APPENDIX. 329 



and OCAC-\- OA = - - } 



A, f- 1 



Hence OG : OA :: OB : OC::\: 1. 



And potential at A : potential at B :: -j : - : X : 1. 



Hence the theorem is proved. 



The laws of the distribution of electricity on spherical con- 

 ductors have been geometrically investigated by Sir William 

 Thomson in a series of papers published in the Cambridge and 

 Dublin Mathematical Journal. See also Thomson and Tait's 

 Natural Philosophy, Arts. 474, 510. 



Note to Art. 12, p. 68. 



In the case of a straight line uniformly covered with elec- 

 tricity, the form of the equipotential surface, and the law of 

 distribution of the electricity over the surface may be investi- 

 gated as follows. 



Denoting the extremities of the straight line by 8, IT, we 

 know that the attraction of the line on p' may be replaced by 

 that of a circular arc of which p is the centre, and which touches 

 SH, and has Sp ', Hp as its bounding radii. Hence the direc- 

 tion of the resultant attraction bisects the angle Sp H, and the 

 equipotential surface is a prolate spheroid of which S, H are 

 the foci. 



dV 

 Again, -j , is the resultant force exerted by the straight line, 



or by the circular arc, and therefore 



dw y 



AT $P' H) 



Now v 



2a 



