68 APPLICATION OF THE PRECEDING RESULTS 



As a second example, we will assume for P r ', the value of 

 the potential function arising from the action of a line uniformly 

 covered with electricity. Let 2a be the length of the line, y the 

 perpendicular falling from any point p' upon it, x the distance 

 of the foot of this perpendicular from the middle of the line, 

 and x that of the element dx from the same point : then taking 

 the element dx ', as the measure of the quantity of electricity it 

 contains, the assumed value of V will be 



V- ( dx> -W a 



~J\% 2 +(*-*Tr S -a 



the integral being taken from x a to x = + a. Making 

 this equal to a constant quantity log 5, we shall have, for the 

 equation of the surface A, 



which by reduction becomes 



= y (i - IJ + x\ 4Z> (1 - IY - ">* & (1 + ) 2 - 



We thus see that this surface is a spheroid produced by the 

 revolution of an ellipsis about its greatest diameter ; the semi- 

 transverse axis being 



1+b -f* 



a T=l-P> 



and semi-conjugate 



By differentiating the general value of V , just given, and 

 substituting for y its value at the surface A 9 we obtain 



1-5 



_ O _ 



afj/' '1 + 6 - 2a/3a; 



of the electric fluid, p, the value of the density near the apex 0, will be determined 



by the formula 



~ T ' n-l 



a being the length of the cone. 



