174 GEODESIC INVESTIGATIONS. 



Problem 10. 



Griven the two azimufhs A', A", and the difference of longitude ci> of the 

 stations, to find tlie two latitudes I', I", and the other entities. 



The co-latitudes I', I", can be easily found from — 



eosijl, — y _ coti(A' + A") 

 cos i{l, -f I.^) tan iw 



cot % + ^ 



'~ b"- __ sin '^A' 



,07 I a" sin '^A" 

 cot H,, 4- — , 



By putting 7f to represent the dexter of the first equation -we can replace 

 it by the following one, which can be easily deriyed from it : — 



K) 



cot^- il, ■ cot^- il, = (f^)' = ^ - 



The second can be put under tlie form — 

 (cotH^. -1)- ^ ^ 



(2 cot il,T- i- _ gi^2 J^r 



(cot^ il„ - ly - + ^ "~ sin- A" 

 (2 cot il,y b- 



(50 



By dereloping this last equation, and making use of equation («'), it is 

 easy to put it in the form — 



M cot^ il, + cot^ U,, _ Jf (4 |! - 2) ^.^, ^, 



(c) 



Mcof^ kl„ + cot^ ^l,- M (^ £ - 2) 



sin- A" 



It is evident that the values of cot^ \l, and cot- \l„ can be easily found 

 from the formula a' and c. 



The resulting values can be arrived at by means of the following set of 

 formulae : — 



tan^= eotH^- + ^-0 W 

 tan ^ 01 



M = io-n^ {^■\- 4.0°) (e) 



a = M (4 ^' — 2) . (sin2^' - siu^^'O : (/) 



U = M sin* A" — sin- A' ; (sf) 



V= M sin2 A' — sin2 A" ; (/^) 



cot^ J. ; _ G-^VG'-\.4 3I.U.V (i) 



cot .,1,, ■ ^—^ 



It is evident M is positive and greater than 1 ; and therefore, since 

 sin- A' is greater than sin- A", it follows that Q and V are also positive. 



