170 GEODESIC INVESTIGATIONS. 



This shows that great caution shoi;ld be obseryed in using inexact formulae 

 or expressions, based on no principle or theorem, as if thej were approxima- 

 tions based on scientific principles ; for although the angle ^is as small as 

 above stated, yet if it were neglected (as many important small terms are 

 often neglected) it would be equivalent to regarding the earth as a perfect 

 sphere. 



ProMem 6. 



Given the azimuth A', the latitude I", and the circular measure 2 and 

 length 5 of the geodesic arc between the stations ; to find — the azimuth A", 

 the latitude I', the difierence of longitude co, and the other entities determined 

 in the solution of problem 1. 



To find the difference of longitude w we have 



s. sin 2 sin A' 



sin 03 = — = — ^ jTT' 



2. H^j cos I 



To find the arc S^,D„ or z,^, we have 



5. sin 2 1 



sm z„ = 



^ (i2„2)2— 4s sin2 i 2 (B„ 2-5> j 



Then in the triangle S,,PD„ we have the side PS„ — l„ — 90° — I" ; the 

 side S^,Dj = z^^ ; and the angle P = w. Therefore 



. -T^ sin I,, sin ca .. . , cos i 2 ■ ai 



sm X>, = :/^ ; it IS also = — = sm A 



sm s^, cos {z^, — I 2) 



tan i A" =4i^i^(k=^ . cot (D -co) 



cos ^ {l„ — z„) , ,j. , . 

 =cosia, + .J-^°t(^' + ") 

 from which to find A,^. And to find l' we have 



sin'^ A' a? 



tan- V = \ tan- r 4- — I — 

 V S- / sn 



sin^ A" 62 

 Or we can find it from 



tan -^l=- ^°^t,^f + j: + "V eot i l„ 

 cos i {^A' -\- A — w) 



The other entities can now be /ound as in problem 1. 



ProMem 7. 



Given the latitude V, the difference of longitude co, and the length * and 

 circular measure 2 of the geodesic arc between tlie stations : to find — the 

 azimuths, the latitude I", and all the other entities determined in the solution 

 of problem 1. 



To find z^ or arc S,D„ we have — 



5. sin 2 



I m, 2)2—4 s. sin2 i 2 (22,2—5) j 



-s^V- 



