Geodesic Lines on the Earth's Surface. 355 



90°—^, 90° — u, of a spherical triangle, and the angles opposite 

 to them, 180°— a. and 180°— a y . Let the third side of this 

 triangle be a, and the third angle w, then 



dcr cos a. — du, 



da sin « = cos u dvr. 



If c/> be the latitude of a point P on the geodesic, so that 

 a tan u = b tan <f>, then 



sin u= sin $(1 — e q cos 2 u)^. 



Also 



from which we get 



d£ sin <j> = dr= —asinuduj 



ds =a(L — e 2 cos 2 u) 2 da, 

 do)= (1 — e 2 cos 2 w)* 



*f 1 • • • (1) 



This completely determines the auxiliary spherical triangle, and 

 through it the latitude and longitude of any point P at a dis- 

 tance s from A measured along a geodesic which has a given 

 initial azimuth. With respect to the angle ot, if we neglect, as 

 we propose to do henceforth, e 4 and higher powers of e 2 , 



e 2 C 

 to = ia — jr 1 cos 2 udiz ; 



but by the consideration of the spherical triangle we see that 



7 7 sin u , sin a. cos u. . 



dtn = da = da ^ - > 



cos u cos z u 



e 2 

 .*. &) = 'S3- — — a sin a. cos u t (2) 



2. 



Now let P B be two points on the geodesic, s s' their dis- 

 tances from A, co co 1 their longitudes, u u' their reduced lati- 

 tudes, then 



sin u = sin u t cos a + cos w y sin a cos a,, 



sin u'= sin w ; cos a 1 + cos w, sin a 1 cos a y ; 



from these, by means of (1), we can express s ands' in terms of 

 a and a 1 , or, inversely, express a and a' in terms of s and s'. 

 Then if -cr, -cr' correspond to co, co 1 . 



-or = a) + x- <r sin a, cos «,, 



e 2 

 -ay' = co' + jr- cr' sin OL i cos w r 



.... (3) 

 2A2 



