^6 On Pr<Jbctical Geodesy. 



K,, cos I,, sin CO 



cos a, = — ^^- r^^— 



k • sin A, . / X 



•(107) 



_ E,, cos l^ sin (o 



k ' sin A,^ 



R, cos I, — E,,, cos I,, (tan Z, cot A, sin w + cos w) 



sm a, = -^ V -, ' 



k. • cos 0, / X 



(los) 



R,, cos I,, — K, cos I, ( tan,, cot A,, sin w + cos w) 



sin a,, = -^^ '- 9 '^j ^ ^ 



A; • cos t,, 



E,,E,,, (cos L cos ^, cos w + (1 — e^) sin ^ sin ^,,) — a^ 



^"'"'^ — ^ — k-i — ■ — ^ - 



(109) 



E.,K,, (cos I, cos I,, cos (o + (1 — e^) sin I, sin ^,,) — a^ 



sin a, = ^ — ■ — ^^ — 



. R, sin A, cot w sin A, + sin I,, cos A, 



tan a = — — — ' 



' R,. • cos I,, sin 0) cos ^ / v 



(110) 



, R,, sin A,, cot w sin A,,4- sin I,, cos A,, 



tan a,, = -r. j • — -^ ' ' 



K^ cos I, sm w cos l^^ 



_ cos I, 'R,, sin A,, cos A,+ R, cos A,, sin A, 



sin A„ ' R,, sin l,,-\- R, sin I, / v 



, cos L, R,; sin A,, cos A,+ R, cos A,, sin A, 



tan a^i = — * - ~ — 



sin A^ R/^ sin l^^-\- R, sin I, 



31. By equating the values of sin a^ given in (108), (109), 

 we have an equation from which we can at once express 

 cot A^ in terms of the two latitudes and the difference of 

 longitude w. And equating the values of sin a^^ given in 

 (108), (109), we can express cot A^^ in terms of the two lati- 

 tudes and difference of longitude. However, we can find 

 other expressions for the cotangents of the azimuths, thus — 



From the spherical triangles S^PD^^, S^^PD^, we have 



, . cot L" cos I, — sin L cos w 



cot A. = 



sm 0) 



cot L' cos L, — sin /,, cos w 



tan a. 



cot A, 



sm w 



And if in these we substitute the values of cot JJ', cot 1/, 



given in (79), we have — 

 "P 



W ■ e^ sin l^ cos I, + (1 — e^) sin I,, cos I, — sin l^ cos l,^ cos co 



cot A, = ^ -, 



cos I,, sin o) / V 



(112) 



T> 



^ • ^ sin l„ cos I,, + (1 — ^) sin l; cos l„ — sin l„ cos I, cos <o 



cot A„ = -L ., —-, : 



COS L. sm <i> 



