2.869 - 2.8 +10.731 + 10.7 - 3.042 - 3.0 - 3.342 - 3.3 + 9.725 - 3.242 + 6. 483 + 6.5 122 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final computation of triangles Symbol Station Observed angle Correc- tion Spheri- cal angle Spher- ical excess Plane angle Loga- rithm Duck-Larrabee © / // " " o / // 3.410111 -1+ 3 Gunner 45 43 29.4 - 2.9 26.5 0.145096 -24'+24 Duck 124 02 01.1 + 9.5 10.6 9.918389 -22+22"" Larrabee 10 14 26.1 - 3.2 22.9 9.249850 + 3.4 Gunner-Larrabee 3.473596' Gunner-Duck 2.805057 Duck-Mam 3.389282 -1+ 4 Gunner 108 41 53.9 - 5.1 48.8 0.023546 Duck 58.8 67.5 57 00 67.5 9.923684 -21+20" Mam Gunner-Mam Gunner-Duck Duck-Lubec Chan- nel Lighthouse 14 17 07.3 - 3.6 03.7 9.392230 3.336512 2.805058-1 3.045619 -1+ 5 Gunner 115 34 42.9 + 2.9 45.8 0.044799 -24"+24 Duck 33 11 41.8 + 9.8 51. G 9.738407 * Lubec Channel 35.3 22.6 31 13 22.6 9.714639 Lighthouse Gunner-Lubec 2.828825+2 Channel Light- house Gunner-Duck 2.805057 Indian Point-Lar- 3.236668 rabee , -2+ 3 Gunner 31 57 41.4 + 5.8 47.2 0.276238 -23'+23 Indian Point 114 00 37.5 + 6.1 43.6 9.960689 -22+22" Larrabee 34 01 35.1 - 5.9 29.2 9.747840 + 6.0 Gunner-Larrabee 3.473595+1 Gunner- Indian 3.260746 Point Indian Point-Mam 3.470097 -2+ 4 Gunner 94 56 05.9 + 3.6 03.5 0.001614 — 23"+23 Indian Point 47 05 36. 5 + 7,4 43.9 9. 864802 -21+20'" Mam 37 58 15.2 - 8.6 06.6 9.789036 + 2.4 Gunner-Mam 3.336513-1 Gunner- Indian 3.260747-1 Point Indian Point-Lu- 3.315762 bec Channel Lighthouse -2+ 5 Gunner 1C1 48 54.9 +11.6 66.5 0.009305 — 23"'+23 Indian Point 18 35 59.1 + 4.7 63.8 9.503759 Lubec Channel 66.0 49.7 59 34 49.7 9.935679 Lighthouse Gunner-Lubec 2.828826+1 Channel Light- house Gunner- Indian 3.260746 Point Larrabee-Mam 3.443126 —3+ 4 Gunner 62 58 24.5 - 2.2 22.3 0.050224 -22'+22 Larrabee 44 10 30.5 + 3.3 39.8 9.843162 -21+20"" Mam 72 50 58.8 - 0.9 57.9 9.980246 + 0.2 Gunner-Mam 3. 336512 Gunner-Larrabee 3.473596 Larrabee-Lubec 3.449573 Channel Light- house -3+ 5 Gunner 69 51 13.5 + 5.8 19.3 0.027415 -22"+22 Larrabee 12 59 26.4 + 6.3 32.7 9.351839 Lubec Channel ' 20.1 08.0 97 09 08.0 9.996608 Lighthouse Gunner-Lubec 2.828827 Channel Light- house Gunner-Larrabee 3.473596 - APPLICATION OF LEAST SQUARES TO TEIANGULATION. Final computation of triangles — Continued 123 Symbol Station Observed angle Correc- tion Spheri- cal angle Spher- ical excess Plane angle Loga- rithm Larra b e e-L u b e o o t it " » Of II 3. 702872 church spire -3+ 7 Gunner 154 32 48.4 +12.2 60.6 0.366814 -22"'+22 Larrabee 10 46 05.9 - 3.1 02.8 9.271431 Lubec church spire 65.7 56.6 14 40 56.6 9.403910 Gunner-Lubec 3.341117 church spire Gurmer-Larrabee 3. 473596 Mam-Lubec Chan- 3. 176968 nel Lighthouse -4+ 5 Gunner 6 52 49.0 + 8.0 57.0 0.921421 -2C+21 Mam 3 04 50.7 + 2.9 53.6 8. 730438 Lubec Channel 20.3 09.4 170 02 09.4 9.238122 Lighthouse Gunner-Lubec 2.828827 Channel Light- house Gunner-Mam 3.336511+ 1 Lubec church spire- 3. 603122 Indian Point +2- 7 Gunner 173 29 30.2 -18.0 12.2 0. 945259 Lubec church spire 11.8 21.0 2 57 21.0 8. 712365 -23+23"" Indian Point Gunner-Ind ian Point Gunner-Lubec church spire Gunner-Lubec Channel Light- house ' 3 33 18.0 + 8.8 26.8 8. 792736 3.260746 3.341117 2.828827 -9+10 Cranberry Point 75 45 12. 8 + 1-4 14.2 0.013565 -5+ 8 Gunner 90 05 58.7 - 4.6 54.1 9. 999999 Lubec Channel 48.5 51.7 14 08 51.7 9.388141 Lighthouse Cranberry Point- 2.842391 Lubec Channel Lighthouse Cranberry Point- 2.230533+ 1 Gunner Gunner-Mam 3.336512 -9+11 Cranberry Point 78 36 65.9 -14.8 51.1 0.008632 —4+ 8 Gunner 96 58 47.7 + 3.4 51.1 9.996769 -20+21 Mam 4 24 21.3 - 3.5 17.8 8.885390 -14.9 Cranberry Point- 3.341913 Manx Cranberry Points 2.230534 Gunner Gunner-Lubec 3.341117 —9+13 church spire Cranberry Point 174 08 32.3 + 3.28 35.58 0.991220 -7+ 8 Gunner 5 24 23.8 -11.02 12.78 8. 973913 Lubec church SDire 03.9 11.64 27 11.64 7. 898195 Cranberry Point- 3. 306250 Lubec church spire Cranberry Points 2.230532+ 2 Gunner Lubec Channel 3. 176968 Lighthouse- -10+11 Mam Cranberry Point Lubec Channel 2 51 53.1 36.3 -16. 2 36.9 58.9 175 48 58.9 1. 301899 8.863046 -20+20' Lighthouse Mam Cranberry Point- Mam Cranberry Point- Lubec Channel Lighthouse 1 19 30.6 - 6.4 24.2 8.363526 3.341913 2. 842339- 2 124 COAST AND GEODETIC STJBVEY SPECIAL PUBLICATION NO. 28. Final computation of triangles — Continued Symbol Station Observed angle Correc- tion Spheri- cal angle Spher- ical excess Plane angle Loga- rithm Lubec church spire- Treats Cranberry Point • , „ ti " . , 3.309982 -13+14 17 44 02.9 - 2.1 00.8 0.5162S3 Lubec church spire 19.0 22.6 144 41 22.6 9.761932 -19+19' Treats Cranberry Point- Treats Cranberry Point- Lubec church spire Lubec church spire-Cranberry Point 17 34 38.1 - 1.5 36.6 9. 479985 3.588197 3.306250 3.306250 -15+16 Telegraph 128 38 59.9 - 6.1 53.8 0. 107352 Lubec church spire Cranberry Point 47.5 49.2 30 51 49.2 9.710115 -12+13 20 29 12.6 + 4.4 17.0 9.544083 T e 1 e g r aph-Cran- 3.123717 berry Point Telegraph-L u b e c church spiro 2.957685-1 Lubec church 3.341117 spiro-Gunner Telegraph -15+17 131 33 53.1 - 6.1 47.0 0. 125967 Lubec church spire 36.1 37.6 30 24 37.6 9. 704315 -6+7 Gunner Telegraph-Gunner Telegraph-L u b e e church spire 18 01 30.8 + 4.6 35.4 9.490600 3. 171399 2.957684 Cranberry Point- 2.230534 Gunner -16+17 Telegraph Cranberry Point 2 54 53.2 0.0 53.2 1.293705 - 9+12 153 39 19.7 - 1.1 18.6 9.647160 -6+8 Gunner Telegraph-Gunner T e 1 e g r aph-Cran- berry Point Cranberry Point- Lubec Channel Lighthouse 23 25 54.6 - 6.4 48.2 9.599478 3. 171399 3.123717 2.842391 -16+18 Telegraph Cranberry Point 29 52 44.5 + 9.3 53.8 0.302588 -10+12 77 54 06.9 - 2.5 04.4 9.990245 Lubec Channel 08.6 01.8 72 13 01.8 9.978738 Lighthouse Telegraph-L u b.e c 3.135224 Channel Light- house T e 1 e g r aph-Cran- 3.123717 berry Point Gunner-Lubec 2.828827 Channel Light- house -17+18 Telegraph 26 57 51.3 + 9.2 60.5 0.343447 -5+6 Gunner 66 40 04.1 + 1.8 05.9 9.962950 Lubec Channel 64. 53.6 86 21 53.6 9.999125 Lighthouse Telegraph-L u b e c _ 3. 135224 Channel Light- house Telegraph-Gunner 3. 171399 APPLICATION OF LEAST SQTJAKES TO TEIASTGULATION. 125 SECOND METETOD. The only difference between the second method of adjustment and the first is in the treatment of the directions taken at the fixed points. At these points observation equations are written for the directions of new points only and the g's are ofmitted. The observations taken over the fixed lines are not used, but the observed directions of the new lines are taken in connection with the adjusted direction of a fixed line, all directions being referred to a common initial line. The equations with s's omitted are the same as if the angle method of adjustment were used. (See p. 196.) In this treatment these equa- tions are given unit weight. Jordan (Vermessungskunde, vol. 1, p. 179, of the third edition) suggests that on some accounts it would be better to assign the equations for observations at fixed points only half weight. The observation equations for directions taken at Gunner, Cran- berry Point, and Telegraph are the same as for the first method given on page 115 and are not repeated here. Below are given the observation equations for the remaining points, formed according to the second method. The assumed azimuths are identical with those used in the first method. As an example, to illustrate the computa- tion of the observed azimuths, take the fine Mam-Gunner. Use the observed direction for the new line and the adjusted one for the fixed line. o / // Fixed azimuth Mam to Indian Point, page 106, =266 17 19. 2 Angle Indian Point to Gunner, page 106 (359° 59' 55".4 to 322° OP" 44".8), =322 01 49.4 Observed azimuth. Mam to Gunner, =228 19 08. 6 or by reckoning from any other fixed line through Mam the same result is reached, thus, Fixed azimuth Mam to Lubec Channel Light- ° ' " house, =225 14 19.0 Angle Lubec Channel Lighthouse to Gunner, page 106 (318° 56' 55". 2 to 322° 01' 44 // .8), = 3 04 49. 6 Observed azimuth Mam to Gunner, =228 19 08. 6 Note that the coefficients of the d3 *X 3 1 V 19 20 21 22 23 24 - 2.19 + 0.54 + 2.54 + 8.67 - 1.39 - 1.47 - 1.71 - 3.51 — 0.64 - 2.01 + 1.08 -1.48 +0.43 +0.64 +0.11 -0.11 +0.25 -0.19 + 0.87 - 2.85 -r 3.47 - 1.04 + 1.08 + 4.97 Normal equations 1 2 3 4 5 6 1 2 +1014.5374 -863.2282 +918. 6459 —822.5130 +876.9117 +924. 4414 +781.8841 —837.3829 —831.3291 +843. 4555 + 8.0389 -13.2644 -39. 8513 +36.7824 +43. 7028 - 0.2265 + 3.9464 +18. 6471 -15. 9112 -33. 6441 +41.7793 - 8.5215 + 5.8167 + 3.7521 + 2. 6609 -18. 8752 +30. 2371 +109.9712 + 91.4452 +130. 0589 - 19.8403 - 17.1109 + 44.8281 91865°— 15- 128 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. 283 Oir- O) O + I i-tCO UJH CIO 00 1-1 ++ + 1 gjtoo COCOH ++ 1 +1 I I++ +1 Oi-H -aieO io HNwmoo 00-31 NOON t- «00 I I.++ I I + ,-t CO N O i-H t* cue QC' -V H © CD 0> HC OlOlO (Oil ooOOiCO os« ■i at & »X 2 Sfa «i a& -0. 89000 +0. 65852 -0. 76711 -0. 10591 -0. 08850 +0. 02328 +0.02035 +0. 15131 -0. 03376 -0. 06259 +0.00779 +0.01814 -0.00379 -0. 15993 -0.07244 +0.00840 -0.00020 +0.00086 +0. 13189 +0.06108 -0. 17888 -0. 89000 -0. 10859 ' -0. 17113 +0.07534 -0. 21023 +0.02315 Computation of corrections 1 2 3 4 5 6 7 8 + 2.007 + 7.377 -10.753 + 0.588 + 3.593 -10. 753 - 2.2 + 0.125 + 3.097 -10. 753 + 4.8 - 0.507 + 2.918 -10. 753 + 3.6 - 1.796 + 8.058 -10. 753 + 7.1 - 0.843 - 3.389 - 3.956 +14. 347 -10. 753 + 9.7 - 0.433 - 3.322 -10. 753 +25.5 - 4.934 -46.059 +16.057 +37. 493 -10. 753 + 7.1 - 1.369 - 1.4 - 8.772 - 8.8 - 2.731 - 2.8 - 4.742 - 4.8 + 2.609 + 2.6 \ +10.992 +11.0 + 5.106 + 5.1 — 1.096 - 1.1 2l 9 10 11 12 13 14 »2 - 5.793 -27. 727 +16. 057 +37. 493 - 3.956 +14.347 +55.6 - 4.934 -46. 059 +16.057 +37. 493 - 1.223 - 4.530 + 8.402 - 1.223 - 1.514 + 2.541 - 1.223 -13.1 - 3.156 - 2.824 - 4.549 +14.685 - 1.223 - 2.7 - 1.541 - 2.911 - 1.223 + 11.5 - 0.480 - 1.845 - 1.223 + 6.8 - 4.934 -46. 059 + 4.836 +40.857 - 4.549 +14.685 + 2.5 + 2.649 + 2.7 -13.296 -13.3 + 5.825 + 5.8 + 3.252 + 3.2 + 1.334 + 1.3 + 0.233 + 0.2 +86.021 -10. 753 + 7.336 - 1.223 15 16 17 18 23 19 20 21 + 1.282 -43. 806 +44. 790 - 3.156 - 2.824 - 4.549 +14.685 +44. 790 -52.6 - 0.843 - 3.389 - 3.956 +14.347 +44. 790 -54.8 - 2.620 +25. 267 +44. 790 -62.2 - 0.843 - 3.889 - 3.156 - 2.824 - 9.843 + 10.493 -169.6 - 0.480 - 1.845 + 4.3 - 1.514 + 2.541 , + 6.4 - 0.507 + 2.918 + 1-1 + 2.266 + 2.2 + 1.975 + 2.0 + 7.427 + 7.5 + 3. 511 + 3.5 + 5.237 + 5.3 - 3.654 - 3.7 - 3.851 - 3.9 -179. 162 + 44.790 22 23 24 + 0.125 + 3.097 - 1.1 + 0.588 + 3.593 + 2.5 + 2.007 + 7.377 - 1.9 + 2.122 + 2.1 + 6.681 + 6.7 + 7.484 + 7.5 130 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. Final computation of triangles 25. Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane angle Loga- rithm -1+3 +24 -22 Duck-Larrabee Gunner Duck Larrahee Gunner-Larrabee Gunner-Duck O t 11 45 43 29.4 124 02 00.9 10 14 25.7 - 1.4 + 7.5 - 2.1 28.0 08.4 23.6 3.410111 0. 145093 9.918392 9.249858 3. 473596-1 2.805062 + 4.0 -1+ 4' +24 -21 Duck-Mam Gunner Duck Mam Gunner-Mam Gunner-Duck 108 41 53.9 57 00 57. 8 14 17 07. 7 - 3.4 + 7.5 - 3.5 50.5 65.3 04.2 3.389282 0.023547 9.923681 9.392234 3.336510 2.805063-' + 0.6 Duck-Lubec Channel Light- house Gunner Duck Lubec Channel Lighthouse Gunner-Lubec Channel Lighthouse Gunner-Duck 3.015619 -1+5 +24 115 34 42.9 33 11 41.9 35.2 + 4.0 + 7.5 46.9 49.4 23.7 31 13 23.7 0.044800 9. 738400 9. 714643 2.828819 2.805062 -2+ 3 +23 -22 Indian Point-Larrabee Gunner Indian Point Larrabee Gunner-Larrabee Gunner-Indian Point 31 57 41.4 114 00 36.0 34 01 32.0 + 6.0 + 6.7 - 2.1 47.4 42.7 29.9 3.236668 0.276237 9.960690 9. 747842 3. 473595 3.. 260747 ,+10.6 -2+ 4 +23 -21 Indian Point-Mam Gunner Indian Point Mam Gunner-Mam Gunner-Indian Point 94 56 05.9 47 05 36. 3 37 58 10.6 + 4.0 + 6.7 - 3.5 09.9 43.0 07.1 3.470997 0.001614 9.864800 9.789037 3.336511-1 3.260748-' + 7.2 -2+ 5 +23 Indian Point-Lubec Channel Lighthouse Gunner Indian Point Lubec Channel Lighthouse Gunner-Lubec Channel Lighthouse Gunner-Indian Point 101 48 54.9 18 35 56.3 68.8 +11.4 + 6.7 66.3 63.0 50.7 59 34 50.7 3.315762 0.009305 9.503754 9.935680 2.828821-1 3.260747 APPLICATION OP LEAST SQUARES TO TBIANGULATION. 131 Final computation of triangles — Continued Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane angle Loga- rithm -3+ 4 +22 -21 Larrabee-Mam Gunner Larrabee Mam Gunner-Mam Gunner-Larrabee O / It 62 58 24.6 44 10 37.0 72 50 01.9 - 2.0 + 2.1 - 3.5 n 22.5 39.1 58.4 O I It 3.443126 0.050224 9.843160 9. 980246 3.336510 3. 473596-' - 3.4 -3+ 5 +22 Larrabee-Lubec .Channel Lighthouse Gunner Larrabee Lubec Channel Lighthouse Gunner-Lubee Channel Lighthouse Gunner-Larrabee 69 51 13.5 12 59 29.9 16.6 + 5.4 + 2.1 18.9 32.0 09.1 97 09 09.1 3. 149573 0.027415 9.351833 9.996607 2.828821-1 3. 473595 -3+ 7 +22- Larrabee-Lubee church spire Gunner Larrabee Lubec church spire Gunner-Lubec church spire Gunner-Larrabee 154 32 48. 4 10 46 00. 71.6 +13.8 + 2.1 62.2 02.1 55.7 14 40 55.7 3.702872 0.366819 9.271423 9. 403903 3.341114+1 3. 473594+1 ; -4+ S . +21 t' Mam-Lubec Channel Light- house Gunner Mam Lubec Channel Lighthouse Gunner-Lubec Channel Light- house Gunner-Mam 6 52 49.0 3 04 49. 6 21.4 + 7.4 + 3.5 56.4 53.1 10.5 170 02 10.5 3. 176968 0. 921432 ■ 8.730418 9.238109 2.828818+2 3.336509+1 Lubec church spire-Indian Point Gunner Lubec church spire Indian Point 3. 603122 +2- 7 -23 173 29 30.2 56 55.4 3 33 34. 4 -19.8 - 6.7 10.4 81.9 27.7 2 57 21.9 0.945226 8. 712401 8. 792767 Gunner-Indian Point Gunner-Lubec church spire 3.260749-« 3.341115 -9+10 -5+ 8 Gunner-Lubec Channel Light- house Cranberry Point Gunner Lubec Channel Lighthouse Cranberry Point-Lubec Chan- nel Lighthouse Cranberry Point-Gunner 75 45 12.8 90 05 58. 