traced upon the Surface of the Sphere. 403 



Let EC be the straight line whose equation is sought, and OE be a per- 



pendicular from O upon it. Let F be a point in the line, and put 



OF = r ; 



and COF = 8 ; 

 Then OE = OFcosEOF 



or r = ttsec(8 /?) (1.) 



which is the equation sought. 

 It may also be written, 



cos(0 /3) = (2.) 



2. To find the equation of a straight line through two given points. 

 Let r, 8, and r u 8,, be the points : then inserting these in (2) we get 



cos(0,-/3) = - 



(3.) 



From which 



r a _ cos (3 cos 6, + sin /3 sin 6, cos 6, -f tan /3 sin 6, 



r t ~~ coS /3 cos 6 U + sin /3 sin $ cos <? + tan /3 sin ^, 



or tan / = 



r, cos 0, r /y cos 8 a 



r sin f, r u sin V a 



r, cos 6, r u cos 8 a 



sin p = qr __^ = 



* r* 2 r, r,, cos 8, Q,, -f- r,, 



r sn r sn 



cos = 



(4.) 



VOL. XII. PART II. 



3r 



