64 LECTURES AND ESSAYS [1869- 



festly great circles through it, and the equipotential lines 

 small circles, of which it is the pole. 



If the radius of the sphere is a, the circumference of the 

 small circle, whose angular radius is 0, is lira sin 0. Hence 



if u be the potential, 



du i 

 JO^sm^d 



i , i - cos 



2 i + cos 

 For any number of sources the potential will be 



i/ i - cos 

 -( 2 log 



2\ I + COS 



and the equation of the equipotential lines, 



i - cos 0-, i - cos 9 ~ i - cos i - cos 2 



i . ^ I JL ^ 



I + COS0J I + COS0 2 &quot; I + COS0 1 I + COS0 / 2 &quot; 



the accented angles belonging to sinks. 



For the lines of flow we have, precisely as in a plane, 

 2( c/&amp;gt;) = c, where $ is the angle between the great circle 

 through a source and a point on the line, and a fixed great 

 circle through the source. 



Let us take, as an example, the case of one source and 

 one sink. Let the co-ordinates of these points be h, k, ; 

 h, - k, 0, and those of any point on an equipotential line, 

 x,y, z. 



We have for the equation of this line, 



I - COS I - COS0 



i + cos i + cos & ~ 

 where 



hx + ky n , hx - kv 

 cos = s-A cos = s-A 



Hence the projections of the equipotential lines on the 



plane of xy have as equation, 



( a 2 _ ^ _ ky) (a 2 + hx- ky) + A(# 2 -hx + ky) (a 2 + hx + ky) = 0, 



or 



- 2 



