28 GENERAL PRELIMINARY RESULTS. 



x, y, z. Then, if V be the value of this function for any other 

 point p exterior to this surface, we shall have 



v= f P d(T . 



%, 97, f being the co-ordinates of da; and 

 F/= f pd<r 



the integrals relative to cZcr extending over the whole surface of 

 the body. 



It might appear at first view, that to obtain the value of V 

 from that of F, we should merely have to change x, y, z into 

 # ', y', ' : but, this is by no means the case ; for, the form of the 

 potential function changes suddenly, in passing from the space 

 within to that without the surface. Of this, we may give a very 

 simple example, by supposing the surface to be a sphere whose 

 radius is a and centre at the origin of the co-ordinates ; then, if 

 the density p be constant, we shall have 



which are essentially distinct functions. 



With respect to the functions F and V in the general case, 

 it is clear that each of them will satisfy LAPLACE'S equation, and 

 consequently 



= SFand 0=S'F': 



moreover, neither of them will have singular values; for any 

 point of the spaces to which they respectively belong, and at the 

 surface itself, we shall have 



the horizontal lines over the quantities indicating that they be- 

 long to the surface. At an infinite distance from this surface, 

 we shall likewise have 



F' = 0. 



