206 
MPw G. H. BRYAN ON THE WAVES ON A ROTATING 
as the ordinary spherical harmonics, while qn{C), {i) can be expressed in finite 
terms of cot~^ ^ i.e., of a ; hence, the function can be calculated for any 
value of By Horner’s method we may then approximate to the values of the 
roots of the equation in k in the simpler cases. The j^eriods of the waves are the 
corresponding values of tt/zcw, ^vhile <y is expressed in terms of p by equation (50).t 
To obtain some idea of the relative frequencies of the various waves, I have 
tabulated the values of k thus calculated for harmonics of the second, third, and 
fourth degrees for a spheroid in which ^ — 1 or a = 77 / 4 , the eccentricity being, there¬ 
fore, \ ^2. The results are embodied in the accompanying Table. As already 
stated, the positive roots correspond to waves rotating more slowly than the liquid, or 
relatively in the direction opposite to that of rotation of the mass, while those having 
the double sign correspond to symmetrical oscillations of the liquid. 
Tables of the Values of 
K 
for Waves on a Spheroid whose Eccentricity 
. TT 
sm — 
4 
2 
Rank of Harmonic. 
I. Harmonics of the Second Degree. 
2 (sectorial) 
1-2126108, 
- 0-2126108. 
1 
1-280776, 
- 0-780776. 
0 (zonal) 
+ 1-128465. 
Oscillatory waves 
11. Harmonics of the Third Degree. 
3 (sectorial) 
1-569830, 
- 0-569830. 
2 
1-677377, 
0-423263, 
- 1-100640. 
1 
1-6008928, 
0-8267846, 
- 0-0733654, - 1-3543120. 
0 (zonal) 
Oscillatory waves 
+ 1-5178954, 
+ 0-5122368. 
III. Harmonics of the Fourth Degree. 
4 (sectorial) 
1-852560, 
- 0-852560. 
3 
1-806374, 
0-366650, 
- 1-173024. 
2 
1-921878, 
0-730910, 
- 0-107365, - 1-545423. 
1 
1-924662, 
0-890193, 
0-585670, - 0-654623, - 1-745902. 
0 (zonal) 
+ 1-994751, 
+ 0-685895. 
I find that the period of the symmetrical or zonal harmonic oscillation of the 
second degree (in which the surface remains spheroidal) is, in this spheroid, 0’8599258 
* “ On the Expression of Spherical Harmonics of the Second Kind in a Finite Form,” ‘ Cambridge 
Philosophical Proceedings,’ December, 1888. 
t See Thomson and Tait’s ‘ Natural Philosophy,’ yol. 2, § 772. 
