50 



II. PARTICULAR MOTIONS OF A POINT. 



For if we consider two points P, P f of azimuths qp, y>' lying on the 

 same parallel and on opposite sides of a vertex A (Fig. 14), 



cldz 



and since the radical changes sign on passing through a vertex, 



cldz 



Therefore the points P, P' are symmetrical about A and the times 



cc 



of traveling the arcs PA and AP r are equal to / - In like 



J v*o*) 



$ 



manner it can be shown that the path is symmetrical about an upper 



vertex B. The path is 

 accordingly composed 

 of equal parts continu- 

 ally repeated. It of 

 course is not generally 

 true that the path will 



be reentrant after 

 Fig. 14. gi n g once around the 



sphere. 

 We will now consider the horizontal projection of the path. 



Fig. 15 a. 



Fig. 15 b. 



1. Suppose both limiting parallels are below the equator, the 

 projection of the circle e = cc is within that of z = /3, and the path 



