LIQUID SPHEROID OF FINITE ELLIPTICITT. 207 
of the corresponding time of oscillation in a non-rotating' spherical mass of liquid of 
the same density. 
Sectorial Harmonic Waves. 
18. When 5 = u, (a) is numerical, and D'’(t-) is zero; thus, the period- 
equation reduces to 
4/c (k — 1) = {^)/(/3 {Q .(73), 
of which the roots are given by 
= i [1 ± ^/(l + ?2'K/(C)/2'3 (^))}.(74). 
The condition that these roots may be real is that 
23(0 (^) > 0 .(^5)5 
that is 
p, (i) 2 , (i) -1,!‘ (0 %!■ (0 - \ (0 ?I (0 - (C) “ 1 ' (0} 
must be positive. 
These results have been obtained previously by Poincare in the special case in 
which n = 2, but in his investigation an extraneous factor has been introduced into 
the period-equation, giving a third root [k — l) which does not properly belong to it. 
The expression for xjj in (21) is here projDortional to 
(1 _ + 
that is, in Cartesian coordinates, to 
{x + ty)" 
and is independent of 2. 
Thus, the motion of the liquid is “ two dimensional,” and takes place in planes 
parallel to the equatorial plane of the spheroid. By the laws of vortex motion the 
molecular rotation or spin of the actual motion of the liquid is therefore everywhere 
constant and equal to w, being that due to the rotation of the liquid. In other wmrds, 
the wave motion of the liquid relative to the rotating axes is irrotational. 
Small Free Precession of the Sj^heroicl. 
19. Another case of some interest is when the harmonics determining the small 
periodic relative motions are of the second degree and first rank. Putting 7i= 2, 5= 1, 
the period-equation (66) reduces to 
4/c^ (k — 1) (cot a) — [(/c — 1) sec" a k] (cot a) = 0 . . (76). 