7 48.5 + 1.4 - 3.7 14.2 55.0 50.8 14 08 50.8 2.828820 0.013565 9.999999 9.388133 2.842384 2.230518 132 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final computation of triangles— Continued Symbol Station Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane angle Loga- rithm -9+11 -4+ 8 -20+21 Gunner-Mam Cranberry Point Gunner Mam Cranberry Point-Mam Cranberry Point-Gunner 78 36 65.9 96 58 47. 7 4 24 21.3 -14.6 + 3.7 - 4.0 51.3 51.4 17.3 3.336510 0.008632 9.996768 8.885376 3.341910 2.230518 -14.9 -9+13 -7+ 8 Gunner-Lubec church spire Cranberry Point Gunner Lubec church spire Cranberry Point-Lubec church spire Cranberry Point-Gunner Lubec Channel Lighthouse- Mam Cranberry Point Lubec Channel Lighthouse Mam Cranberry Point-Mam Cranberry Point-Lubec Chan- nel Lighthouse 174 08 32.3 5 24 23.8 03.9 + 4.5 -12.1 36.8 11.7 11.5 27 11. 5 3.341115 0.991245 8.973889 7. 898156 3.306249 3.230516+* 3. 176968 -10+11 -20 2 51 53. 1 35.2 1 19 31.7 -16.0 - 7.5 37.1 58.7 24.2 175 48 58. 7 1.301891 8.863051 8.363526 3.241910 2.842385-1 -13+14 -19 Lubec church spire-Treats Cranberry Point Lubec church spire Treatj Cranberry Point-Treats Cranberry Point-Lubec church spire 17 44 02.9 19.0 17 34 38. 1 - 2.6 - 2.0 00.3 23.6 36.1 144 41 23.6 3.309982 0.516286 9.761929 9.479981 3.588197 3.306249 -15+16 -12+13 Lubec church spire-Cran- berry Point Telegraph Lubec church spire Cranberry Point Telegraph-Cranberry Point Telegraph-Lubec church spire 128 38 59.9 47.5 20 29 12.6 - 5.9 + 5.6 54.0 47.8 18.2 30 51 47. 8 3.306249 0.107352 9.710110 9.544090 3. 123711 2. 957691 -1S+17 -6+ 7 Lubec church spire-Gunner Telegraph Lubec church spire Gunner Telegraph-Gunner Telegraph-Lubec church spire 131 33 53. 1 36.1 18 01 30. 8 - 6.1 + 5.9 47.0 36.3 36.7 30 24 36.3 3.341115 0. 125967 9. 704310 9.490609 3. 171392 2.957691 -16+17 -9+12 -6+ 8 Cranberry Point-Gunner Telegraph Cranberry Point Gunner Telegraph-Gunner Telegraph-Cranberry Point 2 54 53.2 153 39 19. 7 23 25 54. 6 - 0.2 - 1.1 - 6.2 53.0 18.6 48.4 2.230518 1.293713 9. 647161 9.599479 3. 171392 3. 123710+ 1 - 7.5 APPLICATION OF LEAST SQTJAEES TO TEIANGULATION. Final computation of triangles — Continued 133 Symbol Station. Observed angle Correc- tion Spher- ical angle Spher- ical excess Plane angle Loga- rithm Cranberry Point-Lubec Chan- nel Lighthouse . ; „ " " . , „ 2. 842384 -16+18 Telegraph Cranberry Point 29 52 44. 5 + 9.0 53.5 0.302589 -10+12 77 54 06. 9 - 2.5 04.4 9. 990245 Lubec Channel Lighthouse 08.6 02.1 72 13 02. 1 9. 978738 Telegraph-Lubec Channel Lighthouse 3. 135218 Telegraph-Cranberry Point 3.123711 Gunner-Lubec Channel 2.828820 Lighthouse -17+18 Telegraph 26 57 51. 3 + 9.2 60.5 0.343447 -6+6 Gunner 66 40 04. 1 + 2.5 06.6 9. 962951 Lubec Channel Lighthouse 64.6 52.9 86 21 52.9 9. 999125 Telegraph-Lubec Channel Lighthouse 3. 135218 Telegraph-Gunner 3. 171392 134 COAST AND GEODETIC SXJEVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION GUNNER Second angle a Aa J 1(0+0') 1st term 2d term Duck to Lubec Channel Lighthouse Lubeo Channel Lighthouse and Gunner Duck to Gunner Gunner to Duck 44 44 44 50 -10. 2277 + 0.0008 33. 8SG 10. 227 44. 113 First angle of triangle Duck Gunner X 80 +33 180 299 115 -10. 2269 sin a A' 2. 805062 9. 694237 8. 510480 sin 2 a C sec0' JX 2. 805062 8. 508994 0. 149348 1. 402490 +25.2633 1.009779 sin 1(0+0') 5.6101 9.8782 1. 4016 1. 402490 9.848301 1. 250791 +17.82 11 38 42.1 49.4 31.5 17.8 13.7 46.9 47.822 25.263 13.085 STATION CRANBERRY POINT Second angle 0' 1(0+0') 1st term 2d term Gunner to Lubec Channel Lighthouse Lubec Channel Lighthouse and Cranberry Point Gunner to Cranberry Point • Cranberry Point to Gunner First angle of triangle 44 50 44 44 50 46 -4. 5293 +0.0000 50 44. 113 04. 529 48. 642 Gunner Cranberry Point X 55 + 90 145 180 325 75 sin a A' JX COS a B 2.230518 9. 755156 2. 230518 9.915029 8.510480 sin 2 or C 0. 149357 0.644025 +4. 4058 0. 656027 4X sin 1(0+0') 4.4610 9.5103 1. 4016 0.644025 9. 848315 0. 492340 +3.1 18 58 00.6 55.0 55.6 03.1 52.5 14.2 13.085 04.406 17. 491 APPLICATION' OP LEAST SQUARES TO TBIANGULATION. 135 secondary triangulation - STATION GUNNER Third angle a 4> H 1st term 2d term Lubec Channel Lighthouse to Duck Gunner and Duck Lubec Channel Lighthouse to Gunner Gunner to Lubeo Channel Lighthouse 44 50 44 O I 44 50 -12.4620 h 0.0008 50 31. 652 12.461 44.113 Lubec Channel Lighthouse -12. 4612 Sill a A' sec$' J\ s COS a B 2. 828819 9.914485 8. 508994 0. 149348 2. 828819 9. 756288 8. 510480 Gunner sin2<« C 1. 401646 II -25. 2142 1.095587 sin i(4>+') X X' 235 180 55 5. 6576 9.8290 1. 4016 1. 401646 9. 848294 1. 249940 tf -17. 78 58 06.5 23.7 42.8 17.8 00.6 38.299 25.214 STATION CRANBERRY POINT Third angle Of a' 1st term 2d term -J$ Lubec Channel Lighthouse to Gunner Cranberry Point and Gunner Lubec Channel Lighthouse to Cranberry Point Cranberry Point to Lubec Channel Lighthouse 31. 652 16.990 48. 642 Lubec Channel Lighthouse Cranberry Point -16.9910 + 0.0005 -16.9905 s sin a A' J\ COS a B 2.842384 9.817504 2. 842384 9.877355 8. 510480 sin 2 a C 0. 149357 1. 318239 // -20.8084 1. 230219 4\ ■ sin i(*+0'> -4a X JX X' 5. 6848 9.6350 1.4016 235 - 14 221 ISO 41 6. 7214 1.318239 9.848303 1. 166542 it -14.67 12 58 58 42.8 50.8 52.0 14.7 06.7 38.299 20.808 17. 491 136 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION TELEGRAPH Secqnd angle Aa 4 A 4' M+*') 1st term 2d term Cranberry Point to Lubec Channel Lighthouse Lubec Channel Lighthouse and Telegraph Cranberry Point to Telegraph Telegraph to Cranberry Point 44 60 51 48.642 20.858 09.500 +1 44 50 59 -20. 8617 + 0.0034 sin a A' see^' cos a B 3. 123711 9.941946 8. 508994 0. 149401 1. 724052 +52. 9728 First angle of triangle Cranberry Point Telegraph X a + 77 CO 3. 123711 9. 685157 8. 510480 sin 2 a C 1. 319348 sin 4(#+*') 0.2474 9.8839 1. 4016 1. 724052 1. 572395 It +37.36 58 58 59 06.7 04.4 11.1 37.4 33.7 53.4 17. 491 52.973 10. 464 APPLICATION OP LEAST SQUABES TO TRIANGULATION. 137 secondary triangulation — Continued . STATION TELEGRAPH Third angle a ia iW+tf') 1st term 2d term -JJ, Lubec Channel Lighthouse to Cranberry Point Telegraph and Cranberry Point Lubec Channel Lighthouse to Telegraph Telegraph to Lubec Channel Lighthouse 50 31. 652 37.849 Lubec Channel Lighthouse Telegraph 44 50 50 -37. 8500 + 0.0012 -37. 8488 Kin a. A' sec^' J\ COS a B 3. 135218 9. 713762 0. 149401 1. 507375 +32. 1644 3. 135218 9. 932367 8. 510480 *2 sin 2 a C sin i(tj>+ A^, JW+tf') 1st term 2d and 3d \ terms / East base to west base West base and Cedar East base to Cedar Cedar to east base First, angle of triangle 30 56. 379 51. 941 48.320 +1 Dauphin Island east Cedaj 30 17 52 -351.9476 + 0.0063 -351. 9413 sin « A' sec^' cos a B 4. 9. 995568 8. 511556 *2 sin 2 a C 4. 039354 9. 152706 8. 509351 0. 063997 1. 765408 -58.2650 2. 546478 A\ sin JW+0') X A\ X' 8.0787 t 8.3054 1. 1712 7. 5553 +0. 0036 1. 765408 9. 702857 1. 468265 -29. 39 84 +103 188 180 8 37 10 5.093 2.332 7.425 +0.0027 41.9 35.5 17.4 29.4 00.00 '46.8 32.1 14.288 58.265 16.023 APPLICATION OF LEAST SQUARES TO TRIANGULATION. secondary triangulation 145 STATION CAT Third angle a J m+4>') 1st term 2d, 3d, and 4th terms West base to east 1 Cat and east base West base to Cat Cat to west base 30 O / It 30 16 41 -279.0808 + 0.0231 21. 492 39. 058 Dauphin Island west base Cat -279.0577 8 sin a A' see^' cos a B 3. 972892 9.961281 8. 511557 2. 445730 sin 2 a C 7.9458 9. 2130 1. 1711 J\ 3. 972892 9.606514 8. 509352 0. 063865 2. 152623 -142. 1095 sin $(tf+tf') +0.0214 2. 152623 9. 702600 -71.65 X X' 264 ISO 23 4.891 2.331 7.222 +0. 0017 22.1 11.8 10.3 11.7 -h a 2 sin 2 a E -00.00 22.0 51. 034 22. 109 28. 925 2.445 7.159 5.917 5.521 STATION CEDAR Third c J4 r w+tf') 1st term 2d, 3d, and \ 4th terms / West base to east base Cedar and east base West base to Cedar Cedar to west base 30 20 48. 321 Dauphin Island west Cedar X X' 264 - 38 07 30 17 35 8 COS a B h 4.230999 9. 845213 8. 511557 s 2 sin 2 a C 8.4620 9. 7073 1. 1711 h 2 D 5.176 2.331 -h s 2 sin 2 a E -387. 0517 + 0.2227 2. 587769 9. 3404 +0. 2190 7.507 +0. 0032 -386. 8290 22.1 52.7 29.4 49.5 00.00 18.9 51.034 35. 011 16.023 2.588 8.169 5.917 6.674 +0.0005 sin a A' sec$' 4. 230999 9.853675 8. 509351 0.063997 2. 658022 sin i($+') 2. 658022 9. 702795 2.360817 -229. 52 146 COAST AND GEODETIC SUKVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION PINS Second angle a 1W+*') 1st term 2d, 3d, and\ 4th terms / Cedar to west base West base and Pins Cedar to Pins Pins to Cedar 30 -96. 4175 + 0.4916 48.321 35. 926 -95. 9259 sua A' sec $' COS a B 4. 264921 9. 207685 8. 511550 4. 264921 9. 994274 8. 509350 0. 064116 2. 832661 +680. 2382 First angle of triangle Cedar Pins sin 2 a C 8.5298 9.9885 1.1729 sin i(+$') 9. 6912 +0. 4911 2. 832661 9. 703663 +343.81 45 + 53 180 279 58 37 3.968 2.332 6.300 +0.0002 -h s 2 sin 2 a ,E 18.9 41.5 00.4 43.8 00.00 16.6 23.2 16.023 20.238 36.261 +1 1.984 8.518 5.919 6.421 +0.0003 STATION GRAND Second angle a Aa a' 1st term 2d, 3d, andl 4th terms / Pins to west base West base and Grand Pins to Grand Grand to Pins 30 24.247 04. 658 19.589 30 20 52 +184.4693 + 0.1884 +184.6577 sin a A' sec^' cos a B 4. 099768 9. 654608 8. 511548 JX 4. 099768 9. 950510 8. 509352 0. 063888 2.623518 +420. 2600 First angle of triangle Pins Grand sin 2 a 8. 1995 9. 9010 1. 1734 sin JW+tf') 9.2739 +0.1879 2.623518 9. 703504 +212. 33 X 337 + 85 180 243 54 4.532 2.333 6.865 +0.0007 25 -h s a sin 2 <* E 39.8 48.1 32.3 00.00 15.8 00.2 36. 262 00.260 36.522 2.266 8.100 5.919 6.285 -0.0002 APPLICATION- OF LEAST SQTJABES TO TEIANGULATION. 147 secondary triangulalicm — Continued STATION PINS a Third angle a Aa a' * $' itt+tf') 1st term 2d, 3d, and 1 4th terms J —A& West base to Cedar Pins and Cedar 225 - 67 33 34 29.4 55.9 West base to Pins Pins to west base 157 58 1 33.5 53.7 180 337 00 56 00.00 39.8 30 + 14 8 21. 492 02. 754 Dauphin Island west base X A\ 88 + 14 3 51. 034 45.228 30 O 1 It 30 18 2; -482. 8131 + 0.0585 22 i CO I 24 s .246 + 1 4.205 9.967 8.511 1.10 092 557 P S 2 sin 2 <* C ins 8.4103 9.1480 1.1711 V h 2 D 88 5.368 2.331 18 -h s 2 sin 2 a E 36. 262 2.684 7.558 5.917 h 4. 205130 9. 574026 8. 509350 0. 064116 2. 683779 JX sin i($ 8.7294 +0. 0534 2.352622 9. 702967 7.699 +0.0050 6.159 +0. 0001 -482. 7544 s sin a A' see $' A\ +*') 2. 352622 +225. 2278 -Aa 2.055589 +113. 66 STATION GRAND a Third angle a Aa a' 4- At *' JW+tf') 1st term 2d, 3d, and 1 4th terms / -A$ West base to Pins Grand and Pins West base to Grand Grand to west base 157 - 39 58 54 33.5 52.0 118 03 5 41.5 25.5 180 297 00 58 00.00 16.0 30 + 14 4 21. 492 58. 097 Dauphin Island west base X A\ 88 + 14 10 51. 034 45. 488 30 O 1 if 30 16 50 -298. 5403 + 0.4436 19 s cos E 19 a 589 4.290 1 9.672 8.511 )61 85 >57 Gr * 2 sin 2 a C and 8.5819 9. 8914 1. 1711 V h.2 D 88 4.950 2.331 25 -h s 2 sin 2 a E 36. 522 2.475 8.473 5.917 h 4.290861 9. 945687 8. 509352 0.063888 2.475003 A\ sinKtf 9. 6444 +0. 4410 2.809888 9. 702633 7.281 +0.0019 6.865 +0.0O07 -298.0967 s sin a A' see^' A\ W 2.809888 +645. 4877 -Ac t 2. 512521 II +325.48 148 COAST AND GEODETIC SUBVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION PETIT Second angle d «*+#') 1st term 2d and 3d terms Grand to west base West base and Petit Grand to Petit Petit to Grand 30 30 30 15 58 +404. 0853 + 0.0041 19. 589 44.089 +404.0894 sm a A' sec ' dl COS a B 4.095452 4. 095452 9.999470 8. 511551 2.606473 8.509354 0. 063392 1. 361821 -23.0049 First angle of triangle Grand Petit X JX sin 2 a C 8.1909 7. 3872 1. 1725 6. 7506 +0.0006 sin J(#+tf') 1.361821 9. 702445 1. 064266 -11.59 297 357 180 177 81 10 25 25 5.213 2.332 7.545 0.0035 16.0 52.8 08.8 11.6 00.00 20.4 28.3 36.522 23.005 13. 517 STATION HORN Second angle j i«+tf') 1st term 2d. 3d, and \ 4th. terms/ West base to Grand Petit and Grand West base to Petit Petit to west base 30 30 13 28 +105.5810 + 0.4109 12 21. 492 45.992 35.500 Dauphin Island west base Petit +105.9919 8 sin a A' sec $' s cos a B 4.229508 9.282521 8.511557 2. 023586 sin 8 a C 8.4590 9.9837 1. 1711 4.229508 9.991874 8.509354 0. 063392 2. 794128 +622. 4837 sin i(#+tf') 9.6138 +0.4109 2. 794128 9.701905 +313. 35 X A\ 118 180 258 4.047 2.331 6.378 +0.0002 25 -h 2 sin 2 a E 41.5 39.4 02.1 13.4 00.00 48.7 51.034 22. 484 13.518 -1 2.024 8.443 5.917 6.384 -0.0002 STATION HOEN Third angle a Ac, A$ *«+*') 1st term 2d, 3d, and \ 4th terms J -J(t> Petit to Grand Grand and Horn Petit to Horn Horn to Petit 30 30 13 35. 500 04.511 40. 011 Petit Horn X A\ 177 - 80 180 276 00 O f It 30 13 08 S COS a B h 4.245589 9.055530 8.511559 * 2 sin 2 a C 8.4912 9.9943 1. 1706 h 2 D 3.625 2.331 -h s- sin 2 a E -64.9648 + 0.4533 1.812678 9.6561 +0.4530 5.956 +0.0001 -64. 5115 20.4 49.8 30.6 29.2 00.00 01.4 13.517 54.021 07.538 -1 1.813 8.485 5.916 6.214 +0.0002 < sin a A' see$' AX 4.245589 9.997178 8.509354 0. 063471 2. 815592 +654.0214 AX sin JW+#') 2.815592 9. 701830 2. 517422 +329. 17 150 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION PASCAGOULA a Second angle a Aa at' * ' Ito+tf') 1st term 2d, 3d, and \ 4th terms/ -Ad) Grand to Horn Horn and Pascagoula 58 + 43 14 14 46.3 20.1 Grand to Pascagoula Pascagoula to Grand First angle of dangle X A\ 101 29 3 06.4 52.8 180 281 97 00 25 55 00.00 13.6 57.5 30 + • 1 19 1 19.589 21.006 Grand 88 + 25 7 36.522 40.922 30 o / 30 20 00 20 s cos I 4 a 1 0.595 4.0991 9.299 8.511 )72 LOO 551 Pasca; sin 2 a C 'oula 8. 1981 9. 9824 1. 1725 X' h2 D 88 3.820 2.332 33 -h E 17.444 1.910 8.180 5.918 -81. 