﻿lA' D L A.N A VNL \ EKS IT Y 



8. The Derivation of Poisson's Equation by Means of Gauss's 

 Theorem of the Arithmetic Mean. 



By Kenneth P. Williams, A. M. 



The equation 



L^v = — ^ ^ ^ H — ^ = — 4^^', 



dx" d?r d^" 

 where Y is the Newtonian potential function and f the density of 

 the distribution, has been the subject of investigation by different 

 mathematicians since it was first given bj^ Poisson in 1813. 



The proofs of this equation that are usually met with in trea- 

 tises on the potential function can be divided into three classes: 

 (1) proofs depending on a particular case, (2) proofs by the aid 

 of Gauss's theorem of the integral of the normal component of the 

 force over any closed surface, (3) proofs by direct differentiation 

 of the integral giving the potential. 



It is the purpose of this discussion to review briefly these meth- 

 ods and to give a proof depending on Gauss's theorem of the arith- 

 metic mean of the potential over a sphere. 



I. 



1. Proofs depending on a particular case. The character- 

 istic of this method is that the potential of a distribution of some 

 special form is actually calculated, after which it is possible to find 

 the desired second derivatives. Historically, Poisson 's proof was 

 of this nature. 



The special distribution selected is that of a sphere of constant 

 density p. It can easily be shown that at an interior point 



7 _ _ _ _ ^^^^ 



d.T~ a?/^ a.e" 5 



whence, 



The general case is now treated by dividing the distribution 

 into two parts, a small sphere enclosing the point in question, and 

 the portion outside this. The latter by Laplace's equation con- 

 tributes nothing toAF. We have then merely to find AFj , where 7 1 

 is the potential due to the matter within the sphere. In case the 

 distribution is such, that the density can be said to be constant as 



