CONTEMPORARY ADVANCES IN PHYSICS 295 



which, of course, is merely the definition of /. The essential thing is 

 that if the sphere is infinitesimal, then in the limit/ becomes the value 

 /o which the function / possesses at the centre of the sphere. If /o is 

 finite at the origin, A vanishes in the limit; but if/ varies inversely 

 as the square of the distance from the origin, A approaches in the limit 

 a finite value differing from zero. Upon this property our demonstra- 

 tion will depend. 



Developing by means of (10 a, b, c) the expression given in (11) for 

 div W, we find: 



r \ dx" dy'^ 0. 



div IF = - ( ^TT + ^TT + .,2 



r^ \ dxdr dydr dzdr / 



I / du , du , du\ 



r \ dx dy dz / 



The second and third terms on the right may next be beneficially trans- 

 formed by means of (10 d), using first U and then dU/dr as the argu- 

 ment of the derivatives in that equation : 



xdU , ydU , zdU dU dU ... . 



r dx r dy r dz dr dr 



xdH^ ydHJ^ zdHJ_^d_dU_dHI .^^ ^. 



r dxdr r dydr r dzdr dr dr dr- ' 



and so finally we arrive at 



^'''^ = 7[^'^l^^^-^)'^?d-rV-di-^) ^'^^ 



as the expression to be integrated over the volume between 5 and the 

 infinitesimal sphere. 



Now owing to the nature of the junction U, the first term of the expression 

 vanishes. This is responsible for our theorem ; for the volume-integrals 

 of the remaining terms can easily be translated into surface-integrals 

 over 5 and the infinitesimal sphere, from which it will follow that the 

 value of 5 at the origin is determined by the values of U and its deriva- 

 tives over S; but if this first term should remain, its volume-integral 

 could not be thus transformed, and we should find that the value of 5 

 at the origin was influenced by the values of U all through the space 

 which 5 encloses. 