2312 + 0.2256 h 1.909723 9.3530 +0. 2254 2. 663627 9. 703317 6.152 +0.0001 6.008 +0.0001 -81.005 S sin a A' sec$' 5 4.099072 9. 991216 8. 509351 0.063988 sin £(# +-t>') A\ 2.663627 +460.9215 -A * 2. 366944 +232. 78 STATION BELLE Second angle a Aa 4 Ad, «#+*') 1st term 2d term Pascagoula to Horn Horn and Belle Pascagoula to Belle Belle to Pascagoula 30 20 40. 595 8.730 31. 865 — 1 First angle of triangle Pascagoula Belle X 30 20 36 +8.3511 +0. 3790 +8. 7301 sin a A' sec£' cos a B 8.207256 8. 511550 A\ 4.202937 9.999944 8. 509351 0. 063977 0. 921743 A\ sin J(<£+#') «2 sin 2 a C 8. 4059 9.9999 1.1729 9. 5787 +0.3790 2. 776209 +597.3226 2. 776209 9. 703447 2. 479656 It +301. 76 180 268 48 11.1 24.6 35.7 01.8 00.00 33.9 48.0 17.444 57.323 14.767 APPLICATION OF LEAST SQUARES TO TKIANGULATION. 151 secondary triang illation — Continued STATION PASCAGOULA a Third angle a Act a' 4- Atj, -t>' 1st term 2d, 3d, and\ 4th terms / -A$ Horn to Grand Fascagoula and Grand 238 - 38 09 49 28.1 42.8 Horn to Pascagoula Pasoagoula to Horn X 199 + 19 1 45.3 25.8 180 19 00 21 00.00 11.1 ' 30 + 13 7 tt 40. 011 00.584 Horn 88 36 2 07. 539 50. 094 30 O / 30 17 it ft 10 20 > CO I 40 i 595 4.137 9.974 8.511 527 S03 558 Pascag sill- a C oula 8.2750 9.0396 1.1709 X' h" D 88 5.248 2.331 33 -h s2 sin^ a E 17. 445 -1 2.624 7.315 5.917 -420. 6182 + 0.0345 h 4. 137527 9.519824 8.509351 0. 063988 2.623888 4X sin $(^ 8. 4855 +0.0306 2.230690 9. 702706 7.579 +0.0038 5.856 +0.0001 -420.5837 8 sin a A' sec$' A\ +#') 2. 230690 -170.0944 -Aa 1. 933396 -85.78 STATION BELLE a. Third angle a Act a' * At JW+tf') 1st term 2d, 3d, andl 4th terms / -A$ Horn to Pascagoula Belle and Pascagoula Horn to Belle Belle to Horn X JX 199 - 61 19 17 45.3 47.9 138 01 3 57.4 35.5 180 317 00 58 00.00 21.9 o 30 + t 13 6 it 40. 011 51.853 Horn 88 + 36 7 07. 539 07. 228 30 O I 30 17 it 06 20 i cos I 31 a 1 864 4.2321 9.871 8.511 396 296 558 Bel sin 2 a C le 8. 4642 9.6505 1.1709 X' h2 D 88 5.230 2.331 43 -h & sins a E 14. 767 2.615 8.115 5.917 -412.0501 + 0.1976 h 4.232096 9. 825236 8. 509351 0.063977 2. 614950 A\ sin£(# 9.2856 +0. 1936 2. 630660 9. 702690 7.561 +0.0036 6.647 +0.0004 -411. 8525 8 sin a A' sectf' +#') A\ 2. 630660 +427. 2283 -A X 2.333350 +215. 45 152 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computatidn, STATION CLUB Second angle iW+*') 1st term 2d and 3d\ terms / Belle to Horn Horn and Club Belle to Club Club to Belle 31. 864 19. 886 +319. 8836 + 0.0023 +319. 8859 sma A' sec^' cos a B 3. 993596 9. 999846 8.511550 JX 3. 993596 8. 425207 8. 509353 0. 063584 0. 991740 First angle of triangle Belle Club X 4X 82 sin 2 a C .7. 9872 6.8504 1.1729 Sin J(£+0') 6. 0105 +0.0001 0. 991740 9. 702856 317 + 40 358 180 178 105 5.010 2.332 7.342 +0.0022 58 28 43 21.9 06.7 28.6 05.0 00.00 33.6 56.5 14.767 09.812 04. 955 +1 STATION DEER Second angle 4> «#+*') 1st term 2d and 3dl terms J Belle to Club Club and Deer Belle to Deer Deer to Belle 30 31. 864 55. 356 27.220 -55. 4695 + 0. 1137 COS a B 3. 949436 9. 283068 8. 511550 sin a A' sec^' JX 3.949436 9. 991853 8. 509351 0. 064045 2. 514685 +327. 1033 First angle of triangle Belle Deer sin^a C 1. 744054 sin £(0+0') 7.8989 9.9837 1.1729 X JX 9. 0555 +0. 1136 2. 514685 9. 703531 +165.28 358 +102 101 180 281 3.488 2.332 5.820 +0. 0001 35 48 20.0 +.1 11.4 14. 767 27.103 41. 870 APPLICATION OF LEAST SQUARES TO TBIANGULATION. 153 secondary triangulation — Contin ued STATION CLUB Third angle a Aa. 4> a$ *' }(#+*') 1st term 2d, 3d, and \ 4tn terms J -Aj, Horn to Belle Club and Belle Horn to Club Club to Horn 30 30 14 26 -92. 1514 + 0. 1848 40. Oil 31. 967 -91.9666 sin a A' sec$' COS a B 4. 061239 9. 391705 8. 511558 1. 964502 A\ 4.061239 9.986395 8. 509353 0.063584 2. 620571 +417. 4179 Horn Club Sin 2 < C 8.1225 9.9728 1. 1709 A\ sin i(.+4>') 9. 2662 +0. 1845 2. 620571 9. 702113 2. 322684 +210. 22 X A\ 138 - 33 104 180 284 3.929 2.331 6.260 +0.0002 43 -h S 2 Shl 2 a E . 57.4 57.1 00.3 30.2 00.00 30.1 07. 539 57. 418 04. 957 -1 1.964 8.095 5.917 5.976 +0.0001 STATION DEER Third angle a Aa a' d(j> i«+tf') 1st term 2d, 3d, and 1 4th terms J Club to Belle Deer and Belle Club to Deer Deer to Club 30 30 30 18 20 -375.3652 f- 0.1233 21 11.978 15. 242 27.220 COS a B 4. 165767 9. 897131 8.511556 2. 574454 sin a A' sec^' A\ 4. 165767 9. 788357 8. 509351 0.064045 2. 527520 +336. 9147 Club Deer sin 2 a C 8.3315 9. 5767 1. 1713 A\ sin H0+0') 9. 0795 +0. 1200 2. 527520 9. 702957 2. 230477 // +170. 01 X 4X h2 D 178 180 322 5.149 2.331 7.480 +0. 0030 48 -h s 2 sin 2 a E 33.6 04.8 50.0 00.00 14.8 04. 956 36.915 41. 871 -1 2.574 7.908 5.918 6.400 +0.0003 154 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION SHIP Second angle JW+*') 1st term 2d and 3d terms Deer to Club Club and Ship Deer to Ship Ship to Deer 27. 220 17.200 30 18 19 +377. 1734 + 0.0265 +377. 1999 SHI a A' sec tjS COS a B 4.088917 9. 976075 8. 511549 JX 4. 088917 9.509200 8. 509353 0. 063581 2. 171051 +148.2692 First angle of triangle Deer Ship S 2 sin 2 a C 8.1778 9. 0184 1. 1731 sin h(+4>') +0.0234 2. 171051 9. 702952 X +74.82 322 + 56 18 180 198 70 5.153 2.332 7.485 +0. 0031 14.8 24.7 39.5 14.8 00.00 24.7 34.5 41.870 10. 139 STATION BILOXI LIGHTHOUSE Second angle m+') 1st term 2d, 3d, and \ 4th terms / Deer to Ship Ship and Biloxi Lighthouse Deer to Biloxi Lighthouse Biloxi Lighthouse to Deer 27.220 12.200 ;9.420 -1 First angle of triangle Deer Biloxi Lighthouse -132. 3104 + 0.1107 -132.1997 sin a A' sec ij>' cos a B 3. 978395 9. 631650 8. 511549 2. 121594 sin 2 a C 7. 9568 9.9120 1. 1731 A\ 3.978395 9.956016 8. 509350 0. 064209 2. 507970 +322. 0846 sin JW+tf') 9. 0419 +0. 1102 2. 507970 9. 703868 +162. 87 X 180 295 48 243 332 +0. 575 0004 -h 2 sin 2 a E 33.5 13.0 42.9 00.00 30.1 21.0 41. 870 22. 085 03. 955 -1 2.121 . 7.869 5.919 5.909 +0.0001 APPLICATION OF LEAST SQUARES TO TBIANGULATION. 155 secondary triangulation — Continued STATION SHIP a Third angle a Ja a.' Aj> *' «#+*') 1st term 2d term Club to Deer Ship and Deer 142 - 52 06 20 04.8 01.2 Club to Ship Ship to Club 89 46 4 03.6 04.4 180 269 00 41 00.00 59.2 30 15 11. 978 01. 958 Club X A\ X' K) 3 88 + 43 8 04. 956 05. 182 30 30 15 11 15 CO. I 1 a i 0.020 4.1 7.6 8.5 12975 07988 11556 Ship 52 sin 2 a C 8.22f 0.00( 1.171 88 51 10. 138 +1 +1. 7081 +0. 2496 h 4. 11297 9.99999 8.50935 0. 06358 0.232519 9.39i +0. 24S 385905 702275 3 6 +1. 9577 5 sin a A' sec $' 5 3 3 1 AX sin J(^+ f) 2. 9. A\ 2. 685905 +485.1824 -Aa 2. 388180 +244. 44 STATION BILOXI LIGHTHOUSE Third angle a Ja »(«+*') 1st term 2d, 3d, and 4th terms -Atj, Ship to Deer Biloxi Lighthouse and Deer Ship to Biloxi Lighthouse Biloxi Lighthouse to Ship 30 + 15 30 Fixed value o r tr 30 19 25 -509. 4364 + 0.0376 23 10.020 29.399 39. 419 39.419 Ship Biloxi Lighthouse -509.3988 sin a A' see^' cos a B 4. 213748 9. 981786 8. 511556 2. 707090 sin 2 a C 8. 4275 8. 9056 1. 1713 A\ 4. 213748 9. 452781 8.509350 0.064209 2. 240088 +173. 8153 A\ Sin U+$') 8.5044 +0.0319 2. 240088 9. 703190 1. 943278 +S7. 7ti - 35 163 180 5.414 2.331 7.745 +0. 0056 -h s 2 sin 2 a E 24.7 05.8 18.9 27.8 00.00 51.1 10. 139 53.815 03.954 03. 820 2.707 7.333 5.918 5.958 +0. 0001 \ 156 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Preliminary position computation, STATION SHIP ISLAND LIGHTHOUSE Second angle a Act. !(*+#') 1st term 2d, 3d, mid 1 4th terms j Biloxi Lighthouse to Ship Ship and Ship Island Lighthouse Biloxi Lighthouse to Ship Island Lighthouse Ship Island Lighthouse to Biloxi Lighthouse First angle of triangle 30 39. 419 54.077 45. 342 +1 Biloxi Lighthouse Ship Island Light- house X 313 h 33 180 197 50 57 30 18 12 s COS a B h 4.323989 9.980049 8. 511546 S 2 sin 2 a C 8.6480 8. 9434 1. 1738 h 2 D 5.631 2.332 -h s 2 sin 2 ct E +654. 0094 + 0.0673 2. 815584 8. 7652 +0. 0583 7.963 +0.0092 +654.0767 51.1 13.3 04.4 57.9 00.00 06.5 40.3 03. 954 53.590 57.544 2.816 7.592 5.919 6.327 -0.0002 sin a A' sec#' J\ 4.323989 9. 471708 8.509354 0. 063404 2. 368455 +233. 5904 sin iW+^') 2. 368455 9. 702930 2. 071385 n +117.87 Fixed a Biloxi Lighthouse to Ship Island Lighthouse, 17° 14' 17. 6". APPLICATION OF LEAST SQUARES TO TEIANGULATION. 157 secondary triangulation — Continued STATION SHIP ISLAND LIGHTHOUSE Third a Ja J,/, iW+*') 1st term 2d, 3d, and! 4th term J Ship to Biloxi Lighthouse Ship Island Lighthouse and Biloxi Lighthouse Ship to Ship Island Lighthouse Ship Island Lighthouse to Ship 30 30 Fixed value O I II 30 13 58 a +144.5010 + 0.1764 12 10. 020 24. 677 45. 343 45. 311 Ship Ship Island Light- house +144. 6774 sin a A' sec<£' COS a B 4.070759 9. 577556 8. 511556 2. 159871 sin 2 a C 8. 1415 9. 9330 1. 1713 A\ 4.070759 9.966509 8.509354 0.063404 2. 610026 +407. 4046 sin i(+4') 9.2458 +0. 1761 2. 610020 9. 702011 +205. 13 X 95 180 247 4.320 2.331 6.651 +0.0004 -h 2sin2a E 18.9 06.9 12.0 25.1 00.00 46.9 -.1 10. 139 47. 405 57. 544 57. 464 2.160 8.075 5.918 6.153 -0.0001 Biloxi L.H- Deer (I I J Belle CS>) _e> PascagoulQ (7) Pins (3) Shipld.LH. Club(e) ^, Horn Cedar(l) Petiti,. Dauphin Id- wfiase r Dauphin d- e- base ft- Morgan Fig. 7. 158 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. FORMATION OF OBSERVATION EQUATIONS The position computation was carried westward from the fixed lines at the eastern end of the scheme and the observation equations were formed in the same order. The treatment of fixed lines is the same as in the first adjustment of figure 6, pages 105 et seq. No new detail arises until the points Deer and Ship are reached, which have lines connecting them with the fixed points Biloxi Lighthouse and Ship Island Lighthouse. Suppose for the moment that Biloxi Lighthouse were not fixed but that its latitude and longitude were to receive corrections of d 12 - 248 i2 - 248(M 12 + 0.0 which with the use of the above values of d

n and 8X 13 then for the line Biloxi Lighthouse to Ship Island Lighthouse the formula gives, d« = 89( ls - 8 n ) + 250 ( 13 = -0.002, §4> 12 = 0.000, dX 13 = -0.080, and ! 0= 0.0+z l -v l 'l=2!+0. v 19 22. 9+da 19 22.Q-z l +v 2 0=+0. Z+z 1 -v 2 +2293 r v 2 =z 1 +229S 1 -2778A l +0. 3 ■277^! DAUPHIN ISLAND EAST BASE Assumed azimuth Observed azimuth Equation Station observed O 1 II 84 14 41.9 137 52 06. 3 188 10 17. 4 281 42 17.9 O 1 It 84 14 41.9-z s +!>3 137 52 06.9-z 2 +!>i 188 10 18. 3-Z2+B5 281 42 17. 9-Z2+U3a V3 -Z2+0.0 V4 =+z 2 +420^2— 4O30X 2 — 0.6 vs = +z 2 — 820,fc— 4980X].— 0.9 D3O=Z2+0.0 West base Cat Cedar Fort Morgan (1) CEDAR 324 16 25. 324 16 25. 0-Za+Vw, »4 13 »J3= zs— 2660^1— 2260Xi—O.2 Cedar 918fiK°_lf 160 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. (3) PINS Assumed azimuth Observed azimuth Equation Station observed 279 11 16. 6 302 33 55. 1 337 56 39. 8 30 22 49. 2 63 09 48. 1 O 1 It 279 11 16. 6-08+ois 302 33 56.9— Zs+Via 337 56 42. 6—za+vx 30 22 52. 1— z s +»a 63 09 49. 6-Zo+»s2 fli8=Ze-34100i+480Xi+341003-480X3+O.O eiB=Z6-459002+2550X 2 +45900a-2550X3-1.8 »2O=Z6+148003-3180X3-2.8 Va=Zt— 153303— 2280X 3 +153004+22S0X4-2.9 B22=Z6-45O00s-1980X3+45O00s+1980X B -1.5 Cedar Cat West base Petit Grand (4) PETIT 96 31 30.6 139 08 41. 2 177 10 20. 4 210 19 28. 8 258 51 48. 7 96 31 30.6— Zj+va 139 08 40.9-Zr+»» 177 10 19.5-Z7+B30 210 19 28. 2-z 7 +»3l 258 51 47. 7-z,+Vsi 7)28=27—358004+360X4+358006—360X6+0.0 »29-27-2O9004+2110X4+2O9007-2110X 7 +O.3 «lao=Z7-25004+4420X4+25005-4420X6+O.9 »3i=Z7-153003-2280X 3 + 153004+2280X4+0.6 !)82=Z7+367*04+62JX 1 +l.O Horn Pascagoula Grand Pins West base (5) GRAND 243 06 15. 8 243 06 15. 8-Z8+U23 »23=zs-45O003-1980Xs+45O00s+1980X 6 +O.O Pins 297 58 16.0 297 58 17. 4— za+vtt »24=Z8+287005-1330X 5 -1.4 West base 357 10 08. 8 357 10 09. 4-z 8 +t>25 »26=Z8— 2500i+4420X4+2500 B — 4420X 6 — 0.6 Petit 58 14 46. 3 58 14 46. 6-Z8+7J26 »26=Z8-27200 5 -1460X B +272006+1460X6-O.3 Horn 101 29 06. 4 101 29 05.5-Z8+D27 U27=Zs-49500 B +870X 6 +495007-870X7+O.9 Pascagoula (6) HORN 104 16 00. 3 104 16 00. 3-ZB+S38 i »38=Z9-535006+1180X6+53500a-1180X8+O.O Club 138 01 57. 4 138 01 54.0-Z8+7J89 !>S9=Z9— 249006+2403X6+2490*)— 24O0X 9 +3.4 Belle 199 19 45. 3 199 19 47. 5-Z9+l>« 1>4O=Z9+15300 6 +3790X 6 — 153007— 3790X 7 — 2.2 Pascagoula Grand 238 09 28. 1 238 09 26. 5-ZB+Sfl !)4i=Z9-27200 B -1460X B +27200 s +1460X 6 +1.6 276 26 01.4 276 26 00. 7-z»+!)« !) (2 =Z9—35800 1 +360X 4 +358006— 360X 8 +O.7 Petit (7) PASCAGOULA 281 25 13. 6 281 25 13. 6-Zio+»sa »ss=Zio-495006+870X 6 +495007— 870X7+0.0 Grand 319 04 37. 2 319 04 34. 2— z M +»34 » 3 <=Zio-2O900 1 +2110X 4 +2O9007-211aX7+3.O Petit 19 21 11. 1 19 21 07.6— Z M +!>3S f36=zi0+ 153006+3790X6- 153007— 3790X7+3.5 Horn 57 14 14. 4 57 14 12. 2— Zio+»s6 »36=zw-286007— 16O0X7+286008+16O0X s +2.2 Club 89 04 35. 7 89 04 35. 6-Z10+C37 D37=zio-398007-60X7+398009+60X 9 +O.l Belle (9) BELLE 268 59 33.9 268 59 33.9— zu+043 »43=zu— 398007— 60X 7 +398008+60X9+O.O »44=zu-24900a+24O0X 8 + 249009— 24O0Xs-1.8 Pascagoula 317 68 21.9 317 58 23. 7-Zu+Vu Horn 358 28 28.6 358 28 29. 2-Zu+t>« »45=zn-17008+5590X 8 +17009-5590X9-O.6 »46=Zu— 311009— 21l0X9+31100io+2110Xio-2.O Club 52 04 22. 8 52 04 24.8— Zu+Dm Ship 101 03 48. 6 101 03 47.7-ZU+C47 »«=Zu-7OO009+1190X9+7OO00ii— 1190Xu+O.9 Deer (8) CLUB 89 46 03.6 89 46 03. 6-zi!+7>48 »48-Zu-49O00 8 -20X 8 +49O00io+ 20Xio+O.O Ship 142 06 04. 8 142 06 07. 1— ZU+U49 !)i8=zl!-266008+2970X8+26B00u-2970X u -2.3 Deer 178 28 33. 6 178 28 33. 9-Zi!+a 6 o »50=zi2— 17008+5590X8+17009—5590X9-0.3 Belle 237 09 18.0 237 09 17.2-Zia+t>6i l'si=Zl!-286007-16O0X,+286008+16O0X8+O.8 Pascagoula 284 12 30. 1 284 12 30. 