202 
MR. G[. H. BRYAN ON THE WAVES ON A ROTATING 
and also 
xjj = {fji') T.J (v) + 
A = C/cttT/> (u) e-‘G^ + 2-« 
which combine to give the real motions 
xjj = A;/ T,/®) (/a') T,/ {v) sin {s(j) + ’2o)Kt — C;/) 
h = C/ CT (/a) sin {s(ji + 2o}Kt — e/) 
(G8). 
These represent a system of waves travelling round the axis of the spheroid with 
relative angular velocity — 2(okIs. But it must be remembered that the coordinate 
axes to which we have referred the wave motion are themselves rotating with angular 
velocity oj. Hence, the angular velocity of the waves in space is &» (1 — 2k/s). 
According to our convention, positive values of k give waves rotating more slowly than 
the liquid, and vice versd. 
There are n — 5+2 such waves determmed by harmonics of degree n and rank s, 
and, since the values of k are not equal and opposite in pairs, these waves do not 
combing into oscillations fixed relatively to the moving axes. As in the symmetrical 
oscillations, the form of the corrugations is the same for all the waves, but the motion 
of the fluid particles different in each. 
14. If K^, k. 2 , . . ., /c«_i + 3 be the roots of (66), it is obvious that 
(69). 
"b ^.3 "b • • • + k,i_s + 2 — 1 
Hence, the mean relative angular velocity of all the different harmonic waves of 
degree n and rank s is 
(71 — s + 2) s’ 
in direction opposite to that of rotation of the liquid, whilst their mean actual angular 
velocity in space is 
Analysis of the Period-Equations. 
15. From Poincare’s investigations it appears that the spheroid will be secularly 
stable, even if the liquid be viscous, provided that the coefficients which are here 
denoted by K,/ ( ^q) or 
Ihii) (h{i) - {i) Unfl) 
