172 
MR. J. II. JEANS ON THE CONFIGURATIONS 
where \fs is any function whatever. Equation (63) can be solved in the form 
x 2 /a 2 + y 2 /b 3 +z 2 /c 2 —l—r)=0, .(65) 
and on comparison with equation (62) the deformation is seen to consist of the addition 
of a thin homeoidal shell of uniform density. By a well-known theorem, the potential 
of such a shell is constant at all internal points, so that for the particular deformation 
represented by equation (64), we have AV;(l) = a constant. Incidentally, we may 
note that $Y l (1) is also constant, and so on to all orders. 
15. Returning to equation (59), it appears that P" 0 must satisfy 
-i/DP"+*/ 2 D a P"]A-eP/' 
and from the considerations brought forward in the last section, it is clear that the 
solution is 
so that 
1(2 — 1 
2 v a 2 ) ’ 
. ( 66 ) 
L" = &c. ; p" = q" = r" = 0. 
We are left with the problem of determining P', which must satisfy 
jV-i/DP'+A/W]^|-0P' O 
= i(ZJ M *‘+22J Bc ^-J) + A»(a?+»’).(67) 
Unfortunately there is no simple means of dealing with this equation, and the 
general solution obtained by direct algebraic treatment is so complicated as to convey 
no meaning at all to the mind. Our method will be to consider first an approximate 
solution of a simple form, this having reference only to the fourth degree terms in 
P'; we shall then attempt to estimate the amount of error involved, giving detailed 
calculations and precise solutions for two configurations of special importance. 
16. The approximate solution we shall consider is 
a UU j 
2 0 Jbc ’ 
&c 
(68) 
This satisfies equation (67) as regards terms of fourth degree except for the integral 
on the left-hand, so that the error of the solution is roughly measured by the value 
of this integral. 
Now from the definition of the integrals J AA , &c., it is clear that the greater part 
of the value of these integrals arises from contributions from comparatively small 
values of X, so that to a moderate approximation' we shall have 
•U.Y _ •! BB _ Jcc 
_L 1 1 
70 
J 
BC _ _ 
= k, say 
a 
c 
b 2 c 2 
(69) 
b' 