8-Zi2+i)58 W52-Z12-535006+1180X8+53500S- 1180X 8 -O.7 Horn APPLICATION OP LEAST SQUARES TO TRIANGULATION. (11) DEER 161 Assumed azimuth Observed azimuth Equation Station observed o / // 281 01 03. 4 322 03 14. 8 18 50 39. 5 42 46 50. 3 115 21 13. o / // 281 01 03.4-zi 8 +t>,a 322 03 14. 1-zu+vn 18 50 41. 3-218+DS5 42 46 46. l-Zia+ass 115 21 12. 5-ZU+V57 t)53=Zl3-700«X9+700^ u -119*Xii+0.0 »64=Zis— 266^8+297aX8+266«^ u -297*X u +0.7 !)D5=zi3+167#^io+425«Xio-167^n-425*X u -1.8 »56=zi3-197^u-185*Xu-11.0 !)67=Zi3-6O3^u+2480Xn+33.7 Belle Club Ship Ship Island Lighthouse. Biloxi Lighthouse (10) SHIP 67 47 12. 163 31 18.9 198 49 24. 7 232 00 23. 1 269 41 59. 2 67 47 12. 0-Zu+t>58 163 31 19. 0-ZH+to 198 49 23. 4-Zu+vso 232 00 23. 6— zh+Wi 269 41 58. 4— zn4-%2 »58= Zi4-5003&o-177,jXio-15.2 »59=Zu-110^io+323«Xio+43.2 7>60=zu+167*£io+4253Xio— 167^11— 42&JXu+1.3 »6i=Zn-311^-211aX 9 +311^io+211«Xio-0.5 f62=Zi4-490^s-23X8+490^io+2JXio+0.8 Ship Island Lighthouse Biloxi Lighthouse Deer Belle Club BILOXI LIGHTHOUSE 295 18 30. 1 343 29 51. 1 17 14 04. 4 295 18 30. 1— Z15+B63 343 29 47. 5— Zi5+t)o< 17 14 06. 1-Z15+DG5 !>63=zl5-6O3a#ii+2480Xu+33.2 »«= Zi6- 110^io+323aXio+ 46.9 %5=Zi5+11.6 Deer Ship Ship Island Lighthouse SHIP ISLAND LIGHTHOUSE 197 12 06. 5 222 42 10. 247 43 46. 8 197 12 06. 5-Zie+Va, 222 42 08.6— Zi6+1)67 247 43 47.7— Zie+»68 »w=zi6+13.3 »87=Zw— 197^ii— 185tfXu— 13.8 »68=Zio-500^io— 177aXio— 16.1 Biloxi Lighthouse Deer Ship In order to get the quantities on a better relative basis, it is best to adopt lOOc^, 100(5^, etc., as unknowns in the equations. The coefficients throughout will then be divided by 100, and from the solution we shall determine one hundred times the corrections in seconds to the various latitudes and longitudes. 162 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. S ■», o o iB O ©' o o © o w. i i i i iii B- ©COCO ©cocs©»o OOOStHO or-oso ©■* o-qococn.-i OCOOQO>00 O CO OS O"© CO ooo ++ 0000--H 1 1 1 1 + 1 1 O CN i-I V 1 1 1 ©©©.-.©©CN +++ 1 1 + 0"« MHO) 1 1 1 1 I dddo'Hfi +++++ - CN CN 1 1 <*s OS OS s s + + CO to CO CO 1 1 88- S CO CO + + CO CO C-3 CO 7 7 COCO 55 os rH r-i + + CN IN — . -f- 1 1 I- I-. + + oo io i : ++ IO iQ CN CN + + ts CM CN CO CO + + a a cm cn + + COrHtNCOtNOJ CO i-i "& CN CO t- tN tf N en ++++++ CO co + + CO CO in m + + 00OJincOf-tN COCN rH CO II 1 ++ 1 COCO 1 1 CO CO CO CO 1 1 co > o oo co co r- ~

o IO 7 7 'cs CO CO 7 T co to CN CN N«ONHN N00O«JfH CN CNCOC0.-H + 1 1 1 |7 CO * + 1 It + 1 o i> »o oi ira io +++ 1 1 1 T-H 1 00 1 1 o to 7 164 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. T3 o a s ft, © d d d d d *^ rH i-H i-H i-H i-H rtfo 1 rH rH rH r-1 i— 1 Hr> 1 i-H i-l i-H i-H i-l Hho 1 rH rH rH rH rH »*> 1 i-H rH i-H i-H T-H Hk" 1 ^-1 r-C i-H i-H i-H ri>0 1 R- o ■*»* CM co t- »« OOU3NH00 OOOcOOOSiO oeoeooot-»o or-oo onco tM IN CO »0 00 CO d co* cn i-h d eo" + 1 + + + OCOCOCNOOO +++++ OrHOCNOCO 1 1 1 + 1 dcN'oddo! i i + ii ooH ssa + ii ++ idco'i-H odd 7++I++ OS OS 77 i i os r- »o >o oo oo HffiiMOOrtiN H«mHWt» 1 1 II + 1 1 i oo ++ CD CD CO CD CN tN + + OCONNCOH 0(000)0 t-tN HiHcO ++ 1 1 1 1 CO to 7 7 7-t I-H o» Oi-Hi-HOO CN >OCN i-l 00 +1 1 1 + 1 OS OS IO "5 1 1 OS OS iH i-H + + 1 1 ■8 •to + + 00 00 ■-. -..-.■ CO CO + + OOfflNHON OS ^* ?H i—l O ^ cool cor-co +++ 1 1 1 r-i i-H + + s s 1 1 rH i-H rH i-H CO CO 1 1 00 00 1-1 tH 1 1 o o Osoaoco OS tO CD i-H OS CN lOrHi-H 0O 1 +++ 1 + a s + + ococo 00i— It-CD TP CNCOi-H 00 1 II 1 1 1 CO CO 1 1 o o CD CD rH i-i 1 1 *0 CO CO 7 7 •OCA CO CO 00 CO OS&IOOQOSCO ++7 u7 00 CO OS OS CO CO 1 1 CO CO 1 1 «3 XOO>CD(CN ++++ 1+ os os t- !>• CO CO + + o o 3 SI + + ooao i-H i-H iH i-H + + >oa>nc>iciOH CO^JfiO t»io lO « rH N C<3 1 I+++ 1 CO CO io »o i-l i-H + + OS o» -* ■•* 1 1 lOiO COCO e H 00M«pl>WONOONOOM«0)'<)!OO«N QQOQM © O © e» K — VOQOig ddddMO^^rlciwwc^T^r^TiiNoaiioc^dHTjHN'rt'a'do'dd^ridooooo i ii i i i i i + 1 i ii i i +++ in i i i i i +++++ + 1 i ++ fS- i— lira c co c5 *<3 w C0«<0CftNOO0J'*Ht-ONa©VNQ0(DNMOQ0 00C»i0OO«Cft J- +2.87 +1.17i +4.50 +2.01i +0.25 +0. Hi +4.50 +2.87 +0.25 -2.72 -4.95 -0. 02l 00 +0.62 +0.25! +2.28 +1.02! +0.36 +2.11 +4.42 +2.28 +0.62 +4.38! +4.42 +1. 98* I— +3.67 +1.50! +1.53 +0. 68l -3.58 -2.09 -0.25 +1.53 +3.67 -0. 32! -0.25 -0. Hi CO •to -0.48 -0. 21i -3.18 -1.30i -0.48 -2.55 -3.18 -2.28 -1.98 —4.68s -2.28 -1. 02i -1.98 -0. 89i lO +3.41 +1.52i +1.48 +0. 60i +3.41 +4.59 +1.48 -1.53 -4.50 +1.54i -1.53 -0.68i -4.50 -2. Oli - -4.03 -2.02i +2.26 + 1.0U -5.37 +2.26 -4.03 -4. 12i -5.37 -2. 19i +2.55 +1. 14i CO +4.20 +2. lOi +6.57 +2. 94i -2.73 +6.57 +4.20 +4.64! -2.73 -1. Hi -4.59 -2.05i N -2.77 -1.96i -4.98 -2. 49i -2.77 -4.98 -2.26 -2.26 +0.48 -5.27i -2.26 -1.302 -2.26 -0.92! +0.48 +0. 21i - ^ a> <55 N *-i a* eq t0«)rtWaMM'5Q0Hs0OOOM©0C(D dccMi^OfHOcococ^c5«o?^oc>ioi^oNoc5ofHc3ow^ocdrJu3 HW CN-tf rH CN •* rH "O >H i-H CN i-t 4- 1 +++ +-M-++ 1 1 I + 1 1 1 + 1 1 + 1 1 ++ 1 ++ 1 +++++++ 1 1 1 B- CO CO CO CI ■** CO rH •«> 1 , «!OMOOOlOP^Ha>0(»»00)l50«MCO^HONMO^• l ElN«MU)MC5NOi!DO>^5XHOO dMMHOHOMWNdridHCJNciHdNdddHCJdrtHed . »d co" i-i d d co eo to »-* cn" co" co cd . +i+++ +++++ iii+i ii+ii + m ++7++I+++++++77T CN CM dw n eo eft t— in w oo oo U5 ooo co >C i-^ iHUJ Cft CO >-< OS CN 00 TJi rji CN OS'S 1 iji 00 O rH © CN Hi-i CN ■"* rH CN CO -* i-HGN i-H r-H i-I III M 1 11+ 1 1 1+ +11 CH *t> ow id eft o to i>- r- co t- wS co » n '5 Ot-H CO HO-icfto eft r- iooot^-i-ioc|oco ooo >— i CO Cft i— t CO l^OH!DHCJ)S i-H CO ooo eo *-I -r cn rA o»rii-4>-Ico , ^i-4i-HO* >o cn + ++ + + +ii ++++ i i ii zc O r» cdcqcooohOco eft ooi "co .-i'-Si "^ O OOOi"OrtHO) »0 lOiH iO H O) cn" i-i dddw'irfNi-Iri »o oii-i o" cno i i +++ iii+i i i+ + ii I— cft i-H cocooocfiNHOio i>- ooo co i—i en -* r-l Cft t- CO -eft i-t i-( O tO i-H OO rt tH CO c4 r-t ni-JoswdriNH d o'r^ CO CO* r-J + + +++++ III + +1 1 II 2 0O CO © CN Cft CJNNOlOOOH N CO CN 3 H W5 CO r»- «3 U50C410COHO Cft CO OO >-i o i-id 'o c4dc4ioHHii oi *-i oo 1 1 + + + + 1 +++ 1 + + + II IO «3 i*3 eft toco t- oooonisich c "en © "cn CO CO 00 CN i-l O OS tO T-i 0O CO CN CO i-H Cft i-l »o cn cn i-i d d •** cn d cn" id d cn e4 -i*cn + + + + 1 MM + + + 1 1 M ~t> Cft cft't-i-icftoco^eo co o "cn r- eoMrfr-coot-o o co r— co H*6(NMH'dmo o i-4 d i i i ii i i i i i ii CO CO 00 «3 Cft CO CO 00 Cft 00 00 CO 0Q IO eO Cft O W 00 Cft lO Cft I> 00 CN i-« d^cNi-JcNeodco .-J n rt 1 I++ M M 1 1 II CN oooeftcococft cb cft o I s - ooco H-ri..-rWI- I— CO ^ O i-H»0 WCicOrHdeO CO i-H pS 1-4 HO ++++I+ + + + + ++ " loeftcotNoo co oo cft i5 mos CO-*»Ot-10 »0 tO "ttl rH COCO lOC'iiHNCO rH ON i-4 >OtN 1 1 +++ + + 1 1 II o to ION Cft ■*t» tooo CO h eJei d 11+ + o N c5<0 3 t" tNOJ CN CN* i-I -l" CN III 1 00 < COCO tH "cH MH rH Cft dd cn d ++ + + i- oo *c5 cft "co IOCO O Cft co^h cn d III 1 168 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. i-mnotoooNH OJt-CCO-<: h*SSc thOOMOi-hoo +++i +11+ MMMiflNooiNinooosHQHi-fiNggHmgr- OHiooJCMHMoaJiniOQBMNnNsffliSBJHr; roia-j'iowojMfflciiO'jiSiiisHffir-iOHWNWco i ++++++ i ++ i i i ++ i ++++T+ « 00 «ai 00 CO N + I + I I I + I ++ OQCOiOHHlOCCnNQCO r^c3»OCOiO-*'T-1I--«OC'J(N«3< ecr-Ic4oi-^coc4o6r^gw5|rf I+II+I+IMI+ ineOCNC»aQQ0i- *-* c> c> ■>£ +++ I I + I + jt-cW«Q001ONMh SOsoOOOoOOOOOOiCs HHHOOIVOI -+ I +++ I I ssssssss n .-5 eo c* i>: e$ o> ++++ 1 1 + t-QOr^OOCN^COOOQ r-CDClsDOS'^'i^-OOO + I ++ I I I I + so i-H 1-1 eo eo a» i i + i ++ i+i i+i i i+i i+ so»nr^r-co i + 1++ i-lQOOt^OiCOfNQOOO'* ■VQONOHMlOrtOOOSO) Mi-5>ooirfcdc^odr-Jcoe^ T-4 CO l-H Tt< h + + I ++ I + I + I S©e0C0C0C>l* co oo 25 i-h Oi 5 oo eo O co 00 *3i ^ CO I s - 00 ■* i-l SO HHNOrtfQOOiCrtW II + II I M + + iOi0C4OC5 + 1 HiO r-r— 8§ rid + 1 »nto OS w os© OO + 1 CON O1H00 CO 00 w TPCOO cioo 1 ++ ■* as* oso eo o oo o'o* 1 + Tin iOeo« ocsod ++++ OS l- 00"^ tow wO w'o + 1 i—( os os com oo op to wn NONON oddcsd 1 + 1 1 1 & gs COO do 1 + io r- n ■* *n ci O^HM^HO^r HOOOHO oeidddci 1 ++ 1 ++ t-N toco 38 do + 1 -*• i-H 7 wl cn'cn ■qio wd 1 + OS I> m o 8 3 «-5 d 1 + sl wo w*d 1 + CO in i^ o m + ml tJICO t- 1— oo md + 1 m n m to U *° os •<*< n d + 1 COOS OS to -*co NO + 1 N os + N t-00 ss wd + 1 S I d ©* + 1 m to ss O w coo + 1 s 1 *o CO + NCO OCO tt m mo cod + i w % tr~ CO n d + 1 m eo CON NCO OS CO N'd + 1 o o Oa N* -*co COO r-o COIQ COCO too N OS WOO asm £© in to § s osn too OS lOrf ^J< WCOCN W S3 ooo cow NO lOtOCONO NI—NNOS COOOSNCO WWW CO-* 1 os co ON r— Tf l>CO MflNHHHOO NCOCOCOCO W co mesmosos WOOONO 00 w mos cSeo do + 1 OO 1 I OO 1 + WOO ++ 1 wO + 1 WCJOO 1 + 1 1 NO 1 + soooo 1 ++++ 1 + 00OOOON 1 + 1 + 1 + too 1 + os »nw O00 CO-* coo CO 00 ■V o wo tDCN N N WQ COO in w NHN OS 00 00 w COO * t— o CO r-- ©■* tow OSO WW gst^eooo ooo to to r— o»o co CNOWCO O to HN OSN ■•PCO t-o oQOseowos NOS WCICO tOON Q T Nto woco to N OO coos W i-H W CO oo co os m too r-eo-tp hoin OS O i-H CO N t» t- CO O N N I- OSN W H ■CFO + 1 do + 1 oo + 1 «CNO +++ coo + 1 OCNOO + + + 1 wo + 1 -Addend T+ 1 + 1 wo T + ddddoeo i +++++ NO 1 + 00 osn as COO oo »n lO"W woo oo 00 cow 88 mNcn com w to too WHO CO * t— NCO CO 1 * lOO CO cot— O iraiococo w t cs o> tOOOOCN t- "*tO Osw COO OO ■^t—^ OS cO OS to oo cot— 1»- N O N O 8 wS cot-cocoi--o hNMhhv CONON w w OS wo WOO o OS to OS om co to WO WO + 1 oo + 1 do" 1 + N W© ++ 1 coo + 1 OOOO 1 + 1 1 OO 1 + WOO wO 1 + + + + oo 1 + ++ i + i + WO ++ - to "* COCO OO CSS CON COO NO + 1 Nr- t— co S3H wO* + 1 8 WtO CO tfi WIN WO WO + 1 OOOCO OJIOH lOON OC-© WHO +++ to HN COOS oo m r-o «*d + i CN w'oO + + + 1 CO CO COX OO) COCO r-o COO + 1 r-oooo w-hi MON COCO cor— osmco OcoNt-r- t^dd wd 1 + 1 + I N OO OCO OO) t— CO OSO i + t— OS to ^* to-* CO CO CO ■* OS CO to 00 COCO CO 1 * nwow -win 1 +++++ 00w ooo 1 + CO CO OS© or- eoo coo 1 + t-co ©N i-iO 1 + CO as -*CO con cog wo ! + WTTO + 1 1 * t-CO COtH r— co r-o NO 1 + MtOO-^! ci t n m OO* w CO CO OS iocs COCNOO 1 I ! + in r- w oeo o w i>-o 1 + QOOSOOt- os r- co to i— NCOOS Ht» NCOCOOCO tOOO w w + 1 + I + CO CO tOCO oS wo + 1 MOOMNiO to wosmoso m ■* "ran co os-wooosio + 1 i i i i to 4- lO s COCO too Nm 38 t^W W 1 + r- 00 00 to to-* CS-* lOCO irfo" W 1 + CO-* W iocs oo 1000 03 t-CON + 1 1 »n OS wr- 00 CO CO 00 t-o too - + 1 t-t-O w CNr-cor- oto t— to r- co com COOSkOO* t- w + 111 OS + WNCO-* CO o w IQN 00 os coco into r-O 1 + Nr- COCO into CON 1 + W Cjw CO 1-4 coo oo 1 + csNin wosco tDt*H inc-o s«° + 1 1 o + WCNCO N 3 Oeo M to W© ■*d + i NCO ■* w idcJ ■* + i a* ■* + HN i-H s 3 w 170 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 1+ I I I + + HO OSCO Nt- PKowonNHn *rc «)OONW©HSOO OO dddHNeovdiaw r-p oooooc 33838888 ■«* t- qo o o OHO oo" + I + I + -rtIO 00 CO i>o OO I + OOO •&<£> IQO OSH epoj nh OO OSO O OO I 1 + conn or- ■flil-N -*04 St-OS Ht- VO OO do© ho* I I + I + g 3 1 OO tfjd + 1 "* NO CJ CO iO ■oo © HO + 1 + ss O0O0H VN OSHIO t~^l HMI- H"Ct< HMO OO OS'*' t-»o i--*ra Nw CN IO ooco HO OO OOO HO OOO OO I ++ I + gOOO N CD Tuoom oooo OO-Cf HO qon OOH TTO CIO OOOO NO II ++ I + OS CO OOO© H «-l r-ocs^r-- coco QOHtfiQON Nifl woow oseo OOOOO ti"o" I 1 + I I 1 + om ot- U5H ow -Tr-I NH oo osd ++ 1 + t-cDt* tHH HH CD t-CO HCOOS -O CI OHO i>o odd do + 11+ I HCOOOO OO MOHO do ++i i 7 + I>HCOOH ON Osioh cNira cocp coinNeON co© r-00N©OS COOS ho©©-*" n© ++I+ I +1 HO COH do do + ++_!_ OQHH tor- NOOO OSO ON© ■"»■ 00 OOH OSO odd no + 1 + I + ^"^•COIO »T3CO HI>O0M OCT T COt-O O ■* t-OOH »c*© ddc4d cod ++++_+ 1 OOiOOSO coo oor-osoo »oos hovcow r-os "TflHtMCOCO t-H dd©N© NO +++ 1+1+ OO gj© ++ I + 00^- OS *3d + 11 I + IOHNH H00 HCOH Tjl OOCO fflffiON OOOS HNCCN -^H novo uid ++ 1 + 7 + NMNOIO OSiO «inHL)iD -*os Hcor-eoco w* w co io re io oco HOOCOH ++ 1 I I OSO I + OJWNOtOOl 1 CC fl H O IO t Ol 00OOH.-ION c i ac g?3 n- o WlflHCONtNtO NtP t-HONOuJHH NO lOOOOOOVO CO00 HNOS CON OOOOONO + I + I + I I £-0000000 r-o "l I I I ] I + I I + 3HCOOSOOOCOU3QO CO -< aoooooHoosr- os;£ CDOOOO-cJiOOO t^-O + 1 + i +7 i i + i + COOOOOt^NOHO CO + 1 I I I I I I I I + ■i -ctH N t- N CO N ■*© HOQHiniMioM r-o J N H CO N OS © OO r-r»-<_ „ . . , OJOOOiOOO'VCN OOCO OOOCOOOH^I HO §CO IO I— CO OS V CD H 00 CCVHNHOOHH rH O O CO IO IO CO CO *■*< 00 m HNO OOOtJHNCO O 00-©. HNCOVlOOMMffl c-ooooco© I I I I I I I HO H > + HOOOOOOO HO + I + i + I + I + t OSOOOOupON© OS + 1 I I I I I I I + HNCO^lOCDt-00 OONMNN 8 S3 „ in ^-ij^ 3©© COO CBOSHlOHVfflW CD fOOVOOt'tp 00 -f NOOHOOOO O^ OOOOOOO OO I I + I + I + I + ■HOOOOOOO TM +1111111+ OCHtTOCOON 00 -I tP cDcC HNMt IO ON OSOOOOOO 00* »o us + I! II I I + APPLICATION OP LEAST SQUARES TO TEIANGULATION. 