GENERAL PRELIMINARY RESULTS. 23 



dV , dV , 



= -=- ace + -7- ay 



dx dy dz 



which equation being integrated gives 



F = const. 



This value of V being substituted in the equation (1) of the 

 preceding number gives 



,0 = 0, 



and consequently shows, that the density of the electricity at 

 any point in the interior of any body in the system is equal to 

 zero. 



The same equation (1) will give the value of p the density 

 of the electricity in the interior of any of the bodies, when there 

 are not perfect conductors, provided we can ascertain the value 

 of the potential function F in their interior. 



(3.) Before proceeding to make known some relations which 

 exist between the density of the electric fluid at the surfaces of 

 bodies, and the corresponding values of the potential functions 

 within and without those surfaces, the electric fluid being con- 

 fined to them alone, we shall in the first place, lay down a 

 general theorem which will afterwards be very useful to us. 

 This theorem may be thus enunciated: 



Let U and F be two continuous functions of the rectangular 

 co-ordinates x, y, z, whose differential co-efficients do not become 

 infinite at any point within a solid "body of any form whatever ; 

 then will 



the triple integrals extending over the whole interior of the 

 body, and those relative to d<r, over its surface, of which d<r 

 represents an element: dw being an infinitely small line per- 

 pendicular to the surface, and measured from this surface towards 

 the interior of the body. 



