traced upon the Surface of the Sphere. 



295 



cos (p = sin 0, cos & 

 sind) = -h J~ 



sn 



cos 



VI sin 2 (f)' cos 2 



4. If, besides changing the origin to N, it be required to change the 

 direction also of the prime meridian to NU, which makes with PN an 

 angle PNU = p, then we have only to write in equation (5.) for & the 

 angle 6' + /*, and the transformation will be completed. 



5. We may particularly specify the case where the transformation is to 

 the opposite pole of the equator, a case of frequent occurrence in the inves- 

 tigation of different theorems, and still more frequently in the examination 

 of the course of the curve whose equation we may desire to examine. 



Here a = IT, and p. = 0, & = 6. Hence, 



sin (f) = sin (f)' ; cos (p cos <p' ; tan <f> = tan (f>' : and 

 sin 6 = sin & ; cos 6 = cos 6' ; tan 6 = tan & 



&c. &c. &c. 



Sc/iol. 1. We may illustrate this transformation in the case of the circle. 

 Let its equation be 



cot (p = tan X cos (6 K) 



cos <b sin X . ., 



or, L- (cos 6 cos K + sin 6 sin K) (6.) 



sin (p < * 



cos X 



P p 2 



