206 
MR. S. S. HOUGH OH THE APPLICATTOH OF HARMOHIC 
to supply in another paper,''' ljut mathematical difficulties have compelled me, in 
treating of friction, to omit from consideration the important influences due to the 
rotation. We have already called attention to the fact that the free steady motions 
on a non-rotating globe are far less restricted in character than those on a rotating 
globe, while, in that the latter essentially violate vdiat appears to be a necessary 
condition when the water is viscous, namely, that there can be no slipping at the 
bottom, it seeins to me to be probable that even the limited forms of steady motion 
here dealt with would be no longer possible if the water were viscous, but that, if 
they were started by any means, they would at once give place to periodic motions 
of comparatively short period.t This conclusion has been forced on me by the 
apparent impossibility of satisfying the equations of motion of a viscous ocean on a 
rotating globe by means of slowly declining current-motions. If such should be the 
case, it follows that no stable currents can exist without variations in the density of 
the water. As however I have not a.s yet been able to support this vnew by 
anything approaching a rigorous mathematical treatment, the cjuestion must for the 
present remain open. 
^ 1. Di fferen tial Equation!^ for the Vihration of a Rotating Mass (f Liquid. 
Suppo.se we are dealing with the small oscillations of a mass of liquid about a state 
of steady motion consisting of a rotation as a rigid body with angular velocity co about 
a certain axis. 
Take this axis as axis of 2 , and refer to a set of rectangular axes rotating about it 
with uniform angular velocity w. Then, in the steady motion supposed, the fluid will 
have no motion relatively to these axes. 
Let It, V, w denote the relative velocity-components at the point x, y, z due to the 
small oscillations. The actual velocity-components parallel to the instantaneous 
positions of the moving axes will then be 
u ■— (oy, V -f- (jiX, w, 
and, therefore, if we suppose the amplitude of the vibrations sufficiently small to allow 
of our neglecting squares and products of the small quantities u, v, u\ the differential 
equations of motion of the liquid may be written in the form| 
* Loc. cit., ante. 
t The condition that there can be no slipping at the bottom Avill reduce the number of degrees of 
freedom of the system, and hence we may anticipate that certain tyjics of motion which were possible 
before this condition was imjjosed will no longer exist afterwards. 
J Basset, ‘ Hydrodynamics,’ vol. 1, p. 22. 
