GEODESIC INYESTIGATIONS. 



155 



^ 

 O 



From the equations (5) we have- 

 sin A' cos ^ 2' sin ^^ 

 sin . '" - -" 



cos i 2" sin cpj, 



But it is evident that $lA z= ^,^^' ^ r^^cosXj; 

 sm <|)^^ i?/ cos I r, cos A 



And since with respect to any pair of mutually visihle stations, we may 



cos i 2' , 1 

 „ _. -j^ . ^Q have — 



cos \ 2" 



sin A! 

 sin A!' 

 sin A' 



R„ cos I" 



a, cos I' 



r„ cos a" 



1 



sin A" r, cos A' 



(10) 



The rigorously true formulae, for any two stations on the earth, being — 



-S. 



cos 



I" 



R, 



cos 



V 



r^i 



cos 



A" 



cos \ 



2" 



cos i 



2' 



cos \ 



2" 



sin ^' 

 sin A!' 

 sin ^' 

 sin ^" r, cos A' ' cos \ 2' 



From these we easily deduce — 



sin^ A! _ (l-e2) tan ^Z' + 1 cos^ ^ 2" 



1 



(11) 



sin^ A!' 



(1— e2) tan3 1" + 1 

 52 



tan" Z" + 



cos- \ 2' 



cos^ ^ 2" 

 cos^ i 2' 



sin* A! 



1 cos2 i 2" 



1 sec- A" — 1 cos'^ 



52 tan2 A' + 1 . cos* \ 2" 



J 



(12) 



(13) 



(13) 



^ tan* A" + 1 cos* i 2' 

 5* 



And if we find m! and m" such, that tan* m' = (1— e*) tan* Z' ; tan* m" ■=■ 

 (1-e*) tan* I" ; then will 



sin A! cos m" cos \ 2" 



sin ^" cos m' 

 If we find an angle % such, that 

 tan* X 



cos i 2' 

 _ (1— e*) tan* Z' + 1 . cos* \ 2" 



(1- 

 then 



e*) tan* I" + 1 cos* 

 sin A' tan % 



2' 



sin A" 

 tan ^ (^' — A!'') 



tan (x — 45°) 



tan* {A!-\-A!') 

 tan \ {A'— A") = tan i (A' + A") tan (x— 45°) ... 



tan i {A'-A") =f?i|-g=J) • (tan x-45°) cot io. 



(14) 

 (15) 



