34 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



X = cons. We shall now limit this amount of freedom by assuming that the region 

 at infinity is made to coincide with the surface X = + . 



Consider a new function V, defined at any point of space x', y', z', by 



(22) 



o 



then 



(23) 



by equation (18). Hence, if (as will be .the case in all our applications) the value 

 of the limit when X' = oo of the term in square brackets is zero, we shall have 



V-'V, = - 47T P ........... (24) 



At infinity V u must vanish, so that at infinity, by equation (21), 



v, = w = I V wyv, ?/, *', x) d\ = v, 



Jo 



Thus V; V, vanishes at infinity, and satisfies V- (V, V,) = at all points of 

 space ; whence (except for a possible singularity at the origin, which will be found 

 not to cause trouble) we must have V^ = V,, so that V,- is the internal potential. 

 Knowing V { and W we find Vu immediately by equation (l), and have 



,!/,z',\)<lX ........ (25) 



) i/ > z f ,\)d\ ........ (26) 



To recapitulate, the condition that these equations shall give the true values of the 

 potential are 



(i) that V,. shall be finite at the origin, 



/ 7^ \ 2 ^ f 



(ii) that x/, (x) (-=-) ^- shall vanish at infinity. 



\CLlli / C\ 



If these conditions are satisfied, as they will be without trouble in all our 

 applications, then equations (25) and (26) will give the potentials. 



8. If y (x, y, z, 0) is the equation of the boundary, the potential at the boundary 

 will be 



and therefore will contain exactly the same terms in x, y, z as the equation of the 