171 fflOONWlNO'H i-i coo r-t-oseM f OHNHMlO COO I I 1 + I+ + + I CgO-fMifOMO oo O W t- H H CO C-3 O OSW 00 00 CO O W t- 00 CO 090 CM H CM W in CO in t— H CM OOOKJOOHO in O I I I I I I ++ I + t^iOMNWW^COO) COO i-i OS W O W h CM OS CO COO nb-OmtNOOON WW in l> OS 00 i— I F- CM O CO OH COOH WOCMC-COO t-O I I I +++ I + I + I 3W oocot- OS o w (OcaHffiiiH I— CO " -QV3 WO NOOrlOOMrtMO COO +++ I I +++ I I + I (Mcoooor-o oo ONOfNfrt W in NnH^wmin coo mOOOOO© wo OOOiOMN«(ON OS© W CM CMOS CM W inin H W SrtQHHHHCH OO OOOOOOO ho HW r-t CM CO CM H t- CO O-b OMONmSHNCD WP JOOONOf OCOOltt• CO oiiociMOcnt-'aiosOT w w W i— lOOOO CM ©COO WO OOOOOOOOOO OO OOOOOOO OO I + I I I ++ I + OOOOOOOO OO I + I + I I + I I + OOOOOOOOO OO I + I I ++ I I I I + OOOOOOOOOO OO + I ++ I ++ I 1+ I + ON t-in © w t>o cod 1 + © HO I + 1 ©CO HO W © W OS t- CO in © OOO w© OO coo 1+1 + OOO OO I ++ I + coo + 1 OCM W in t> co in co OCM COCO WCM COi-H HO OOO + 1+1 cocpco OSCO w as co i-~ o t-©00 CM CO CO^H 000 OCM© OO + 11 1 + COO Win t-co CM© OOO HO + 1 O OO + I + r-.i> t-00 CO (MO d do + 1 + WO CMO H© COO + 1+1 OJiOH 00 o OHO o'o + I I + I . m o HHtoffl ioa t-OS cpH tjio 0OCM OCM t-o OO Colo OO dodo do I + I I + I t-©r-i_ , OSi— IC0COO COCO CMOOHO t-O dodo© ho +++ I + I + NOCBNOH OCM dododd cmo ii i ++ i 7+ aao W CO 00 CO CM H OS© as©inco©osin coin CM lOOWOQHr-cO OS CO CD ©©©CMOHCM t-W co" odddoH'd w© I ++ I I + 1 + I + ocooh cor- SOHN COCO CMt- 00 t- CO HOH© COO + i + i 7+ O W ©CO W COO COt-t-OsiO t— W OifliO'*'* cor- OOOOO WO I I I I I + 1 Socoioo w eg© CM CO OSOS © 00 CO VtPioNNN COO flOSOHN COCO ©dco'dco© do ++++ I + I + h no co w w co l— cmco NOt-0!DHK3 Mil hohcmoooco t- a dddoHcod do I I + I I ++ + I oooo COO CM I I + I + I U5WOOM O OS © OOOOHU30 I + I I 1+ I + CM 00 © in H in iflN oo co r- as r» f- t-CM NHNHOlia WW ooinosmco hh ©000100 t^d I ++++I+I sss __WCOt- CM OO lOOJOW CO CO COCM OSH WH OOOOCOCOO HO + I + I + I I I + cm* osincoo om K3U3 0SH HOS »f who as in OCOCO W CMO no'h'o wd »n + I ++ I + OONfflO Osep in co ocm eg gso CO CO OSCM© OH OOHCMHCO OOH odddeo cm'o I I I + J I + omwmcMH com CMiOOOOSiQin COCN os co c-i m c5 1^ co w NOWKJCSH HW ddwdcMO cmo +++ 1717+ HNfllOMOCfi l"-0 GO W W CO CM H © LO 00 1— O OS H H H CO w in HOHINlOO"^ 1 CO 00 O O O O t*- H o I I ++ I I + I + H OS CM OO CO t- OS HCO go m t- os w co c— hoo o i- in w cm w as osco COOHWOi'H COH ONHMHOMH OOCM IOIOOMSNOO 000 C I CM © O OS CO CO t- OO fOOMOOOltD CMO cor- os© wo ©i>in hos H COO OOCM -iOCNC r^oooooooo o . + + I I I I I I I + I OOOOOOOOOi-t T i i i++ i + i + ooo I + eooooooooo-^eo oo ++ 1 I + I + I +++ + I COrtOOJ -¥ t Tfl-H U3U3 «C CNOiHO I + 1 I COO + 1 OMtOH i-Joooo +++ 1 + lOO I + t-COftiOOOO ocn COtaiQlAO)H 0"-t i-HQOOSWlQ-^ f-CO CDHOcOCilO OSO ©* O « O w i-I ci O I I I + + + + I tomo + i + i OiO I + oo oo »-i i-t pa T-i o HOHOO rid I I I I I I + S Hoioono ++++ I + I + % o © oseooso coo MiOKJ V (OCA oonm osio l> rH oo-^ I>tH ©odd do +1+1 + I NHOCOW 001OC-O.-H OH(ON1< cocmcmo-* ddddd coo cooa i i i i + i+ ■* to i— r» os o l— CO CO l> 00 t-H ooho*h r»o ++++++ +1 (OIOOOC6 "* iH osi-hcoos r— eo oiQcoco ^eo ■*roci& dd CO + I ++ I + O0>O OS00K5 OS CO l~- OS CO 10.-H IOHN OSO ^H "CM (NcO I I I + I I + OONHOOS OS CO rHcoooooseq 1-1-1 t-5d**deod wad + + +M+ 7 + 00O©t-(OH00«CD S-r if:- -v lT- OS i— ( V. CO ^ O CO CO ^ OS Oh ^h hhShOOhmh codddddd io"^' ++++++ 1 I I - Of ^ "2 2J 9P °° -* w in tm o io woooMftoqaw to „ TOHHNOOfflMC6H m ? HOOOOOONOHH "3^5 OiOOOOflOcOH 7+i 1++1+ i jflOOOOCHHOO + 1 I I I I I I I I o-jcocoi-oor^ira jOOOOgOOt + ! I I I I I I I + t^ooosO'HeqcO'* t-OOOSO^HCNCO'*iO APPLICATION OF LEAST SQUARES TO TBIANGULATION. Solution of normals — Continued 173 18 19 20 21 22 1 J + 61.5179 - 1.0383 - 2.8738 + 31.3248 _ 8. 1976 + 7. 4331 + 7.1924 7 — 0.0484 - 0.0031 + 0.8784 8 - 0.0005 + 0.0005 + 0.0230 9 — 0.0313 + 0.0066 + 0.4053 10 - 0. 0463 + 0.0045 + 0.4581 11 - 0.0001 - 0.0061 - 0.0010 - 0.0057 + 0.0034 + 0.0004 - 0. 0120 12 - 0.1183 - 0.1551 - 0.0650 - 0.2477 + 0. 0726 - 0.0070 - 0.3548 13 - 2.9008 + 1.6151 + 0.4512 + 2.0065 — 0.8315 - 0.0420 + 1. 6955 14 - 2.2294 + 0.4010 - 0.2352 - 0.3850 — 0.3233 — 0.0208 + 1.7425 15 - 0.5441 - 3.2335 + 0.9685 - 0.8120 + 0.2910 + 3.2905 + 3. 4243 16 - 35.3839 - 14.8119 - 4.3115 - 0. 9158 — 3.6707 - 7. 2125 - 7. 9221 17 - 1.3061 - 1.8508 + 0.8318 - 10.5336 + 0. 3480 + 7.0757 + 5. 7131 + 18.9087 - 19.0796 - 5.2350 + 20.4315 _ 12.3081 + 10.5259 + 13.2434 , d\s + 1.009038 + 0.276857 - 1.080534 + 0.650923 - 0.556670 - 0.700387 +106.8200 +27.9899 - 16.8102 _ 7. 0763 + 34.6511 + 85.3611 11 - 0.5579 - 0.0879 - 0.5202 + 0. 3101 + 0.0382 - 1.0954 12 - 0.2034 - 0.0852 - 0.3249 + 0. 0952 - 0.0091 - 0.4653 13 - 0.8992 - 0.2512 - 1. 1172 + 0.4629 + 0. 0234 - 0.9440 14 - 0.0721 + 0.0423 + 0.0693 + 0. 0582 + 0.0037 - 0.3134 15 - 19.2169 + 5.7557 - 4. 8257 + 1.7296 + 19.5557 + 20.3509 16 - 6.2003 - 1.8048 - 0.3834 — 1.5366 - 3.0192 - 3.3162 17 - 2.6226 + 1.1786 — 14.9264 + 0. 4931 + 10.0264 + 8.0956 18 - 19.2520 - 5.2823 + 20.6162 - 12. 4193 + 10.6210 + 13.3631 + 57.7956 +27.4551 - 18.2225 _ 17.8831 + 71.8912 +121.0363 Sfao - 0.475038 + 0. 315292 + 0. 309420 - 1.243887 - 2.094213 +50. 9179 - 9.2141 _ 26. 1027 +167. 6744 +214. 2334 11 - 0.0138 - 0.0819 + 0. 0488 + 0.0060 - 0.1725 12 - 0.0357 - 0.1361 + 0. 0399 - 0.0038 - 0.1949 13 - 0.0702 - 0.3121 + 0. 1293 + 0.0065 - 0.2637 14 - 0.0248 - 0.0406 — 0. 0341 - 0.0022 + 0.1839 15 - 1.7239 + 1. 4454 — 0.5180 - 5.8571 — 6.0953 16 - 0.5253 - 0.1116 — 0. 4473 - 0.8788 - 0.9653 17 - 0.5297 + 6.7081 — 0.2216 - 4.5060 - 3.6382 18 - 1.4493 + 5.6566 — 3.4076 + 2.9142 + 3. 6665 19 -13.0422 + 8.6564 + 8. 4952 - 34.1511 - 57.4968 +33.5030 + 12.5701 _ 22.0181 +125.2021 +149. 2571 0' 1st term 2d and 3d terms East base to west base West base and Cedar East base to Cedar Cedar to east base First angle of triangle o 30 14 56. 379 5 51.937 30 20 48. 316 +1 Dauphin Island east base Cedar 30 17 52 -351.9435 + 0.0063 -351.9372 sin a A' sec^' COS a B 9. 995568 8.511556 2. 546473 sin 2 a C 8.0787 8.3054 1. 1712 Jl 4. 039349 9. 152688 8. 509351 0. 063997 1. 765385 -58. 2620 sin J(#+#') 7. 5553 +0.0036 1. 765385 9. 702857 JX X' 84 +103 188 180 8 37 5.093 2.332 7.425 +0.0027 1. 468242 10 41.9 34.3 16.2 29.4 00.00 45.6 33.6 14.288 58. 262 STATION CAT Second angle ■t. J West base to east base Cedar and east base \ 264 - 38 / 11 37 22.1 52.4 West base to Cedar Cedar to west base 225 + 33 3 29.7 49.5 180 45 00 37 00.00 19.2 30 + 1 14 6 21. 492 26. 825 Dauphin Island west base X 88 14 7 51.034 35.008 30 a r 30 17 It 35 20 CO. I 4 at i 8.317 4.230 9.845 8.511 995 212 557 Ced s 2 sin 2 a C ar 8.4620 9. 7073 1. 1711 X' h 2 D 88 5.176 2.331 07 -h s 2 sin 2 a. E 16.026 2.588 8.169 5.917 -387.0473 + 0.2227 h 4.23099 9. 85367 8. 50935 0. 06399 2. 587764 9. 3404 +0. 2190 2.658019 3. 702795 7.507 +0. 0032 6.674 +0.0005 -386.824 3 sin. a A' see^' 5 5 3 1 7 sini(# +#') J\ 2. 658019 -455.0080 -J y. 2. 360814 tf -229. 52 STATION CAT Third angle i«+tf') 1st term 2d, 3d, and' 4th terms West base to east 1 Cat and east base West base to Cat Cat to west base 30 30 -279. 0776 + 0.0231 21. 492 39.055 Dauphin Island west Cat s COS a B 3.972888 9.961280 8. 511557 2. 445725 sin 2 < C 7. 9458 9.2130 1. 1711 sin a A' see^' A\ 3. 972888 9. 606517 8. 509352 0.063865 2. 152622 -142. 1092 sin JW+tf') -Ja 8.3299 +0.0214 2. 152622 9. 702600 -71. 65 X JX X' 264 180 23 4.891 2.331 7.222 +0.0017 -h s 2 sin 2 a E 22.1 11.1 11.0 11.7 00.00 22.7 51.034 22.109 2.445 7.159 5.917 184 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28, Final position computation, STATION PINS Second angle 4' ifo+tf') 1st term 2d, 3d, and 4tn terms Cedar to west base West base and Pins Cedar to Pins Pins to Cedar First angle of triangle 30 48. 317 35. 914 Cedar Pins 30 21 36 -96. 4055 + 0.4916 COS a ■ B 4. 264911 9.207641 8. 511550 sin 2 t C 9.6912 +0.4911 AX 45 + 53 180 279 58 37 19.2 37.8 57.0 43.8 00.00 13.2 27.5 16.026 20.225 3.968 2.332 6.300 +0.0O02 -h 2 sin 2 a B sin a A' sec tf>' J\ 4. 264911 9. 9942754 8. 5093503 0. 064116 2. 8326527 it +680.225 A\ sin J(#+^') 9. 703663 2. 536316 it +343. 81 36.251 +1 1.984 8.518 5.919 6.421 +0.0003 STATION GRAND Second angle dtjt ' «#+*') 1st term 2d, 3d, and \ ;erms / ■Aij, 4U, C 4th Pins to west base West base and Grand Pins to Grand Grand to Pins 30 22 24.231 04.656 30 20 52 +184.4676 + 0.1884 +184.6560 sin« A' sec $' COS a B 4.099752 9. 654620 8.511548 AX 4.099752 9.9505064 8.509352 0. 063888 2. 6234984 +420. 2409 First angle of triangle Pins Grand sin 2 a C 8.1995 9.9010 1. 1734 AX sin i(.4+4'1 9. 2739 +0. 1879 2. 623498 9. 703504 +212.32 X w 337 + 85 180 243 54 4.532 2.333 6.865 +0. 0007 25 -h « 2 sin 2 a E 40.7 04.4 45.1 32.3 00.00 12.8 03.0 36.252 00.241 36. 493 2.266 8.100 5.919 6.285 -0.0002 APPLICATION OP LEAST SQUABES TO TRIANGULATION. 185 secondary triangulation — Continued STATION PINS Third angle 1st term 2d, 3d, and' 4tn terms . West base to Cedar Pins and Cedar West base to Pins Pins to west base o / n 30 14 21. 492 8 02. 739 30 22 24.231 Dauphin Island west base Pins 30 18 23 tt -482.7974 + 0.0587 -482. 7387 sin a A' COS a B 4.205115 9. 967093 8. 511557 2.083765 s 2 sin 2 a C 8. 4102 8.1480 1.1711 see#' J\ 4. 205115 9. 5740212 8. 5093503 0.064116 2. 3526025 It +225. 2177 4X sin £($+•£') 8.7293 +0. 0536 2.352602 9. 702967 +113.65 X i\ X' 225 - 67 ISO 337 00 5.368 2.331 7.699 +0.0050 -h s 2 sin 2 a E 29.7 55.3 34.4 53.7 00.00 40.7 51.034 45. 218 36.252 2.684 7.558 5.917 6.159 +0.0001 STATION GRAND Third angle a a' *' 1st term 2d, 3d, and 1 4th terms / West base to Pins Grand and Pins West base to Grand Grand to west base 30 30 O I It 30 16 50 -298. 5266 + 0.4435 21. 492 58.083 19 19. 575 Dauphin Island west Grand -298.0831 sin or A' COS a B 4.290942 9.672484 8. 511557 2. 474983 sin 2 a C 8. 5818 9.8914 1. 1711 J\ 4. 290942 9.945687 8.509352 0.063888 2. 809869 +645. 4595 Sin JW+tf') 9. 6443 +0. 4409 2.1 9. 702633 2. 512502 +325. 46 X 4X X' 157 180 297 4.950 2.331 7.281 +0.0019 -h s 2 sin 2 a E 34.4 53.1 41.3 25.5 00.00 15.8 51.034 45.459 36. 493 2.475 8.473 5.917 6.865 +0.0007 186 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION PETIT a Second angle a Acl <*' 1st term 2d and 3d \ terms J -A Grand to west base West base and Petit O 297 + 59 1 58 11 15.8 52.0 Grand to Petit Petit to Grand First angle of triangle 357 + 10 07.8 11.6 180 177 81 00 10 41 00.00 19.4 30.1 30 i 19 6 it 19.575 44.068 Grand X A\ X' h" D 88 25 36.493 23.006 30 o r 30 15 rr 58 12 s cos B C a 5.507 4.0 1 9.9 1 8.5 )5429 »470 U551 Petit sin 2 a C 8.1909 7. 3873 1.1725 88 5.213 2.332 25 13.487 +404.0639 + 0.0041 h 4.09542! 8. 69366t 8.50935* 0.063391 2.606450 6.7507 +0.0006 161841 •02445 7.545 +0.0035 +404.068 s sin a A' sec^' ! A\ sin }(^+ *'.) 1.. 9. A\ 1. 361841 -23.006 -Aa 1.064286 it -11. 59 * STATION HORN Second angle 1st term 2d, 3d, and \ 4tn terms / Grand to Petit Petit and Horn Grand to Horn Horn to Grand 13 19.575 40. 016 30 16 +339. 1338 + 0.4253 +339. 5591 8 sin a A' sec^' cos a B 4. 9. 721214 8. 5115514 2.5303714 JX 4. 297606 9. 929579 8. 509354 0.063471 2.800010 +630. 9719 First angle of triangle Grand Horn sin2« C 8. 5953 9.8592 1.1725 A\ sin §W+0') 9. 6270 +0. 4236 2. 800010 9. 702560 +318. 10 X A\ 357 h 61 58 180 238 5.061 2.332 7.393 +0.0025 -h s" sin 2 a 07.8 37.0 44.8 18.1 00.00 26.7 35.3 07.465 2.530 8.454 5.918 -0.0008 APPLICATION OF LEAST SQUARES TO TEIANGULATION. 187 secondary triangulation — Continued STATION PETIT ct Third angle a Aa a' i West base to Grand Petit and Grand 118 - 39 1 03 06 41.3 38.4 West base to Petit Petit to west base 78 57 5 02.9 13.3 180 258 00 51 00.00 49.6 -.1 30 r 14 21. 492 Dauphin Island west base Petit X 88 14 51. 034 H # - 1 45.985 A\ X' + 10 22. 452 30 12 35.507 88 25 13. 486 +1 1st term 2d, 3d, and \ 4th terms J -A$ q r rf 30 13 2 +105. 573 + 0.410< 5 ! COS a B 1 h ) 4. 229486 9.282513 8. 511557 S2 sin 2 a C 8.4590 9.9837 1.1711 h 2 D 4.047 2.331 -h S 2 Sill 2 a E 2.024 8.443 5.917 2.023556 9. 6138 +0. 4109 6.378 +0.0002 6.384 -0.0002 +105.9841 s sin a A' see ' 4. 229486 9. 991874 8. 509354 0. 063392 AX sin t(+4,') 2. 794106 9. 701905 2. 794106 2. 496011 J\ +622. 4522 -Aa +313. 34 STATION HORN a Third angle a Aa a' 4> 44, 4-' Petit to Grand Horn and Grand Petit to Horn Horn to Petit O 177 - 80 10 38 19.4 48.3 96 31 5 31.1 29.1 180 276 00 26 00.00 02.0 30 + / 12 1 35.507 04.509 Petit Horn X A\ X' 88 + 25 10 13.487 53.978 30 13 40. 016 88 36 07.465 JW+tf') 1st term o i n 30 13 08 -64. 9618 s COS a B h 4.245560 9. 055539 8. 511559 «2 sin 2 a C 8.4912 9.9943 1.1706 h 2 D 3.625 2.331 -h s 2 sin 2 a E 1.813 8.485 5.916 1.812658 9.6561 5.956 6.214 2d, 3d, and 1 4th terms / -Aif, + 0.4533 +0.4530 +0.0001 +0.0002 -64.5085 S sin a A' seo^' +4 4.245560 9. 9971774 8. 5093541 0.063471 AX sin J(«S+^0 2.815563 9. 701830 2. 815563 2. 517393 A\ +653. 9778 -Aa. +329. 15 188 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION PASCAGOULA Second angle a Ac iW+*') 1st term 2d, 3d, and \ 4th terms / -H Grand to Horn Horn and Fascagoula Grand to Pasoagoula Fascagoula to Grand First angle of triangle 19. 575 20.998 40. 573 Grand Fascagoula A\ 58 + 43 180 281 97 sin a A' sec^' A\ 9. 991216 8. 509351 0. 063988 2. 663593 +460.8859 sin J(^+0') 2. 663593 9. 703317 +232. 76 30 20 00 s COS a B h 4.099038 9.299092 8. 511551 s 2 sin 2 a C 8. 1981 9.9824 1.1725 h 2 D 3.820 2.332 -h s 2 sin 2 a E -81. 2232 + 0.2256 1.909681 9.3530 +0.2254 6.152 +0.0001 -80. 9976 44.8 20.8 05.6 52.8 00.00 12.8 59.1 36. 493 40.886 1.910 8.180 5.918 6.008 +0.0001 STATION BELLE Second angle a Aa a.' * A4> 1st term 2d and 3d \ terms / -A4> Fascagoula to Horn Horn and Belle Fascagoula to Belle Belle to Fascagoula 30 30 20 +8. 3451 +0. 3790 +8. 7241 sin a A' sec^' 20 40.573 08.724 31.849 COS a B 8. 509351 0.063977 2. 776179 +597. 2814 First angle of triangle Fascagoula Belle X A\ 4.202907 8.206975 8.511550 0. 921432 sin 2 a C 8. 4059 9.9999 1.1729 A\ sin i($+it>') 9. 5787 +0. 3790 2.776179 9. 703447 2. 479626 +301. 73 19 + 69 180 268 48 1.844 2.332 4.176 33 43 11.9 26.3 38.2 01.7 00.00 36.5 48.4 17. 379 57.281 14.660 +2 APPLICATION OP LEAST SQUAEES TO TBIANGULATION. 189 secondary triangulation — Continued STATION PASCAGOULA Third angle a Act V M+t') 1st term 2d, 3d, and 1 4th terms / Horn to Grand Pascagoula and Grand Horn to Pascagoula Pascagoula to Horn 30 / // 17 10 -420. 5920 + 0.0345 20 40.016 00. 557 40. 573 -420.5575 Sill ct A' sec#' cos a B 4.137501 9.974802 8. 511558 A\ 4. 137501 9. 519828 8. 509351 0.063988 2. 230668 n -170.0858 Horn Pascagoula S2 in 2 < C 8.2750 9. 0396 1. 1709 A\ sin i(4+4') 8.4855 +0.0306 2.230667 9. 702706 1.933373 -85.78 A\ 238 - 38 199 180 19 5.248 2.331 7.579 +0.0038 -h s 2 sin 2 a E 26.7 40.5 46.2 25.8 00.00 12.0 -.1 07.465 50.086 17. 379 2.624 7.315 5.917 5.856 +0. 