On Practical Geodesy. 45 



This case, in which the given or known latitude l^^ is 

 less than the sought latitude l^, will be intimated to us by 

 the angles A^^ and T>^ ; we shall have the given azimuth A^^ 

 less than the angle D^. If the angle A^^ = D^, then A. = D^, 

 and l^ = l^^, &c. 



Othei^ivise. 

 Case 1°. When l^, A^, s, are given ; to find l^^, A^^, w. 



Find , w, D^^, as indicated in the last solution, and then 

 find A^^ by means of — 



. . cos (z, i 2) . -p, 



sin A,, = ^^-^— - — ^ — '- ' sin Jj,. 



cos J 2, 



And find l^^ from — 



, „ cos A (A, 4- A,. + w) , ■ 



tan 1 r = — w A I A \ • cot 1 Z' 



2 COS J (A, + A,, — w) 2 



;^^ = 90° — I". 



Case 2°. When ^^^, A^^, s, are given; to find l^, A^, w. 



Find z. w, D, as indicated in the last solution, -and then 



find A^ by means of — 



. . cos (z^. ^2) • Tk 



sm A, = ^ , J — ^ • sm D. 



And find I. from — 



cos ^ 2 



cos A (A, + A,. + (u) 



*-4^' = -c-ornA7TAf^--*^^ 

 I, = 90° — r. 



Peoblem 3. 



Given the latitudes l^, l^^, and the azimuth A^ ; to find the 

 azimuth A^^, the difference of longitude w, &;c. 



By equating the values of sin a, as expressed in formulae 

 108, 109, we have— 



R,, cos I,, (cos^ Z, + 1) ij \ — sin^ w 



— (R/ + "r" — -^// * ~2 ' sin l^ sin l,^ cos l^ 



— (R,, cos Z,^ tan l, cot A J sin w 



or, M • n/ 1 — sin^ o> = L — N • sin w 



in which the values of M, L, and N are known. 

 From this we at once obtain 



L N + Vm^ (M^ + N2 — Ln 

 sm <o = ■ — - ^ ' 



