LIQUID SPHEROID OF FINITE ELLIPTICITY. 
201 
This equation is of the degree n — 5 + 2, and involves both odd and even powers 
of K. It therefore has n — s + 2 roots, but in the present case these I’oots do not 
occur in pairs of equal and opposite values. 
Equations (62), (64), (66) are the period-equations of the various free harmonic 
waves or oscillations of the liquid spheroid. Their roots depend on the value of a or 
the eccentricity (sin a) alone. The periods of the waves are the corresponding values 
of Tr/oiK and depend also on w. 
Nature of the Real Oscillations and Waves. 
12. The periodic movements determined by zonal harmonics (s = 0) and those 
determined by tesseral harmonics differ in character considerably. 
The former are symmetrical about the axis. Taking the solution 
i/j = A«P„ (f) P,, (ly) 
h = Q,^ P„ W) 
another solution got by changing the sign of k is given by 
i/; = A„P,, (f) P„ (r) e “ 
h = C„ ter P„ (/X.) e ~ '‘“''h 
Compounding these, we get the real motions of the liquid determined by 
xfj = A„P,i (f) P„ (v) sin (2ojKt — €;,) 
h = (fi) sin (2coKt — e„) 
€« being any constant. 
These are stationary oscillations of the liquid about the spheroidal form. By what 
has already been shown, there are either \n or ^ [n + 1) such free oscillatious, accord¬ 
ing as n is even or odd. In all of these oscillations the expression for h is of the same 
form, that is, the corrugations produced on the surface are similar in each. But this 
will not be the case with the values of xfj, because the auxiliary systems of spheroidal 
coordinates to which they are referred are different for each different value of k. 
Thus, the motions of the fluid particles in the interior of the mass are different for 
each of the oscillations. 
13. Taking next the case when s is different from zero, let us change the sign of 
\/ — 1 everyivhere that it occurs in our investigations. The results will still hold 
good when this is done. Hence, for every root of (66) we get two solutions of the 
equations of oscillation, giving respectively 
MDCCCLXXXIX.—A. 2 D 
