40 GENEEAL PRELIMINARY RESULTS. 



x, y\ z , and that whose co-ordinates are #, y, z, to which p be- 

 longs, also r the distance between p and do-, an element of the 

 surface of the body : V being evidently a function of x', y , z . 

 If now F be what V becomes by changing x ', #', z into x, y, z, 

 it is clear from art. 1, that p will be given by 



Substituting for p ', the value which results from this equation, 

 in that immediately preceding we obtain 



fr, [dxdydzSV , [pd<r 

 V = I - : - 7 -- r I - j 



J 4WT J r 



which, by means of the equation (3) art. 3, becomes 



[ 



- - 



r \ 



[pda- 1 [, v\~r [do- (dV 



\ =7- 1 \da-v \ - 1- \ - -- 



J r 47rJ dw J 



.the horizontal lines over the quantities indicating that they be- 

 long to the surface itself. 



Suppose F, to be the value of the potential function in the 

 space exterior to the body, which, by art. 5, will depend on 

 the value of F at the surface only; and the equation (2) 

 art. 3, applied to this exterior space, will give, since 8F y = 



and 8 - = 0, 

 r 



where dw' is measured from the surface into the exterior space 

 to which F, belongs, as dw is, into the interior space. Conse- 

 quently dw = dw, and therefore 



