



sin 



. &'■ 





-K, 



, COS 



V 







sin 



sin 



G' _ 



G" 



sin A' 

 siu a' 



cos 



COS 



12' 



i2" 



'5 



and, 



since 



COS 



cos 



i2' 

 12" 



sin A' 



- less 



tlian .!i^ 



G' 















GEODESIC INVESTIQATIONS. 163 



Now, if we put G', G", for the angles wliich tlie true geodesic connecting 

 the stations makes with the meridians through S' and S", we know (see notes 

 at end of this paper) that — 



sin G' R,, cos I" 



is greater than 1, we have 



sin A" sin G" 



For any pair of mutually visible stations on the earth's surface, we may 



regard — — ^ — „^^ equal unity, without sensible error in calculated results ; 

 ° cos i 2 



or, which amounts to the same, we may regard -as equal to " 



^ sm A" ^ sin &" 



This is evidently equivalent to regarding A' + A" = a' -f" ^"j the sum of the 

 angles at the base of the triangle S^PS,,. By actually calculating the angles 

 a', a", from the two sides and included angle ai of this triangle, we shall find 

 (A' + A") — (a' + a") to be inappreciable in the actual practice of the most 

 accurate trigonometrical surveying. 



Otherwise, 



When the station S', S", are mutually visible from each other. 



To find the azimuths, we have — 



tan^ I' + ^' 



tan^ 4' = o~ 



tauH" 4-p 



1. \ r Ai \ Aiix COS \ {V I") „„. 1 



tan ^ {A + A") = -^—~ j, , j„( . cot J w 



tan i (A' — A") = tan i {A' + A"), tan {^p — 45°). 

 We have , sin i {V + I") . ^ 



cos i V = 1-7-^7—^ TTTT " Sm i « 



COS i {A' + A ) ^ 



from which to find v and H. Then to find 2, k, s, A, we have — 

 tan I 2 = tan i v . cos fl 

 -E, cos v. sin <a 



h = 



sin A" cos i 2 



. 2 



2 sin i 2 

 tan ^ A = sin ^ 2 tan fl 

 sin i A = sin | p sin n 



