GENERAL PRELIMINARY RESULTS. 29 



We will now show, that if any two functions whatever are 

 taken, satisfying these conditions, it will always be in our power 

 to assign one, and only one value of /o, which will produce them 

 for corresponding potential functions. For this we may remark, 

 that the Equation (3) art. 3 being applied to the space within 



the body, becomes, by making U - , 



do- dV 



snce 



= - , has but one singular point, viz. p ; and, we have 



also S V and S - = : r being the distance between the point 



p to which V belongs, and the element dcr. 



If now, we conceive a surface inclosing the body at an in- 

 finite distance from it, we shall have, by applying the formula 

 (2) of the same article to the space between the surface of the 

 body and this imaginary exterior surface (seeing that here 



- = U has no singular value) 



since the part due to the infinite surface may be neglected, be- 

 cause V is there equal to zero. In this last equation, it is 

 evident that dw is measured from the surface, into the exterior 

 space, and hence 



x d/~ W/ ' ' \^/ + \^/ ; 



which equation reduces the sum of the two just given to 



In exactly the same way, for the point p' exterior to the surface, 

 we shall obtain 



