356 Captain A. R. Clarke on the Course of 



By spherical trigonometry we have these two equations : 



cot ot l sin -cr = cos u { tan u — sin u l cos -or, 



cot « ; sin -sr' = cos u t tan u' — sin u { cos <*r', 



which by the elimination of a i give 



sin w' tan u = sin ot tan w' + tan M i sin [^ — «r) . . . (4) 



Expressing next -or and ot' in terms of co and &)', and writing 

 for brevity co if 'st / , cr l for &)'—-&), ot 1 — -dt, <t' — c, 



. * 2 • 

 sin ot = sin go + — cr sin a y cos u t cos &>, 



e 2 

 sin -cr' = sin w' + -^ a 1 sin a i cos u, cos &/, 



e 2 

 sin -cr , = sin o) ; + -jr o"/ sin « y cos w y cos co r 



The next step is to substitute these expressions in (4) . Putting 

 a! for the azimuth of the geodesic at B, and making use of the 

 equations 



sin a cos a y = cos u, sin u' — sin u t cos u' cos ot, 



— sin c- cos a' = cos w' sin u { — sin i/ cos u t cos «r, 



and effecting other obvious simplifications, we get 



sin co' tan u = sin co tan w' + sin co f tan w, 



e 2 

 + -~- { <r cos u l sin g> ; — cr j cos a sin co}. 



But since we are to neglect e 4 , we may put within the parenthesis, 



. cosw.sincr. 



sma>,= sin&) 



, cos u' sin cr 

 sin ct> = sm ft)' 



(5) 



cos w sin o 1 

 Also, for brevity, put 



, sin ft) sin ft). 



tan u a = tan w' — 7 + tan w, - f ; 



sm &)' ' sm ft)' 



and since tanw— tanw =(w — w ) sec 2 w, we get, finally, 



, Q (cr sin o-, o\ sin cr . A . x 



m = w + 1 e^ cos u ( — : j 1 cos w, cos . '- j— cos u' cos a' V (6) 



02 \ sin cr' ' ' sin a' ) v ' 



This determines the latitude of any point P on the geodesic 

 joining AB, the longitude of P being co. 



