﻿DERIVATION OF POISSON^S EQUATION 



65 



this sphere becomes smaller and smaller we can apply the result 

 obtained for a homogeneous sphere and we have immediately 



AV = - 4^p, 



where p is the density at the given point. 



Proofs of this nature are given by Green\ Priced and Routh^. 



2. Proofs by means of Gauss's theorem. One of the simplest 

 ways of proving Poisson's equation is that due to Stokes*. It is 

 based upon the theorem due to Gauss, that if V is the potential 

 and S any closed surface, then 



ff~ds = - 4rTM, 



where n is the exterior normal, and J/ the mass within the surface. 



When applied to a rectangular parallelopiped with edges dx, dy, 

 dz, G-auss's theorem becomes 



d^V d^F a^F 



( — ^ + — + — )dx. dy. dz= — 4^pdx. dy. dz, 

 dx" dy^ dz'^ 



where p is the average density. Now letting the parallelopiped be- 

 come infinitely small, and dividing by dx. dy. d.z, we find, 



where () is the density at the point around which the parallelopiped 

 vanishes. 



This proof is given by English and American writers^. 

 If in connection with Gauss's theorem we use the modified form 

 of Green's theorem that 



ffjAVdr =JJ .^d^, 



T S dn 



where T represents the volume enclosed by the surface S, and n is 

 the exterior normal, we have a simple means of proving the theorem, 

 which is given by Webster^ and Peirce'. 



3, Proofs by direct differentiation. Proofs of this nature are 

 given by the German and Frencli writers. They show rigorously the 



1 George Green, Mathematical Papers, p. 20. 



2 Bartholomew Price, Infinitesimal Calculus, Vol. Ill, p. 320. 

 " E. J. Routh, Analytical Statics, Vol. II, Art. 80. 



^George Stokes, Cambridge and Dublin Mathematical Journal, Vol. IV, p. 215. 



5 B. O. Peirce, Netvtonian Potential Function, p. 61 ; A. G. Webster, Dynamics 

 of a Particle and of Rigid Elastic and Fluid Bodies, p. 361 ; E. J. Routh, Analytical 

 Statics, Vol. 2, Art. 83 ; Bartholomew Price, Infinitesimal Calculus, Vol. Ill, p. 328. 



^ Webster, Dynamics, p. 360. 



^Peirce, Newt. Pot. Funct., p. 66, 



