A Geodesic on a Spheroid. 



As y goes through the series of vahies from o to jtt, so does .r, 

 or we take .r goes from .v„ to 2- + .\\,, and in this increase in ic = 



2-+.\\ 



-I'o >/ I — ^-sin-'v. 



d-^ 



\/ I — /:-sin--k!/ 



= 4K 



and II (ii\ K + //3) taken between the hmits 

 = JI CcC' + 4K, K + //3) - [[(ic. K+ij]) 



/'ic' + 4k 



4K 



r4K 



41^ 



I k- sn (K + //3) en (K + //3) dn (K + ip) sn- \ii\ \ 



i 



^ — k-sn-(K + i|3}sn^\ 



k- sn (K + //3) en (K 4- i^) dn (K 4- /"/3) sn- \J\ 

 T — ^- sn2 (K + //3) sn-\ 



= 411 (K, K + //3) ; 



, , , I • ^ 4 cos //osin 0(1— i^2) -. , TT/i- 1- , • •>\ 

 .-. total change m ^ = 3 — , ^ ^ ^ K + 4/ IT (k, K + //3). 



s/ 1 — 6'-cos- //,| sin- o 



II(//. A) — iiE(A) = IICI, n) — AE(ii), 



and II(K + //3. K) = 



.-. II(K, K + //3) = k£(K + //3) — (K4-//3)£. 



Applv the addition theorem for /:(// + <■) to /:(K4-'/3), and turn 

 £(//3) into terms of £(/3, ^ ), 



and we tind II(K. K + //3) = //3(K-£) + '^^ ^l^:::^^}i^^^l^ - /K£(p\ k). 



c 



. • . total cliange in ^ 



4cos//osina(l— t--) r iKs/'(i—e^){c-- — k-) .-c-/^aa 



VI — c'-cos-//|,sm-« I e 



and 



cos //,, sm a 



v/ 1 — t'^ cos^ //,) sin- a c V 1—^2 

 . • . total change in f 



=4)K£(/5.-t')-/3(K-£)} 

 where /3 is given by sn (K + //3) = or dn (/3. k) = i\ 



