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



Thus equation (31) is equivalent to 



x 3t? dv\\d\ 4v /or> v 



T5 - + ~- U == ...... (33) 



A fix 3X7 J A A A= 



This must be satisfied for all values of X, so that we must have (as is clear on 

 putting X = in the equation) v = when X = 0. 



It will be remembered that the boundary of the distorted figure is given by 



r 2 ?/ z 2 



T + S + T-1+0A.O-0, 



a* V c* 



and it is now clear that A = reduces to A = n . Thus the generality of the boundary 

 must be involved in the generality of u, and provided u is kept general, we shall 

 obtain a general solution of the problem, even if we take the simplest possible value 

 for r. The most general way of satisfying equation (33) is to take 



....... (34) 



where x may be any function of x, y, z, and X which vanishes for X or 0, but the 

 simplest way of satisfying the equation is to take 



........ (35) 



In each of these equations the sign of identity ( = ) is used in place of the sign 

 of equality, because in the integrand of equation (33) the value of X is not the same 

 as the value of X in the upper limit of the integral, which is determined by the values 

 of x, y, z. 



12. To shorten the algebra we may change to a new set of variables X, ij, f 

 connected with the old variables X, x, y, z by the relations 



x 



Differentiation with respect to the new variable X is given by 



_3_ _3_ _, Sx 3 



OX new 3X ohl 3X dx 



*L 



where, since x = (a a + X) f, we have ~ = g = . and so 



oX A 



- - 



3Xnew 3Xc,d + A8X 