0001 STATION BELLE Third angle a Act *«+*') 1st term 2d, 3d, and 4th terms Horn to Pascagoula Belle and Pascagoula Horn to Belle Belle to Horn 30 30 20 40.016 51.834 31.850 -1 Horn Belle X A\ 199 - 61 138 180 317 43 o t tr 30 17 06 S cos a B h 4. 232071 9.871302 8.511558 *2 sin 2 a C 8. 4642 9.6505 1.1709 h* D 5.230 2.331 -h s 2 gin 2 a E -412.0321 + 0.1976 2. 614931 9.2856 +0. 1936 7.561 +0.0036 -411. 8345 46.2 45.8 00.4 35.4 00.00 25.0 -.1 07.465 07. 197 14. 662 2.615 8.115 5.917 6.647 +0.0004 S sin a A' A\ 4.232071 9. 825229 8. 509351 0. 063977 2. 630628 ii +427. 1968 A\ sin i(4+$') —Aa 2. 630628 9. 702690 2.333318 +215.44 190 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION CLUB Second angle a A* JW+flS') 1st term 2d and 3d -Aj> Belle to Horn Horn and Club Belle to Club Club to Belle 31. 849 19. 873 30 17 52 +319.8704 + 0.0023 +319.8727 S sin a A' sec^' s cos a B 3.993578 9.999846 8.511550 2.504974 A\ 3.993578 8.424985 8. 509353 0. 063584 0.991500 it -9.8062 First angle of triangle Belle Club sin 2 a C 7.9872 6.8500 1. 1729 A\ sin $(#+.£') -Act 6.0101 +0.0001 0.991500 9. 702856 X JX 0.694356 -4.95 317 +40 358 + 180 178 105 5.010 2.332 7.342 +0.0022 24.9 06.4 31.3 05.0 00.00 36.3 54.1 14.662 09.806 04.856 STATION DEEE Second angle §«+« 1st term 2d and 3d terms -A$ Belle to Club Club and Deer Belle to Deer Deer to Belle + 20 30 o / 30 21 00 -55.4752 + 0.1137 21 31.849 55. 361 27. 210 -55.3615 COS a B 3.949417 9. 283133 8.511550 1. 744100 Sin a A' sec^' A\ 3.949417 9.991850 8.509351 0. 064045 2. 514663 +327. 0872 First angle of triangle Belle Deer S 2 sin 2 a C 7.8989 9.9837 1.1729 X A\ sin $(*+#') 9.0555 +0. 1136 2.514663 9. 703531 2. 218194 u +165.27 358 +102 101 281 41 3.488 5.820 +0.0001 '35 31.3 23.3 54.6 45.3 00.00 09.3 09.8 14. 662 27. 087 APPLICATION OF LEAST SQUARES TO TRIAKGULATION. 191 secondary triangulation — Continued STATION CLUB Third angle At, *«+*') 1st term 2d, 3d, and \ 4th terms J -A$ Horn to Belle Club and Belle Horn to Club Club to Horn 30 30 14 15 40. 016 31. 961 11.977 -1 -92. 1459 + 0.1848 -91.9611 s sin a A' sec $' COS a B 4.061211 9.391708 8.511558 1. 964477 4.061211 9.986395 8. 509353 0. 063584 2. 620543 +417.3909 Horn Club S 2 sin 2 a C 8.1225 9. 9728- 1.1709 A\ sin JW+tf') 9. 2662 +0.1845 2. 620543 9. 702113 +210.21 X A\ 138 180 284 2.331 6.260 +0.0002 43 -h S 2 sin 2 a E 00.4 59.8 00.6 30.2 00.00 30.4 07.465 57.391 04.856 1.964 8.095 5.917 5.976 +0.0001 STATION DEER Third angle a Aa At m+4>') 1st term 2d, 3d, andl 4th terms / -A$ Club to Belle Deer and Belle Club to Deer Deer to Club 30 O / 30 18 20 it -375.3574 + 0.1233 11. 976 15.234 27. 210 -375. 2341 sin a A' COS a B 4. 165751 9. 897138 8.511556 2.574445 A\ 4. 165751 9. 788345 8. 509351 0.064045 2. 527492 Club Deer C 8.3315 9. 5767 1.1713 A\ sin i(*+tf') -Aa 9.0795 +0.1200 2.527492 9. 702957 2. 230449 + 170.00 X A\ h 2 D 178 142 180 322 5.149 2.331 7.480 +0.0030 48 -h a 2 sin 2 a E 36.3 27.1 09.2 50.0 00.00 19.2 -.1 04.856 41. 749 2.574 7.908 5.918 6.400 +0.0003 91865°— 15 13 192 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION SHIP Second angle a Aa 4> i«+tf') 1st term 2d and 3d \ terms / -J Deer to Club Club and Ship Deer to Ship Ship to Deer 30 30 21 15 27.210 17. 189 10. 021 First angle of triangle Deer Ship Sin a A' sec $' 4.088909 8.509353 0.063581 2. 171073 +148. 2767 sin 4(tf+#') 2. 171073 9. 702952 1. 874025 +74.8 X q r rr 30 18 19 ft s COS a B h 4.088909 9.976071 8.511549 s 2 sin 2 ' a C 8. 1778 9.0184 1. 1731 h 2 D +377. 1629 + 0. 0265 2. 576529 8. 3693 +0.0234 +377. 1894 322 + 56 18 180 198 70 5.153 2.332 7.485 +0.0031 47 19.1 25.3 44.4 14.8 00.00 29.6 30.8 41. 749 28.277 10. 026 STATION BILOXI LIGHTHOUSE Second angle a Aa 4 A $ $«+*') 1st term 2d, 3d, and 4th terms Deer to Ship Ship and Biloxi Lighthouse Deer to Biloxi Lighthouse Biloxi Lighthouse to Deer + 21 30 30 22 -132.3198 + 0.1107 27. 210 12.209 39. 419 First angle of triangle Deer Biloxi Lighthouse -132. 2091 sin a A' sec^' cos a B 3.978386 9. 631690 8.511549 2. 121625 s 2 sin 2 a C 7. 9568 9.9120 1. 1731 3.978386 9.956007 8. 509350 0. 064209 2. 507952 +322. 0713 A\ sin «*+#') 9. 0419 +0.1102 2.507952 9. 703868 2. 211820 +162. 86 X A\ 18 + 96 115 180 295 48 44.4 37.5 21.9 42.9 00.00 39.0 18.7 41. 749 22.071 4.243 2.332 6.575 +0.0004 -h 2 sin 2 a E 03.820 2.121 7.869 5.919 5.909 +0. 0001 APPLICATION OF LEAST SQUARES TO TBIANGULATION. secondary triangulation — Continued STATION SHIP 193 Third angle J<6 }«+*') v 1st term 2d term Club to Deer Ship and Deer Club to Ship Ship to Club 15 30 9 t It 30 15 11 +1. 7054 +0. 2496 11.976 01. 955 10.021 +1.9550 s sin a A' sec $' cos a B 4.112963 7. 607314 8. 511556 0.231833 J\ 4. 112963 8.5093531 0. 0635812 2.6858937 II +485. 1697 Club Ship 8 2 sin 2 a C 8.2260 0.0000 1. 1713 sin i(0+tf') 9. 3973 +0.2496 2. 685894 9. 702275 2. 388169 n +244. 44 142 - 52 89 04.3 04.9 04.4 00.00 00.5 -.1 04.856 05. 170 10.026 STATION BILOXI LIGHTHOUSE Third angle a iCL a' 1st term 2d, 3d, and 4th terms Ship to Deer Biloxi Lighthouse and Deer Ship to Biloxi Lighthouse Biloxi Lighthouse to Ship 30 + 15 Fixed value O I II 30 19 25 -509.4353 + 0.0376 10. 021 39. 419 39.419 Ship Biloxi Lighthouse -509.3977 sin a A' see#' cos a B 4. 213743 9. 981790 8.511556 2. 707089 sin 2 < C 8. 4275 8.9056 1. 1713 J\ 4. 213743 9. 452733 8. 509350 0. 064209 2.240035 ii +173.7941 ■A sin i(*+tf') 8.5044 +0.0319 2.240035 9.703190 1. 943225 +87. 74 X h 2 D 198 - 35 163 180 343 5.414 2.331 7.745 +0. 0056 54 -h 2 sin 2 a E 29.6 04.1 25.5 27.7 00.00 57.8 -.1 10.026 53.794 03.820 03. 820 2.707 7.333 5.918 5.958 +0. 0001 194 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Final position computation, STATION SHIP ISLAND LIGHTHOUSE Second angle J4> 1st term 2d, 3d, and 4th terms -A Biloxi Lighthouse to Ship Ship and Ship Island Lighthouse Biloxi Lighthouse to Ship Island Lighthouse Ship Island Lighthouse to Biloxi Lighthouse First angle of triangle 30 30 18 12 +654. 0109 + 0.0673 39.419 54. 078 45.341 Biloxi Lighthouse Ship Island Light- house +654.0782 s sin a A' sec#' cos a B 4. 9.980040 8.511546 2.815585 8« sin 2 a C 8.6480 8. 9436 1. 1738 J\ 4. 9. 471798 8. 509354 0. 063404 2. 368555 +233. 6442 sin«tf+tf') 8.7654 +0.0583 2.368555 9. 702930 343 + 33 17 180 197 50 44 5.631 2.332 7.963 +0.0092 -h sSsin'a E 57.7 19.9 17.6 67.9 00.00 19.7 31.4 53.644 2.071485 rr +117. 89 2.816 7.592 5.919 6.327 -0.0002 APPLICATION OP LEAST SQUARES TO TRIANGULATION. 195 secondary triangulation — Continued STATION SHIP ISLAND LIGHTHOUSE Third angle a ia. *(*+*') 1st term 2d, 3d, and 4th terms -J 1 + 1800dX 1 -2.2 u 2 = v 3 -v 2 =- 1997^! + 236^ + 7.0 In a similar way, u 3 = - 2732<50, + 85^! - 1 .2 u t = -5565^-2445^ + 3.5 it 5 = + 5886&fc + 5413^ + 18.4 These contain no s's and the normal equations may be formed in the usual way. Observation equations of this kind would arise when at an unknown point angles are taken on known points, as for example when angles are taken with a sextant from a point off-shore to determine its posi- tion, and for such observations the angle method is both easier and more logical than the direction method. * Wright and Hayford, Adjustment of observations, p. 180. APPLICATION OF LEAST SQUARES TO TRIANGULATION. 197 ■Stack Bosley Pollt/wog Pack-saddle ADJUSTMENT OF VERTICAL OBSERVATIONS GENERAL STATEMENT When reciprocal vertical observations are made over the lines of a triangulation scheme a computation of the differences of elevations is made by the usual Coast and Geodetic Survey formula. For an account of these observa- tions and of the method of computation, see United States Coast and Geodetic Survey Special Publication No. 19, page 140 et seq. As there are always several lines from each station, rigid conditions are present in the figure. Thus it becomes necessary to make an adjustment of the observed values by the method of least squares. In the following fig- ure the differences of eleva- tions as observed are first com- puted and then the results are adjusted by the method em- ployed in the United States Coast and Geodetic Survey. The formula used in the fol- lowing computations is the one given in Special Publication No. 19, mentioned above. On pages 205 et seq. there is given a new development of the for- mula that takes into account some of the small terms that are needed in computation over longer and higher lines. The final form of the new formula differs slightly from the one used in this computation. High Divide Long Ridge Gordon Rattle Red Mountain Redding Rock water level Fig. 8. 198 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Computation of elevations from reciprocal observations Station 1 Pollywog Pollywog Pack Sad- dle High Di- Pack Sad- dle Elk Elk Elk Station 2 Pack Sad- Eft Long Bidge High Di- dle vide vide t n 1 // 1 a 1 it e 1 11 a 1 II Ci 90 04 13 90 57 44 90 30 00 91 53 50 88 33 16 89 15 25 Cs 90 04 59 89 11 15 89 37 33 88 10 59 91 36 54 90 51 04 Ca-Ci + 46" - 1 46 29 - 52 27 - 3 42 51 + 3 03 38 +1 35 39 i(Ca-CO + 23 - S3 14 26 14 - 1 51 26 + 1 31 49 4- 47 50 tan «C«-Ci) 6.04732 8. 18994 7.88258 8.51090 8.42675 8. 14348 log* 4.27444 4.29253 4. 15543 3.98141 4.31524 4.17150 log s tan J(Ca-Ci) 0.32176 2.48247 2.03801 2. 49234 3.74199 2.31498 8 tan i(Ca-Ci) +2.10 -303. 72 -109.15 -310. 70 +552. 06 +206.53 Second term 0.00 0.00 0.00 0.00 0.00 0.00 Third term 0.00 0.03 0.01 0.03 0.06 0.02 I12— hi +2.10 -303.75 -109. 16 -310. 73 +552. 12 +206.55 2 log S 8.549 8.585 8.311 7.963 8.630 8.343 logp=9— 2 logs p of hj— hi 0.451 0.415 0.689 1.037 0.370 0.657 2.82 2.60 4.89 10.89 2.34 4.54 Station 1 High Di- vide Bald Hill High Di- vide Gordon High Di- vide LongEidge Long Bidge LongEidge Gordon Station 2 Gordon Bald Hill Bald Hill Ci a I 11 89 29 43 1 a 88 28 23 88 12 34 89 14 01 1 11 91 27 22 1 11 92 45 09 Cs 89 37 50 91 40 22 91 53 03 90 52 54 88 41 46 87 21 10 Ca-Ci - 51 53 + 3 11 59 + 3 40 31 + 1 38 53 - 2 45 36 -5 23 59 tan iKa-Ci) - 25 56 + 1 36 00 + 1 50 16 + 49 26 - 1 22 48 -2 42 00 7.87759 8.44611 8.50632 8. 15777 8.38184 8.67357 log* 4.21546 4.29195 4.03443 4. 14252 4.29013 4. 15348 logs tan l(Ca-Ci) 2.09305 2. 73806 2.54075 2.30029 2.67197 2.82705 stani(c 2 -Ci) -123.89 +547. 09 +347.34 +199. 66 -469.86 -671.51 Second term 0.00 0.00 0.00 0.00 0.00 0.00 Third term 0.01 0.08 0.05 0.03 0.06 0.07 ha— hi -123. 90 +547. 17 +347.39 +199.69 -469.92 -671.58 2 logs 8.431 8.584 8.069 8.285 8.580 8.307 logp =9—2 logs j> of hs— hi 0.569 0.416 0.931 0.715 0.420 0.693 3.71 2.61 8.53 5.19 2.63 4.93 Station 1 Gordon Gordon Gordon Child Child Battle Station 2 Bed Moun- tain Battle Child Battle Bed Moun- tain Bed Moun- tain 1 a 1 a a 1 II t 11 1 a 1 11 Ci 90 03 28 90 31 05 91 56 00 87 56 00 88 33 47 89 07 26 Ca 90 10 44 89 38 48 88 11 27 92 09 03 91 36 13 90 57 56 ' C1-C1 + 07 16 - 52 17 - 3 44 33 + 4 13 03 + 3 02 26 + 1 50 30 rani(Cs-Ci) + 03 38 - 26 08.5 - 1 52 16.5 + 2 06 31.5 + 1 31 13 + 55 15 7.02404 7.88106 8.51416 8.56610 8.42390 8.20610 logs 4.48786 4.31274 4.23223 4.03839 4.34662 4.06606 logs tan i(Cs-Ci) s ran J(Ca-Ci) 1.51190 2. 19380 2.74639 2.60449 2.77052 2.27216 +32.50 -156.24 -557.69 +402.24 +589.55 +187. 14 Second term 0.00 0.00 0.00 0.00 0.00 0.00 Third term 0.O1 0.03 0.09 0.06 0.09 0.04 ha— hi +32.51 -156.27 -557.78 +402.30 +589.64 +187.18 2 logs 8.976 8.625 8.464 8.077 8.693 8.132 log v= 9— 2 logs p of ha— hi 0.024 0.375 0.536 0.923 0.307 0.868 1.06 2.37 3.43 8.38 2.03 7.38 APPLICATION OF LEAST SQUARES TO TRIANGULATION. Computation of elevations from nonreciprocal observations 199 Station Occ. 1. Pollywog Pollywog Pollywog Elk Pack Sad- dle Bosley Long Ridge Station Obs. 2. Bosley Stack Craggy Bosley Pack Sad- dle Obj. sighted Off/ O f ft O 1 II O 1 II O 1 II O f ff C 89 11 00 89 40 00 88 04 00 88 40 30 89 37 38 91 01 00 90--C 49 00 20 00 1 56 00 1 19 30 22 22 -1 01 00 90°— c in sees. 2940 1200 6960 4770 1342 3660 log ditto 3.46835 3.07918 3.84261 3.67852 3. 12775 3.56348 T 4.68560 4.68558 4.68574 4.68565 4.68558 4. 68562 log* 4. 17163 4.49539 4.20117 4.33291 4. 42745 4. 15591 log s cot C 2.32558 2.26015 2.72952 2.69708 2.24078 2.40501 a and mean 95 42.2 141 42.3 175.9 42.3 157.0 42.1 137.8 42.1 151.8 42.0 log (0.5-m) 9.64167 9.64167 9.64167 9.63990 9.66260 9.60001 2 logs 8.34326 8.99078 8. 40234 8.66582 8.85490 8. 31182 log (0.5— m) s 2 7.98493 8.63245 8.04401 8.30572 8.51750 7. 97183 logp 6.80535 6.80439 6.80376 6.80399 6.80446 6. 80409 log (2d term) 1.17958 1.82806 1.24025 1.50173 1. 71304 1. 16774 scot C +211.63 +182.03 +536.44 +497.83 +174.09 -254. 10 Second term 15.12 67.31 17.39 31.75 51.65 14.71 Third term 0.01 0.01 0.04 0.04 0.01 0.01 t-o 1.51 X51 1.51 1.45 1.50 1.40 I hj-hi +228.27 +250.86 +555.38 +531.07 +227.25 -237.98 log p= 9— 2 logs 0.657 0.009 0.598 0.334 0.145 0.688 P 4.54 1.02 3.96 2.16 1.40 4.88 Station Occ. 1. Bald Hill Battle Red Moun- tain Station Obs. 2. Red Moun- tain Redding Rock Redding Rock Obj. sighted Water level Water level C O I It 88 41 22.5 O / It 91 51 30 Oil! 92 32 45 90°-C + 1 18 37.5 - 1 51 30 - 2 32 45 tan 90°— { 8.35936 8.51115 8.64799 logs 4.44942 4.56765 4.48393 log s cot c 2.80878 3.07880 3. 13192 a and mean $ 21.4 41.6 31.3 41.5 48.0 41.4 log (0.5-m) 2 logs 9.62941 9.64207 9.62747 8.89884 9.13530 8.96786 log (0.5— m) s 2 8.52825 8. 77737 8.59533 log? 6.80387 6.80410 6.80466 log (2d term) 1. 72438 1.97327 1.79067 scot C +643. 84 -1198.95 -1354.94 Second term + 53.01 + 94.03 + 61.75 Third term 0.06 0.21 0.27 t-o 1.36 1.45 1.43 hs— hi +698.27 -1103.26 —1291. 