GENERAL PRELIMINARY RESULTS. 31 



functions represented by F, would be equal to the number of 

 the bodies, one for each. In this case, if there were given a 

 value of F for each body, together with F' belonging to the ex- 

 terior space ; and moreover, if these functions satisfied to the 

 above mentioned conditions, it would always be possible to 

 determine the density on the surface of each body, so as to 

 produce these values as potential functions, and there would be 

 but one density, viz. that given by 



dw dw 



dV dV 

 which could do so : />, -j and -7-7- belonging to a point on the 



surface of any of these bodies. 



(5.) From what has been before established (art. 3), it is 

 easy to prove, that when the value of the potential function F is 

 given on any closed surface, there is but one function which can 

 satisfy at the same time the equation 



= SF, 



and the condition, that F shall have no singular values within 

 this surface. For the equation (3) art. 3, becomes by sup- 

 posing &U=0, 



dw 



In this equation, U is supposed to have only one singular value 

 within the surface, viz. at the point p' 9 and, infinitely near to 



this point, to be sensibly equal to -; r being the distance 



from p. If now we had a value of U, which, besides satisfying 

 the above written conditions, was equal to zero at the surface 

 itself, we should have U= 0, and this equation would become 



