LIQUID SPHEROID OE FINITE ELLIPTICITT. 
211 
whence I find by trial and error that 
1/^ = tan a = / = 3-1414567 .... 
This result agrees with that found by Riemann, who treated the problem as a special 
case of the general motion of a liquid ellipsoid. 
It does not, however, seem possible to justify the above assumption as to the nature 
of the displacements by a perfectly general rigorous proof. The condition that the 
period-equation should have a pair of equal roots is far too complicated to allow 
of this point being fully proved in the way that Poincahe has done for secular 
stability. It is certain that the spheroid will be unstable for all values of tan a 
greater than 3-1414567 ; it is probable, but not certain, that it will be stable for all 
values less than this limit. 
Spheroids of Small Ellipticity. 
21 . If the eccentricity of the spheroid, and, therefore also, its angular velocity, be 
small, the ]3eriod-equations for the waves are much modified. The value of ^ will 
become very great, and we shall suppose it to be so great that is a small Cjuantity 
that can be neglected. Since a is small, we may put cos a = 1, sin ct = a = If. 
The function if) is proportional to since the other terms in it involve only 
and lower powers of C Hence, to this approximation. 
In {Vj f) — C'" [ + 3 — 2n + 1 
and 
K/(o = rM^- 
moreover. 
_ 2(n-l) 
2 ?i + 1 / 3 ( 2 n + 1) ^ 
whence, as in Thomson and Tait (§771), 
. . . . 
(79) , 
(80) ; 
( 81 ), 
(82). 
Firstly, suppose that the values of k remain finite in the limit. Then v — k 
ultimately, and, since i® negligible in comparison with K/(^), equation (65) gives 
6 -HP„(k) + {k- 1)D* + iP4k) = 0.(83), 
having n ~ s real roots between 1 and — 1. 
In the case of the oscillations symmetrical about the axis {s = 0) the equation for 
K is ultimately 
DP„ (k) = 0 or ^ DP„(k') = 0 .(84), 
according as n is odd or even. 
2 E 2 
