218 
MR. G. H. BRYAN ON THE WAVES ON A ROTATING 
viscosity, the changes clue to secular instability will take place very slowly, but the 
effect of the tide generating force due to an attracting body when the angular velocity 
of one of the free tides coincides with that of the body will become very great. If, 
however, the viscosity be considerable, it will prevent the forced tides from becoming 
large, and will cause the lic[uid to rapidly assume the form of a Jacobian ellipsoid, 
owing to the spheroidal form lieing secularly unstable. 
Conclusion. 
28. The results of the present paper suggest several considerations, which may 
possibly throw further light on the past history of the Solar system. The criteria of 
stability applicable to the two cases where the spheroid is formed of perfect and of 
viscous liquid respectively have been already discussed in § 20. In the last para¬ 
graph I have alluded to the possibility that ec|uilibrium in the spheroidal form may be 
broken up by an attracting body which causes the harmonic tides of the second order 
to increase indefinitely, in accordance with Professor Darwin’s hypothesis. 
The present analysis, however, suggests that the same thing may happen in the 
case of harmonic tides of higher order than the second, and, moreover, the results 
arrived at concerning the number and situation of the roots of the frequency- 
equation rendei-.this hypothesis quite admissible. Except for harmonics of the second 
order, many of the free tides will rotate in the same direction as the liquid, but with 
less angular velocity, even though the spheroid be secularly stable;, and, if an 
attracting body should be rotating about the spheroid with the same angular velocity 
as one of these tides, they would certainly rise to an enormous height, and the liquid 
might, perhaps, ultimately be broken up into two or more detached masses. 
Take, for example, the sectorial harmonic waves of order n. If one of these be 
fixed in space, we must, from § 13, have the corresponding value of /c = n, and, by 
(73), this leads to the condition 
K /(0 = 0 ^ - 2 ) 
If n is greater than 2, this will be satisfied for some secularly stable form of the 
spheroid. Under these circumstances, the presence of a fixed attracting body near 
the spheroid would cause the sectorial harmonic waves of the vf*’ order to increase 
indefinitely. The only obstacle in the way of the present supposition is that when 
the distance of the attracting body is at all considerable, the harmonic components of 
tide generating force of the higher orders become very small in comparison. 
Another question of astronomical interest is whether a rotating spheroid can be 
broken up into one or more rings of rotating liquid. This can only happen if one of 
the zonal harmonic oscillations increase indefinite!}^ or become unstable. 