49 Cor. for reduc- + 0.15 + 0.82 tion to mean sea level hs— hi (corrected) -1103.11 -1290.67 log p=9— 2 log s 0.101 9.865 0.032 P 1.26 0.73 1.07 The adjustment of vertical observations as practiced in the United States Coast and Geodetic Survey is made by means of observation equations and differs somewhat from the method of conditions. Of course condition equations could be employed if it were desired, just as triangulation can be adjusted by observation equations. (See the adjustment by the Variation of Geographic Coordinates, p. 91 et seq.) Elevations for the various stations are assumed somewhat near what the final values will be. To these are added x's to be deter- mined by the adjustment. (See table of assumed elevations on p. 200.) By means of these, observation equations are formed by the com- parison of the assumed \ — \ with that determined by computation. 200 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. The method of formation is shown below and a tabulated form of all of the computation on page 201. Fixed elevations. Meters Boaley 1037. 35 Stack 1062. 69 Craggy 1368.31 Bedding Rock 1 0Q Mean sea level / Assumed and adjusted elevations Station Elevation Assumed + correc- Adjusted tion Meters Meters Pollywog 8II+I1 811.06 Elk 507+ij 504. 61 Pack Saddle 817+13 815.74 High Divide 710+n 708. 77 Long Ridge Bald Hill 1059+ls 1055. 96 589+la 585. 16 Gordon 1259+17 1256. 12 Child 701+ls 698. 19 Rattle 1103+lj 1100.46 Red Mountain 1290+lio 1287. 70 FORMATION OF OBSERVATION EQUATIONS The observation equations are formed as follows : (1) Pollywog, assumed elevation* = 811 +#1 (2) Craggy, fixed elevation \ — \ (assumed)* \ — \ (observed)* Observed — assumed* = 1368.31 = +557.31-^ = + 555.38 + ^ = -1.93 + x 1 +'y, = -v t = - 1.93+2! f = l of 3.96 = 1.32 (1) Elk, assumed elevation* = 507 + cc 2 (2) Pollywog, assumed elevation* = 811 + x t \ — \ (assumed)* \ — \ (observed)* Observed — assumed* = +304, + x 1 -x 2 = + 303.75 +v 5 = -0.25-x x + a; 2 +v 6 = — ?> 5 — 0.25 — x 1 + x 2 j) = 2.60 In a similar manner the remaining equations are formed. These are usually formed as in the following table. The constant term is found in the column "Observed minus assumed," and the remainder of the equation in the column "Symbol." ♦Including symbolio correction. APPLICATION OF LEAST SQUARES TO TBIANGULATION. 201 g. 3 » OtQV <19 a> •OS fl §S°? I, o>3 *a> 11 li -."rsooOT-Hrooi--(NOcit--I*#i- +++++ I + I I + I I I ++ I I I I II ++ I I + + NH(0O-*OOOBN0JCqC>lrtOO01OOHH00mnNt0l> «3 C* C* »0 CO CO N Cfl i-l CO »0 C1 1-4 ■* CO »0 (N « t- lftU)rt'*HH +++++ i + i i + i i i ++ i i i 7 i i ++7 i ++ HOHOWOOWJ^MNHMn-SiNlOOOr ««OOONt- 92 95 If 9S 93 So SBtt- HPn a " ?? ■g.S 2-S2S>2W'B 3 M 3 P P « - J _« .£ ~> J2 £ ~» £3= +1.92+Z, -i> 4 = +0.72+^ — d 6 = — 0.25 — x 1 +x 2 and so on for the rest of the 27 v'a. The function u to be made a minimum is 2p n v n 2 , or M=+1.32(-1.93+a; 1 ) 2 +0.34(-0.83+a; l ) 2 +l-5l'(+1.92+a; 1 ) 2 +0.72(+0.72+2; 2 ) 2 +2.60(-0.25-z 1 +z 2 ) 2 +10.89(-0.73-z 2 +z 3 ) 2 +0.47(+6.90+a;3) 2 +2.82( : f3.90 -x 1 +x 3 ) 2 +4.U(-3.55-x 2 +x i y+4.89(+2.lG-x i +x i ) 2 +8.53(+l.ei-x 4 +x 5 ) 2 +2.34(-0.12-a; 2 +2; 5 ) 2 +1.63(+4.02-a;3+a; s ) 2 +3.71(+2.90-a; 4 +a; 6 ) 2 +2.63(-0.08 -x i +x 6 f+4.93(-l.58-x 6 +x 7 ) 2 +2.61(+1.83-x i +x 7 ) 2 +5.19(+0.Zl-x s +x 7 f +0.36(-0.67+z IO ) 2 +0.42(+2.73-z 6 +Zio) 2 +1.06(-1.51-a; 7 +:K 10 ) 2 +3.43 (-0.22-a7+2; s ) 2 +2.03(+0.64+a; 8 -a; 10 ) 2 +0.24(-0.11+a! 9 ) 2 +8.38(-0.30-2; 8 +a; 9 ) 2 +2.37(+0.27-^+a; 9 ) 2 +7.38(+0.18+z 9 -z 10 ) 2 . The function will be rendered a minimum by equating to zero the partial differential coefficients with respect to x t , x 2 , etc. By this means the following equations are derived: +1.32(-1.93+z 1 )+0.34(-0.83+a; 1 )+1.51(+1.92+z 1 )-2.60(-0.25-z,+^)-2.82 (+3.90-a; 1 +^)=0 +0.72(+0.72+a; 2 )+2.60(-0.25-a; 1 +^)-10.89(-0.73-a! 2 +a3)-4.54(-3.55-^+»4) -2.34(-0.12-z 2 +a; 6 )=0 * This vertical net is not of a high degree of aocuracy, it being a small spur of secondary triangulation that was executed in some haste with slight attention to vertical observations. It was selected on account of its small size. The more accurate work is usually in larger nets. See list of probable errors ranging from ±0.23 m. to ±1.83 m. in United States Coast and Geodetic Survey Special Publication No. 13. APPLICATION OF LEAST SQUARES TO TBIANGULATION. 203 +10.89(-0.73-a; 2 +r (; 3)+0.47(+6.90+a; 3 )+2.82(+3.90-a: 1 + : K3)-4.89(+2.16-^+a; 4 ) -1.63(+4.02-Sj+a; s )=0 +4.54(-3.55-a; 2 +a; 4 )+4.89(+2.16-a;3+2; 4 )-8.53(+1.61-a; 4 +a; 6 )-3.71(+2.90 -Xt+xJ -2.61(+1.83 -x i +x 7 )=0 +8.53(+1.61-a; 4 +a; 5 )+2.34(-0.12-a; 2 +» 5 )+1.63(+4.02-a; 3 +a; 6 )-2.63(-0.08 -Xs+Xs) -5.19(+0.31 -x 5 +x 7 )=0 +3.71(+2.90-a; 4 +a; 6 )+2.63(-0.08-a; 5 +a; 6 )-4.93(-1.58-a; 6 +x 7 )-0.42(-|-2.73 -x a +x 10 )=0 +4.93( -1.58 -Ze+z,) +2.61(+1.83 -z 4 +a; 7 )+5.19(+0.31 -x 6 +Xy) -1.06(-1.51 -z 7 +z 10 )-3.43(-0.22-^+z 8 )-2.37(+0.27-a; 7 +a; 9 )=0 +3.43(-0.22-av+a; 8 )+2.03(+0.64+^-a; 10 )-8.38(-0.30-2; 8 +x 9 )=0 +0.24(-0.11+a; 9 )+8.38(-0.30-a; 8 +a; 9 )+2.37(+0.27-^+2; 9 )+7.38(-|-0.18+a! 9 -z 10 )=0 +0.36(-0.67+a; 10 )-l-0.42(+2.37-a; 6 -l-a; 10 )-|-1.06(-1.51-av+a: 10 )-2.03(- r -0.64 +Z8-Zio)-7.38(+0.18+a; 9 -a;i )=0 By multiplying and collecting, we obtain the following normals: +8.59ii- 2.60zj- 2.82*8 -10.2786=0 — 2.60xi+21.O9ls— 10.8913— 4.54X4— 2.34xs , +24.2159=0 -2.82li-10.89l 2 +20.70t3- 4.89X4- 1.63ai -10.8237=0 — 4.54X2— 4.89X8+24.28X4— 8.53X5— 3.71X5— 2.61x 7 —34.8232=0 — 2.34xj- 1.63X3- 8.53X4+20.32X5— 2.63la- 5.19x 7 +18.6066=0 - 3.71x4- 2.63X8+11.69X1- 4.93x 7 - 0.42ii +17.1914=0 - 2.61X4- 5.19X5- 4.93X6+19.59X7- 3.43x 8 - 2.37i 9 - 1.06xio + 0.3111=0 — 3.43X7+13.84X8— 8.38X9— 2.03xio + 3.0586=0 - 2.37X7- 8.38X8+18.37X8- 7.38xio - 0.5721=0 - 0.42X8- 1.06X7- 2.03x8- 7.38xs+11.25xio — 3.3228=0 (See the table of normals on p. 204.) The normals are most conveniently formed from the table given on page 204. The various observation equations are written along the horizontal lines in the columns of their respective x's. The nor- mals are then formed as in condition equations, except that the constant terms must also be multiplied by each column and the sums taken for the constant terms in the normals, as may be seen from the direct computation of the normals above. After the x's are determined from the solution of the normals, they are added to the assumed elevations, giving the adjusted final elevations. The v's are most easily determined by computing Ti 2 — \ from the adjusted values; if the observed h 2 — \ is subtracted from the adjusted value the respective v results. They could, of course, be computed by substituting the x's in the observation equations, but this would require more work. For a check the I pv at any station should equal zero, with the possible exception of a small amount due to dropping the decimals on the x's. In the table on page 201, use pv from the first column if the x is positive and from the second column if the x is negative. 204 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. Table for formation of normal equations V N 1 2 3 4 5 6 7 8 9 10 ■pN x's 1 1.32 -1.93 +1 - 2. sin 1 +0.0595 2 0.34 -0.83 +1 - 0.2822 2 -2.3925 3 1.51 +1.92 +1 + 2.8992 3 -1.2577 4 0.72 .+0.72 +1 + 0.5184 4 -1.2304 fl 2.60 -0.25 -1 + 1 - 0.6500 5 -3.0402 H 10.89 -0.73 -1 +1 - 7.9497 fi -3.8403 7 0.47 +6.90 +1 + 3. 2430 7 -2. 8757 8 2.82 +3.90 -1 +1 +10.9980 8 -2.8107 9 4.54 -3.55 -1 + 1 -16.1170 9 -2.5441 10 4.89 +2.16 -1 +1 +10.5624 10 -2.2951 11 12 8.53 2.34 +1.61 -0.12 -1 -1 +1 +1 +13. 7333 - 0. 2808 13 1.63 +4.02 -1 +1 + 6.5526 14 3.71 +2.90 -1 +1 +10. 7590 15 2.63 -0.08 -1 +1 - 0.2104 l(i 4.93 -1.58 -1 +1 - 7. 7894 17 2.61 +1.83 -1 +1 + 4. 7763 18 5.19 +0.31 -1 +1 + 1.6089 19 0.36 -0.67 +1 - 0.2412 20 0.42 +2.73 -1 +1- + 1. 1466 21 1.06 -1.51 -1 +1 - 1.6006 22 3.43 -0.22 -1 +1 - 0. 7546 23 2.03 +0.64 +1 -1 + 1.2992 24 0.24 -0.11 +1 - 0.0264 25 8.38 -0.30 -1 +1. - 2.5140 26 2.37 +0.27 -1 +1 + 0.6399 27 7.38 +0.18 +1 -1 + 1.3284 Normal equations 1 2 3 4 5 6 7 8 9 10 1 2 1 +8.59 - 2.60 - 2.82 -10.2786 - 7.1086 2 +21.09 -10. 89 - 4.54 - 2.34 +24. 2159 +24.9359 3 +20.70 - 4.89 - 1.63 -10.8237 -10.3537 4 +24.28 - 8.53 - 3.71 - 2.61 -34.8232 -34. 8232 5 +20.32 - 2.63 - 5.19 +18.6066 +18.6066 6 +11.69 - 4.93 -0.42 +17.1914 +17.1914 7 +19.59 - 3.43 - 2.37 - 1.06 + 0.3111 + 0.3111 8 +13.84 - 8.38 - 2.03 + 3. 0586 + 3. 0586 9 +18.37 - 7.38 - 0.5721 - 0.3321 10 +11.25 - 3.3228 - 2.9628 Solution of normal equations 1 2 3 4 5 6 7 1 I +8.59 II 1 - 2.60 + 0.30268 — 2 82 + o! 32829 -10.2786 + 1.19658 - 7.1086 + 0.82754 +21.09 - 0.7870 +20.3030 1 2 -10.89 - 0.8536 -11.7436 + 0.57842 - 4.54 - 4.54 + 0.22361 - 2.34 » - 2.34 + 0. 11525 +24.2159 - 3.1111 +21. 1048 - 1.03949 +24.9359 - 2.1516 +22.7842 — 1.12221 +20.70 - 0.9258 - 6.7927 +12.9815 a* 2 3 - 4.89 - 2.6260 - 7.5160 + 0.57898 - 1.63 - 1.3535 - 2.9835 + 0.22983 -10. 8237 - 3.3744 +12.2074 - 1.9907 + 0. 15335 -10.3537 - 2.3337 +13.1788 + 0.4913 - 0.03785 +24. 28 - 1.0152 - 4.3516 +18.9132 - 8.53 - 0.5232 - 1.7274 -10. 7806 + 0.570004 -3.71 -3.71 +0. 196159 -2.61 -2.61 +0. 137999 -34. 8232 + 4.7192 - 1.1526 -31.2566 + 1.652634 -34.8232 + 5.0948 + 0.2845 -29. 4440 + 1.556796 APPLICATION OE LEAST SQUARES TO TRIANGULATION. 205 Solution of normal equations — Continued 9 8 10 7 6 5 1 3 +18.37 9 - 8.38 + 0.45618 - 7.38 + 0. 40174 2 37 + o! 129015 - 0.5721 + 0.03114 - 0.3321 + 0.01808 +13.84 - 3.8228 +10.0172 Xs 9 8 * - 2.03 - 3.3666 - 5.3966 + 0.53873 - 3.43 - 1.0811 - 4.5111 + 0.45034 + 3.0586 - 0.2610 + 2.7 16 — 0.27928 + 3.0586 - 0.1515 + 2.9071 - 0.29021 +11.25 - 2.9648 - 2.9073 + 5.3779 Xu> 4 9 8 10 - 1.86 - 0.9521 - 2.4303 - 4.4424 + 0.82605 - 0.42 - 0.42 + 0.07S10 - 3. 3228 - 0.2298 + 1.5072 - 2.0454 + 0.38033 - 2.9628 - 0. 1334 + 1.5661 - 1.5299 + 0.2S448 +19.59 - 0. 3602 - 0. 3058 - 2. 0315 - 3.6696 +13.2229 Xl 4 10 7 - 4.93 - 0.5120 - 0.3469 - 5.7889 + 0. 43779 - 5.19 - 1.4877 - 6.6777 + 0.50501 + 0. 3111 - 4.3134 - 0.0738 + 1.2599 - 1.6896 - 4.5058 + 0.34076 + 0.3111 - 4.0632 - 0.0428 + 1.3092 - 1.2038 - 3.7495 + 0. 28356 +11.69 - 0. 7277 - 0. 0328 - 2.5343 + 8.3952 Xi 2 3 4 7 6 - 2.63 - 2. 1148 - 2.9234 - 7.6681 + 0. 91339 +17.1914 - 6. 1313 - 0. 1597 - 1.9726 + S.9279 - 1.06345 +17. 1914 - 5. 7757 - 0. 1195 - 1.6415 + 9.6550 - 1.15006 +20.32 - 0.2697 - 0. 6857 - 6. 1450 - 3. 3723 - 7.0040 + 2.8433 Is +18.6066 + 2.4323 - 0.4575 -17. 8164 - 2.2755 + 8.1547 + 8.B442 - 3.04020 +18.6066 + 2.6259 + 0. 1129 -16. 7832 - 1.8935 + 8.8188 +11.4875 - 4.04020 Bach solution 5 6 7 10 8 9 4 3 2 1 -3.0402 -1.0634 -2.7769 +0.3408 -1.5353 -1.6812 +0.3803 -0.2999 -2.3755 -0.2793 -1.2950 -1.2364 +0.0311 —0.3710 -0. 9220 -1.2822 +1.6526 — 1. 7329 -0. 7533 -0.396S +0.1534 -0.6987 -0. 7124 -1.0395 -0.3504 -0.2751 -0. 7275 +1.1966 -0. 4129 -0. 7242 -3.0402 -3. 8403 -2.8757 -2.2951 -2.8107 -1.2677 -2.5441 -1.2304 -2.3925 DEVELOPMENT OF FORMULAS FOR TRIGONOMETRIC LEVELING GENERAL STATEMENT The formulas used on pages 198 and 199 in the computation of ver- tical observations were found to be lacking in some of the quantities that were appreciable when the lines were very long and high. Accord- ingly, a' new derivation is now given that takes into account some of these quantities. As a result, the formulas derived in this develop- ment differ slightly from those used in the computation cited above, but they ought to give practically the same result in computing over lines of such length as occur therein. 206 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. The following derivation of the formulas for trigonometric leveling is based on certain approximate assumptions which fall under four general heads: 1. Geometric approximations.— The verticals at the two points (P t the point occupied and P 2 the point sighted on) are treated as if they lay in one plane and the intersection of this plane with the ellipsoid that represents the surface of the earth is treated as the arc of a circle whose radius is the mean radius of curvature of a vertical section through P 1 and P 2 . Helmert (in his Hohere Geodasie, Vol. I, p. 520, and Vol. II, p. 563) investigates the error arising from these assumptions and finds it to be about 1/40 meter at a maximum when the distance P i P 2 is about 100 kilometers. 2. Geodetic approximations. — The difference between the geodetic zenith and the astronomic zenith, i. e., the deflection of the plumb line, is ignored. If these deflections are known, corrections may be applied to the measured zenith distances (which, of course^ are referred to the astronomical zenith) to reduce them to the geodetic zenith. Furthermore, the elevations obtained by trigonometric leveling between two points are referred to an assumed ellipsoid, while spirit leveling gives elevations referred to the geoid, so that the distances between geoid and ellipsoid must be known to make the two kinds of leveling comparable. If trigonometric leveling could be carried out with great precision, its use in connection with spirit leveling would give just this information as to the distance of the ellipsoid from the geoid. The change in the distance from geoid to ellipsoid occurring between P t and P 2 may be found from the deflections of the vertical at those points, provided it is assumed that the deflections vary uniformly between P t and P 2 , an assumption which may be con- siderably in error. 