

1888.] Waves on a Liquid Spheroid of finite Ellipticity. 43 



shown that by a transformation of projection, the determination of 

 the oscillations of any particular period is reducible to finding a suit- 

 able solution of Laplace's equation. 



He then applies Lame's functions to the case of the ellipsoid, show- 

 ing that the differential equations are satisfied by a series of Lame's 

 functions referred to a certain auxiliary ellipsoid ; the boundary con- 

 ditions, however, involving ellipsoidal harmonics referred to both the 

 auxiliary and actual fluid ellipsoids. 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 the auxiliary spheroid required in solving the differential equa- 

 tion. 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 liquid about its axis, the 

 values of K are the roots of a rational algebraic equation, and depend 

 only on the eccentricity of the spheroid as well as the degree and 

 rank of the harmonic, while the number of different free waves de- 

 pends on the degree of the equation in K. At any instant the height 

 of the disturbance at any point of the surface is proportional to the 

 corresponding surface harmonic on the spheroid multiplied by the 

 central perpendicular on the tangent plane, and is of the same form 

 for all waves determined by harmonics of any given degree and rank, 

 whatever be their frequency, but the motions of the fluid particles in 

 the interior will differ in nature in every case. 



Taking first the case of zonal harmonics of the nth degree, we find 

 that according as n is even or odd there will be ^n or ^(n -f- 1), 

 different periodic motions of the liquid. These are essentially oscil- 

 latory in character, and symmetrical about the axis of the spheroid. In 

 all but one of these the value of K is essentially less than unity, that is, 

 the period is greater than the time of a semi-revolution of the liquid. 



