Ix GEODESY. 



a,.! — (i8o° — aj.o), in series proceeding according to powers of the distance s. 

 Formulas of this kind with convenient tables for facilitating the computations 

 are given in the Reports of the U. S. Coast and Geodetic Survey.* 



b. Secondary triangulation. 



For secondary triangulation, wherein the sides are 12 miles (20 kilometres) or 

 less in length, and wherein differences of latitude and longitude are needed to the 

 nearest hundredth of a second only, the following formulas may suffice. Using 

 the same notation as in the preceding section, the formulas are : — 



^2 = ^1 + A<^, 



A, = Ai + AA, (i) 



"2.1 = lSo° -\- a,.; -|- Aa, 



A^ = — aj s cos ai_2 — ^2 ^"^ sin^ aj ,, 



AA = -\- di sec 01 J sin ojo — l>.2 s^ sin qi., cos aj 2, (2) 



Aa = — ^1 tan cf>i s sin aj ^ -j- ^2 ^" sin aj o cos aj ,. 



The constants entering the latter equations are defined by the following 

 expressions, wherein p,„ and p„ are the principal radii of curvature of the spheroid 

 at the point whose latitude is 0, and p" = 206 2 64."8 : 



// 



P_ 



Pm Pn 



a, = '— , />y =c, = ^, 



^ _ p" tan <^i , _ p" sec (^, tan c^i _ p"(i-l-2 tan^c^i) 



"2 ) ^2 o > ^2 o • 



2 Pm Pn Pn' 2 /D„^ 



The logarithms of the factors a^, bi, Ci, a-,, bo, % are given in Table 15 for the 

 English foot as unit, and in Table 16 for the metre as unit, the argument being 

 the initial latitude 4>i for all of them. 



When all of the differences given by (2) are computed, they may be checked 

 by the formula 



sin i(</.2 + </.,) = ^. (3) 



For convenience of reference in numerical applications of the above formulas, 

 (2) may be written thus : 



A0 = ^h + ^2, 

 AA = ^1 + B„ 

 Aa = Ci -)- Cj, 



in which, for example, A^ and A., are the first and second terms respectively of 

 A<^, due regard being paid to the signs of the functions of ajo. 



Ntmicrical example. The following example will serve to illustrate the use of 

 formulas (i) to (3). The value of log s is for s in English feet, J being in this 

 case about 12.3 miles. 



^, 38°54'o8."3S 



A<^ —07' 5o."2i 



</.2 38°46'i8."i7 



i(<^2+<^i) 38°5o'i3."27 



* See Appendix 7, Report of 1884, for latest edition of these tables. 



