168 
MR. J. H. JEANS ON THE CONFIGURATIONS 
The boundary reduces to the ellipsoid 
2 2 2 
°L + !L + *- = 1 
a 2+ b 2 c 2 ’ 
and the value of V, (1), which is the potential of this ellipsoid filled with matter of 
unit density, is 
V t -(l) = —7rabc( J x x 2 + J B y 2 + J c z 2 —J),.(36) 
where the notation is that I have previously used* in which 
Jab - = I A{a 2 + X){b 2 +\)... . (37) 
On substituting this value for V, (l) into equation (35) and equating coefficients 
of x 2 , if, z 2 , we obtain, as the conditions for equilibrium, 
T _ T i _ w — ( oo \ 
v irp.abc 2-irp 0 abc « 2 ’. 
T _ r 2 _ 0)2 — / Q Q \ 
irp () Ctbc ^irp^oJjC 6 “ 
T r 3 _ ^ 
° C 7 — 2 J ... 
7 Tpftbc c 
.(40) 
in which 
n _ K 7 e Po y ~ 2 
irabc 
.(41) 
By addition of the 
corresponding sides of the three 
equations (36) to (38), we 
obtain 
2 _/i i r\ 
.(42) 
7 / ~~ ^(2+72+2)' 
abc Trp/ibc \a b c 1 
It is now clear that equations (38) to (40) together with (41) are simply the 
equations which determine a, b, c , the semi-axes of the ellipsoid which is a figure of 
equilibrium for an incompressible mass. We have, however, found that as far as 
this first approximation the value of p is not necessarily constant throughout the 
ellipsoid ; it is given by 
P = P«-(po-v) ( — + r- 3 + ~ 2 J .(43) 
We have further found the relation connecting 0 with the constants k and y. 
Substituting the value for 0 from equation (41) into equation (32), and equating the 
* ‘ Phil. Trans.,’ A, vol. 215, p. 50. 
