TO THE THEORY OF ELECTRICITY. 59 



It now only remains to determine the value of V from this 

 equation. For this, let /3 now represent the linear radius of s, 

 and y, the distance between its centre C and the foot of the 

 perpendicular z : then if we conceive an infinitely thin oblate 

 spheroid, of uniform density, of which the circular plane s con- 

 stitutes the equator, the value of the potential function at the 

 point P, arising from this spheroid, will be 



T) being the distance do; 0, and k a constant quantity. The 

 attraction exerted by this spheroid, in the direction of the per- 



pendicular z, will be ~~ , and by the known formulae relative 

 to the attractions of homogeneous spheroids, we have 



. 



M representing the mass of the spheroid, and 6 being determined 

 by the equations 



tan 6 = - . 

 a 



Supposing now z very small, since it is to vanish at the end 

 of the calculus, and y < /3, in order that the point P may fall 

 within the limits of s, we shall have by neglecting quantities of 

 the order z z compared with those retained 



and consequently 



V) SMir 





This expression, being differentiated again relative to Zj gives 

 d* , fdo- 





