210 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
allow of no other displacements (such constraints are, of course, purely theoretical). 
The spheroid, if at all viscous, will be secularly stable or unstable according as 
K,/(C) >0 or < 0, 
and we have seen that the latter condition can only hold if n — s be even, that is, if 
the displacement be one symmetrical with respect to the equatorial plane, The 
critical form is that in which 
K,/(0 - 0. 
But since, when K,/{^) changes sign, the roots of the period-equation at first 
continue real, the limits of eccentricity for wdiich the perfect spheroid is “ ordinarily ” 
stable are in every case wider than those consistent with secular stability if the liquid 
be viscid. The critical form is determined by the condition that the period-equation 
must have a pair of equal roots. 
In my paper on “ The Waves on a Viscous Rotating Cylinder 1 have endeavoured 
to further elucidate the difference between “ ordinary ” and “ secular ” stability. 
Assuming the displacement from relative equilibrium to be proportional to 
is always complex for viscous liquid, and the condition that the disturbance may 
not increase with the time is that the real part of must be positive. Both 
real and imaginary parts change sign when = 0, the corrugations becoming 
relatively fixed and the liquid figure becoming a form of “bifurcation.” Relative 
equilibrium is then critical. But, if there be no viscosity, may be purely 
imaginary, as in the present case, when k is real, and the waves will neither 
increase nor diminish in amplitude with the time; thus, a change in’ the sign of 
one of the roots of the period-equation merely implies a change in the relative 
direction of the wave. 
Moreover, it appears that the criteria of ordinary and secular stability will be 
different only if the angular velocities of the waves be different in the two opposite 
directions, and this can only be the case if the liquid be rotating. 
Reverting to the perfect liquid spheroid, the determination of the greatest 
eccentricity consistent with ordinary stability involves the question, if ^ be gradu¬ 
ally diminished, what is the harmonic displacement for which the period-equation of 
the waves first commences to have complex roots ? It appears probable that this 
happens for = 2, s = 2. With this assumption, we see, by (75), that the critical 
value of ^ is given by the equation 
2 { p . (i)qAO- (1) ’V (01 + (0 A (0 - ft (0 (0 =«; 
this leads to 
(0^4 _P 8^3 _{_ 1) cot-1 i - (3^3 ^ ^ 0, 
* ‘ Cambi’idge Philosophical Proceedings,’ 1888. 
