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VI. The Waves on a Rotating Liquid Spheroid of Finite Ellipticity. 
G. H. Bryan, B.A. 
Communicated hy Professor G. H, Darwin, F.R.S. 
Received November 6,—Read November 22, 1888. 
1. The hydrodynaraical problem of finding the waves or oscillations on a gravitating 
mass of liquid which, when undisturbed, is rotating as if rigid with finite angular 
velocity, in the form of an ellipsoid or spheroid, was first successfully attacked by 
/ 
M. Poincare in 1885. In his important memoir, “ Sar TEquilibre d une Masse Fluide 
animee d’un Mouvement de Potation,” * Poincare has (§ 13) obtained the differential 
equations for the oscillations of rotating liquid, and shown that, by a transformation 
of projection, the determination of the oscillations of any particular period is reducible 
to finding a suitable solution of Laplace’s equation. He then applies Lame’s 
functions to the case of the ellipsoid, showing that the differential equations are 
satisfied by a series of Lame’s functions referred to a certain auxiliary ellipsoid, the 
boundary-conditions, however, involving ellipsoidal harmonics, referred to both the 
auxiliary and actual fluid ellipsoid. At the same time, Poincare’s analysis does not 
appear to admit of any definite conclusions being formed as to the nature and 
frequencies of the various periodic free waves. 
The present paper contains an application of Poincare’s methods to the simpler 
case when the fluid ellipsoid is one of revolution (Maclaurin’s spheroid). The 
solution is effected by the use of the ordinary tesseral or zonal harmonics applicable 
to the fluid spheroid and to the auxiliary spheroid required in solving the differential 
equation. The problem is thus freed from the difficulties attending the use of Lame’s 
functions, and is further simplified by the fact that each independent solution contains 
harmonics of only one particular degree and rank. 
By substituting in the conditions to be satisfied at the surface of the spheroid we 
arrive at a single boundary-equation. If we are treating the forced tides due to a 
known periodic disturbing force, this equation determines their amplitude and, hence, 
the elevation of the tide above the mean surface of the spheroid at any point at any 
time. If there be no disturbing force, it determines the frequencies of the various 
free waves determined by harmonics of given order and rank. Denoting by k the 
ratio of the frequency of the free waves to twice the frequency of rotation of the 
* ‘ Acta Mathematica,’ vol. 7. 
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23.3.89 
