212 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
The frequencies of these waves or oscillations are proportional to the angular 
velocity of the liquid. As the' latter is diminished without limit they become 
relatively unimportant, and finally cease to exist, for the limiting case of a mass of 
liquid without rotation oscillating about the spherical form. 
Secondly, suppose the periods of the waves remain finite in the limit. Put 2o)k = X, 
so that 27r/X is the period. Since X remains finite, k will increase without limit as w 
diminishes, and, therefore, equation (66) gives, to the first order of small quantities, 
2^-^\{\-2co) 2(n-l)^ 2(71-1), .X-2(o 
~15 3 (271 + 1) ^ ~ 3 (271 + 1) \ ~ 
whence, by (82), 
X (X — 2o)) = 
871 (71 — 1 ) 
2,(271 + 1) 
1 
(85). 
If we put w = 0, we get 
X3 = 
871 (71 — 1) 
3 (2» + 1) 
( 86 ). 
the well-known result for the oscillations of a liquid sphere. Denoting by the 
expression 
8 71(71 — 1 ) 
3 (2n + 1) 
we find 
X3 - = I" f X3 - , 
whence, substituting X = i A in the small terms, we get 
X=±A-f-w .(87). 
71 
Remembering the expressions found in § 13 for the relative and actual angular 
velocities of the corresponding waves, this result may be stated as follows :—The effect 
of communicating a small angular velocity w to a spherical mass of gravitating liquid 
will be to add an angular velocity (n — 1) co/n to the angular velocities of all the free 
waves which are determined by harmonics of degree n. 
The symmetrical oscillations vdll be unaffected by rotation to this order of approxi¬ 
mation. If we proceed to a higher approximation by taking into account small 
quantities of the second order, the equations become much more complicated. But, for 
a spheroid similar to the Earth, the above approximation would be practically sufficient. 
