GEODESIC i:n"vestigations. 173 



i 1/7- J\ sill "2 i-^' — -^/z) i 1 



tan i (L, - y = ^.^.(^.^j^^y tan -| ., 



cos i{j9, — ^") 

 tan i a. + i.) = ^^^r7^;ip-j77y- tan i z,, 



tan i (Z,, - A) = ^^,^j,_^^^,.y tan i .,, 



cos i (i„ — Z,) . 1 / ^/ I TV \ 



tan i CO = g °^?[;^7f-j . cot K A + .4") 

 cos i [l^, -t L^ 



From these we can find l^, l,^, L^, L^, and cu with precision. 



To obtain the latitudes V, I", we liave 



I' = 90° — I, ; I" = 90° — I,,. 



And the other various entities can be found as in problem 1. 



^^° We can find I" and I" from the two equations — 



tan2 Z' -f 1 /^sin z^^ . sin D A ^ ^^^ 



tan- I" -f 1 ^sin s, • sin 2)^^ 



^ A- sin2 A' 



. o ,,/ , *" sin- A" 



tan-Z 4-y3 



(b) 



"We can also find the latitixdes (independently of z^ , z,, , D, , D^^) by 

 solving the two following equations : — 



tan i 2 sin I' — cos A' cos I' ^ — tan ^ 2 sin I" -\- cos A" cos I" ; 

 UnH' + j3 _ s in^ A' 



They may be written in the forms — ■ 



tan- I' -\- \ (tan ^ 2 . tan I' — cos A')"^ 



iapo? I" + 1 (tan J 2 . tan I" — cos A") 



tan^ V A-~ 



0^ 1 — cos2 A' 



tan^ I" -\.— 1 — cos2 A!' 



(a) 



(^) 



It is evident that by having any three of the five entities A' ^ A", I', I", 2, 

 we can find the remaining two from these equations. 



