332" Prof. Cayley on the Geodesic Lines 



on the meridian, using (instead of the parametric latitude V) the 

 angle (f> determined by the equation 



, sinX' 



COS d>= -. jr> 



T sin /' 



and writing, moreover, s to denote the geodesic distance V P, 

 and A to denote the longitude of P measured from the meridian 

 C A which passes through the vertex V, these are 



ds = dcj> v/CHA^sinVcos 2 ^ 



</A= e ^ll #^ / ^ 2 + A C2 6 2 sm 2 / / cos^ . 

 A 1 — sin 2 /' cos 2 <j> 



which differential expressions are to be integrated from <£> = 0; 

 and the equations then determine X', s, and A, all in terms of 

 the angle <f>, — that is, virtually s and A, the length and longitude, 

 in terms of the parametric latitude X f . 

 Writing, with Legendre, 



A^sin 2 /' 

 C 2 + A 2 S 2 sin 2 /" ~° sin *' 



6 2 =1 — c 2 = - — 1 — £ 2 sin 9 /- 



' C 2 + A 2 S 2 sin 2 /" _1 d sm l ' 



also 



7z=tan 2 /', M=— £_ = <?L_ 

 bA cos / A cos / 



then the formula? become 



C 



b 



ds= ~dcj> v/l~c 2 sin 2 ^, 



,. -»rd6 \/l — c 2 sin 2 6 



di\ ~ M --^j- . . ~ 



1 -f n sm^cp 



Hence integrating from $ = 0, and using the notations F, E, II 

 of elliptic functions, we have 



A=^\(n + c*)Il(n } c,cl>)-c*?(c } cf>)}; 



viz. these belong to any point P whatever on the geodesic line, 

 parametric latitude of vertex = /' ; and if we write herein $ = 90°, 

 then they will refer to the node N, or point of intersection with 

 the equator. 



