traced upon the Surface of the Sphere. 299 



points in AB, are represented by equations (3.), (4.) Those, for instance, of 

 LAHIRE, ARKOWSMITH, NICHOLSON, &c. ; but as there is nothing par- 

 ticularly interesting, in a mathematical point of view, attached to these in- 

 quiries, we shall pass them over. We may just state, however, that when 

 we wish to assign the cone whose intersection with the sphere is any specific 

 spherical curve, and whose centre is at the centre of the sphere, the same 

 equations apply as are employed for the interchange of rectangular and 

 polar co-ordinates in the usual processes of analytical geometry. 



'.!..,''* 



.... ...(9.) 



f- , : sin = H- 1 *# ..... (10.) 



V* +*+** -v - 



The equations (9.) also serve to transform (5, 6, 7, 8.) into rectangular 

 curves. By means of (9.) (10.) it will be remarked, that the spherical equa- 

 tions of the hectemoria were converted into rectangular ones. (See art. XXI. 

 of that paper.) 



I shall not dwell longer on this part of the subject, which, on account 

 of its very elementary character, I would gladly have omitted, but that some 

 of its results are necessary in our future investigations. 



XVI. 



1. The length of any curve on the sphere, referred to the co-ordinates 

 and 6, is 



L -J^dQ . sin + d(j) 2 + C. 



For, let MS be an element of the curve lying between consecutive meri- 

 dians PE, PQ, and let EQ be a corresponding element of the equator. 

 Then, 



MN 2 + SN 2 = MS* ; that is, MS* = EQ 2 sin 2 c/> + SN 2 ; 



d(p, (1.) 



