traced upon the Surface of the Sphere. 413 



The proof of this is very simple. 



We see at once, from the manner in which e and (3 enter into the equa- 

 tion, that the sign of cos e (since e and 8 are themselves mutually indepen- 

 dent) must be totally independent of the value of 8, that is, of the origin of 

 angular ordinates ; and, therefore, as it will considerably simplify our in- 

 vestigation, we shall take 8 = 0. This gives 



cot (f>= cosec e cosec a {cos 6( cos a sin e) + sin 6 (ip cose)} (10.) 



In this we have cot K = 



cos a sin e 



(- COS 



= zt cos a tan e (11.) 



Now, by the quadrantal triangle PLR, we have 



cot LPR = cot = cos a tan (12.) 



And by the quadrantal triangle P'LR, we have 



cot/c = + cos a tan 6 (13.) 



We see, therefore, that in the case of P and the origin of e being on the 

 same or on different sides of L, the upper or lower sign of (11.) is to be 

 taken, in conformity respectively with (18.) and (12.), which is the rule given 

 above. 



The equation of LD, then, as referred to P and P' respectively, is 



( cos 6 (sin 8 cos e + cos 8 cos a sin e)l 



cot (p = cosec a cosec e-J J- (14.) 



( + sin 6 (cos p cos sin p cos a sin 6) ) 



( + cos 6 (sin 8 cos e cos 8 cos a sin e) ) 



cot m = cosec a cosec e-< I (15.) 



( sin 6 (cos p cos e + sin p cos a sin e) ) 



It might be thought, however, for the sake of generality in the subse- 

 quent investigation, to be as well to retain the double form given in (9.) : 

 though, in many cases, our purpose may be fully answered by the adoption 

 of either one or other of its components (14.) or (15.), merely taking care to 

 keep clearly in view the connection between the pole employed and the 

 branch of the meridian from which e is measured. Except contrarily ex- 

 pressed, I shall use the pole P and branch P'L, from its being that adapted 

 to the figures I have most commonly used. This requires the lower sign , 

 and (14.) is the equation of the circle adapted to that assumption. 



3 G 2 