3. Optical approximations. — The path of the ray of light between P x and P 2 is assumed to be the arc of a circle in a vertical plane through P x and P 2 . The angle between the chord P t P 2 and the tangent to the circle at either point is the refraction in zenith distance and it is evidently implied that this refraction is equal at P x and P 2 . If we call (see figure 9) the center of the circle referred to in approxi- mation 1, and call the angle PfiP 2 = 8, the refraction in zenith distance of the angle TPJP 2 ( = Z TPJ 3 ^ is written as md and m is termed the coefficient of refraction. The course of a ray of light through the atmosphere depends on the variations in pressure, temperature and humidity of the medium through which it passes and may be far from circular. Our lack of knowledge of the conditions which govern the refraction is the greatest obstacle to precision in trigonometric leveling. APPLICATION OF LEAST SQUARES TO TRIANGULATIOX. 207 4. Algebraic approximations. — After the approximations mentioned above have been made, there is the further approximation arising from the dropping of small terms after an expansion in series. In the following developments it will be seen that only extremely small terms are dropped, and that in cases arising in practice their effect even on the sixth place of logarithms is unimportant, while in fact logarithms of only five places are commonly used for this sort of com- putation , The accuracy of the developments is confirmed by the numerical agreement between the approximate and the exact for- mulas ill the examples given. {Exact is used in the sense of dispensing with the use of series. The formula is inexact, owing to the first three sets of approximations.) The examples represent rather extreme cases of those arising in practice, and other numerical examples of extreme cases give a similar agreement. DEVELOPMENT OF THE FORMULAS Figure 9 represents the vertical plane of approximation 1 common to P 1 and P 2 , being in fact the plane parallel to both verticals (see Helmert, Hohere [Geodasie, Vol. I, p. 519) on which the several points are projected. The measured zenith dis- tances are assumed equal to and Z V 2 P 2 T = £ 2 . The measurements are not made exactly in this plane, but the error, which is part of that involved in approxima- tion 1, is negligible. The refraction in zenith dis- tance is, according to approxi- mation 3, it= ZTP t P 2 = ZTPJP^md. S t and S 2 are points on the earth's surface in the verticals of P x and P 2 , so that the re- spective elevations of P 1 and P 2 above the surface are \=8 1 P 1 and % 2 =8 2 P 2 . The mean radius of curvature p of approximation 1 is given by P = OS 1 = OS 2 . 91865°— 15 14 FiQ. 9. 208 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. If s denotes the distance PJP 2 measured along the arc and if 6 be expressed in radians, s = p0 or if 6 be in seconds, psml"- There are two cases to be considered according as to whether both or only one of the zenith distances have been measured. Case I. Reciprocal zenith distances In the triangle PfiP^ ZPffi = 180° -c t -iC= 180° -c.-md ZAiy? == 180° - G - J£ = 180° - £ 3 - m0 also 0P 1 =p + 7i i and OP 2 = p + Ti 2 . Therefore by the law of sines p + Ji^sm fe + m<9) p+\ sin (G+mf?)' Treating this as a proportion and taking by division, (p + Ji 2 )-(p + h 1 ) _ sin (G+mfl) - sin fc+mff) p + \ sin(£ 3 +m#) or 2 (p+\) sin (kj^ 8 ) cos (^±^ + me) \-\ = — sin (C 2 +m6)~ ( A ) Since the sum of the angles of a triangle is 180°, 180° -G -m0+ 180° -C 2 -m0 + = 180° which gives also ^p + m0 = 9O°+| G +md =^±- C - 1 - +m0 + ^i = 90° + ~ + ^=^ whence (A) becomes 2(p + \) sinfc^) sin f 7i 2 -Ti,= V^_? I f. (1) The quantity 2 (,0 + ^) sin 2 has a simple geometrical interpretation In the figure make OL 2 = OP 1 and draw OM J_ Pj-Z^- Then P^^L.M^OP, sin P^if = Co +A.J sin |- APPLICATION OP LEAST SQUAEES TO TEIANGTJLATION. 209 Then 2(p + h 1 )sin ^ is the chord P X L 2 or the chord S t S 2 increased to allow for the elevation of P t above the earth's surface. In fact, the relation (1) might have been obtained by applying the law of sines directly to the triangle PJPJL 2 , which makes it evident why P X L 2 appears. For convenient computation* (1) may be transformed as follows: By the sine series »-l->[KG) ,+ ■ ■ ■] -*(^)0-£H^X>-£> « The remai nin g factors of the right-hand side of (1) may be written, (Sty tant^e 1 )sec^ 1 — tan » tan = tan(^)( 1+ f)[ 1+ |tan(&^)) <» n The last transformation comes by expanding sec s in powers of 6 and noting that tan s = 5 nearly, and that the product ~ tan ~ is small, so that, 1 ^V^r 1 ^ tan< ^* very neariy " By combining (2) and (3) and using 6 = -, equation (1) becomes A 3 -^ = S (l + |>an(^)[l + Atan(^)][l +I ^] or h 2 - \ = s tan (^Hy^ 1 ) ^#C (4) * See also note 1, p. 219. 210 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. where A = H--i = correction for elevation of station whose elevation is known, .8=1+^- tan (&^A = correction for approximate difference of elevation, s 2 0= 1 +— — = correction for distance. The logarithms of A, B, and C are given in the tables on pages 218 and 2l9 with the respective arguments h lt log s tan ( 2 ) L and log s. The tables show the limiting values of the respective arguments for which logarithms of A, B, and C become 1, 2, 3, etc., units of the fifth place of decimals. Equation (4) may be compared with the expression more commonly given for ~h 2 — \, h-h-sUn(^)[l + ^ + ^] (5) With the tables here given Jjo) will probably be found slightly more convenient for logarithmic computation than ££)■?" The two forms are equally accurate. Case II. Only one zenith distance (d) observed Where two zenith distances are known, the formula, either (4) or (5) , does not involve the coefficient of refraction (m) explicitly. Where only one zenith distance is known, a value of m must be assumed from the best sources of information available. In the triangle P x L 2 P 2 ZP 1 L 2 P 2 = 9Q°+^ZV 1 P 1 L 2 lP 2 P,L = ZV.P, L 2 - L V, P, P 2 = 9O o + 2-(C 1 +^C)=9O°-C+(*-m)0 For the third angle we find, by subtracting the sum of the other angles from 180° LP,P 2 L = ^-{X-m)6. By the law of sines L 2 P 2 = sin PJP^ P^L 2 sin P X P 2 L 2 or fe 2 -^^p A cos g'-;f- TO ^ . (6) 2 * * 2 sin [£j — (1— md)] The chord P x i 2 = chord S 1 S 2 X- '= chord S l S 2 xA, A having the meaning previously given; chord ( i+ t:)[>-£ + ■ ■ -]-*('-£ + (9) (10) 212 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. The factor cos [& — ($— m) 0] in (6) may be written cos [&-(i-m) 0] = cos £ cos [(i-m) d] + sin & sin [(£-m) 0] Since (A— m) is a small quantity, the series forms for its sine and cosine may be used, giving cos[c 1 -Q-m)0] = cosC 1 [l-(i-m) 2 f+ • • -J + sinc 1 [(i-m)(?-(i-m)»^+ • • •] The third factor on the right-hand side of (6), namely, - — f j- r-T^cosec [Ci— (1 — m)0] may be expanded in powers of (1 —m)0 by Taylor's theorem. /(G) = cosec G f'(Zi) = —cot £ t cosec Ci /"(d) = cosec & (1 + 2 cot 2 Ci) f"(Q = -6 cosec 3 Ci cot d +cot d cosec d- This gives, rin[C 1 -(l-mr ^° COBeo Cl + C0Se ° Cl Cot f' (1 - m) " /)2 + cosec Ci (1+2 cot 2 Q{l-m) 2 -^ t 11 ) 3 + cosec d cot d (6 cosec 2 d — 1)(1— m) 3 -^- + • • • The expressions (9), (10), and (11) for the factors on the right-hand side of (6) are now to be multiplied together. In cases that actually occur, and cot d are small quantities of about the same order of magnitude. If we call cot d a quantity of the first order, it is evident that cosec d differs from unity by a quan- tity of the second order. In forming the product from (9), (10), and (11) it is seen that the product is of the second order, and will more- over contain only terms of even order, so that if terms of the fourth order are retained the error will be of the sixth order, or the propor- tional error (the error as compared with the quantity itself) will be of the fourth order or of the order of ^ part of the difference of eleva- tion, if we suppose a quantity of the first order may be as large as «?>> a liberal allowance. The error, then, of the omitted terms should not affect the fifth place of logarithms and probably not the sixth. It will be seen that the expansions (9), (10), and (11) have been carried out sufficiently far for the purpose in hand, and if these expressions APPLICATION OF LEAST SQUARES TO TEIANGULATION. 213 be multiplied together, retaining in the product no terms of higher order than the fourth, the result may be written ; ft 2 -iWp+ft 1 ){flcote 1 [i+ 6(1 ir )2 ~ 1 - p] + ( My [ 1+ s-y>' . ^] +( i- m) *«*■<» (12) Since # = -, we may write where A = 1 + 6(1 T )2 ~ 1 --/ 1 6 ,o 2 n 5 — 10?n + 4m 2 s : (13) (13a) 12 ,o 2 The factor A has been omitted from the last term as being unneces- sary, the latter being small and A near unity. D ± and D 2 are also near unity. Their logarithms are tabulated in the same manner as the other quantities, the tables showing the limiting values of the argument between which log D Y or log D 2 may be taken as 1, 2, 3, etc., units 'of the fifth decimal. It may be noted that in some European surveys the term s 2 cot 2 f (1— m) — is dropped and the formula for difference of eleva- tion written as s 2 Ti 2 -\=s cot Ci + (£— ™)~ (14) P The dropped terms or factors all represent quantities of the fourth (1 -m)s 2 cot 2 G ■ , iT . order m our expansion. The term — is, however, tne largest of such quantities as a rule, and might be noticeable where D 1 and D 2 would not be. Probably for short lines and small differences of elevation the most convenient formula would be s 2 h 2 -Ti 1 = As cot ^+A{\-m)- (15) and for other lines formula (7). 214 COAST AND GEODETIC SUEVEY SPECIAL PUBLICATION NO. 28. EXAMPLES The data for the following examples, which illustrate the use of the formulas,come from The Transcontinental Triangulation, Special Publication No. 4, page 273, et. seq. At Snow Mountain West At Ross Mountain c a Approx. eley. 91 13 39.1 39 22 38 18 56 IS 2146 meters 89 34 04.8 38 30 20 197 49 29 672 meters logs=5.007341. For mean a and 2 ) The values of log A, log B, etc., were taken successively at 0.5, 1.5, 2.5, etc., units of the fifth place, namely, at the point where the value 218 COAST AND GEODETIC SURVEY SPECIAL PUBLICATION NO. 28. of log A, log B, as rounded off to 5 decimals would change by one in the fifth place. The corresponding values of h, log s tan ( ^ „ j and log s were then computed by the formulas above. These values are carried out far enough so that the values of log A, log B, etc., may be obtained by interpolation to six decimals. In the numerical examples here given the values of log A, log B, etc., were computed independently for the actual values of p and m. These results as used in the example all agree within a unit of the sixth decimal place with those found by interpolating in the tables. The unit of length throughout the tables and formulas is the meter. Tables Elevation of occupied station Ai log A units of fifth place Elevation of occupied station hi log A units of fifth place Meters Meters 0.0 3009 20.5 73 0.5 3156 21.5 220 1.5 3303 22.5 367 2.5 3449 23.5 514 3.5 3596 24.5 661 4.5 3743 25.5 807 5.5 3890 26.5 954 6.5 4036 27.5 1101 7.5 4183 28.5 1248 8.5 4330 29.5 1394 9.5 4477 30.5 1541 10.5 4624 31.5 1688 11.5 4770 32.5 1835 12.5 4917 33.5 1982 13.5 5064 34.5 2128 14.5 5211 35.5 2275 15.5 5357 36.5 2422 16.5 5504 37.5 2569 17.5 5651 38.5 2715 18.5 5798 39.5 2362 19.5 5945 40.5 APPLICATION OF LEAST SQTJAKES TO TRIANGULATION. 219 log A is positive except in the rare case when 7^ indicates a depres- sion below mean sea level. A is used for both reciprocal and nonreciprocal observations. For reciprocal observations only (unless formula, p. — , is used) For nonreciprocal observations log approx- imate difference logJ3 logDi log2> 2 elevation units of logs logC logs units of logs units of =logstan 5th place 5th place 5th place (*£). 0.0 0.0 0.0 0.0 2.167 0.5 4.875 0.5 4.407 0.5 4.552 0.5 2.644 1.5 5.113 1.5 4.646 1.5 4.791 1.5 2.866 2.5 5.224 2.5 4.757 2.5 4.902 2.5 3.011 3.5 5.297 3.6 4.830 • 3.5 4.975 3.5 3.121 4.5 5.352 4.5 4.8S4 4.5 5.029 4.5 3.208 5.5 5.305 5.5 4. 928 5.5 5.073 5.5 3.281 6.5 5.432 6.5 4.964 6.5 5.109 6.5 3.343 7.5 5.463 7.5 4.995 7.5 5.140 7.5 3.397 8.5 5.023 8.5 5. 167 8.5 3.445 9.5 5.047. 9.5 5.192 9.5 3.489 10.5 5.068 10.5 5.213 10.5 3.S28 11.5 5.0S8 11.5 5.233 11.5 3.505 12.5 5.106 12.5 5.251 12.5 3.598 13.5 5.123 13.5 3.629 14.5 5.138 14.5 3.658 15.5 5.153 15.5 3.685 16.5 5.167 16,5 3.711 17.5 5.179 17.5 3.735 18.5 5.191 18.5 3.758 19.5 5.203 19.5 3.779 20.5 5.214 20.5 3.800 21.5 5.224 21.5 3.820 22.5 5.234 22.5 3.839 23.5 5.243 23.5 3.857 24.5 5.252 24.5 3.874 25.5 5.261 25.5 * Or log s cot["ci— (0.5— m) — ^-^Ifor nonreciprocal observations. (See note 2, p. 220.) log B has the same sign as the approximate difference of elevation. log C is always positive. log D t and log D 2 are always positive. NOTES ON THE DEVELOPMENTS Note 1.— The transformation of (1), page 208, may be conducted rather more simply than is there given. e sin ; \ — \ = or \ — \- 2<> + & 1 )sin(^) (1) cos sm 2 cos (^)eo 3 f-sin(^>,n 220 COAST AND GEODETIC STJEVEY SPECIAL PUBLICATION NO. 28. Divide numerator and denominator by cos I ■ 2 ~ ) cos ~, 2 (^ + ^)tan|tan(^=^) K—\=- 1 — tan p tan s tan 1 + C 6 ^)' 2p or expanding tan » in series and using #=-, ^-^ = (l+f)] Substitute in (4) \ - \ = s cot [A - (0.5 -m) — ^—y,] ABC 1 S1 ,0 sin 1" J for nonreciprocal observations analogous to l 2 -\ = s tan (^5—) -A-BC for reciprocal observations. B should be taken from table with argument log s cot & - (0.5 -m) — A-7-, j. L , ^sm 1"J This is the present Coast and Geodetic Survey formula for non- reciprocal observations. o