TO THE THEORY OF ELECTRICITY. 65 



It may "be proper to give an example of each of these cases. 

 In the first place, let us take the general expression given by 

 LAPLACE, 



then, by confining ourselves to the two first terms, the assumed 

 value of V will be 



r being the distance of p from the origin of the co-ordinates, 

 and U (0} , U (l) , &c. functions of the two other polar co-ordinates 

 6 and tzr. This expression by changing the direction of the 

 axes, may always be reduced to the form 



IT i _ ^ a ^ cos ^ 



V I 



r ' r 2 



a and k being two constant quantities, which we will suppose 

 positive. Then if b be a very small positive quantity, the form 

 of the surface given by the equation V b, will differ but little 



from a sphere, whose radius is -,- : by gradually increasing b, 



the difference becomes greater, until b = ^ ; and afterwards, the 

 form assigned by F=&, becomes improper for our purpose. 

 Making therefore b = p , in order to have a surface differing as 



much from a sphere, as the assumed value of V admits, the 

 equation of the surface A becomes 



-r r ,_2a If cos _ a* 



From which we obtain 



r = 

 If now <f) represents the angle formed by dr and dw, we have 



Q 



, V 2 sin - 



-dr y 2 



2^2 cos? 



