248 
MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 
whence in the limiting case 
u = Y/o), V = — X cu. 
Hence both the velocity-components tend to finite limits while the amphtudes of 
vibration of both coordinates increase without limit. 
The essential cbaracteristics of both cases are (i) that one or more of the generalized 
coordinates does not appear explicitly in the equations of motion but only the 
corresponding velocity-component, and hence (ii) that X = 0 is one root of the 
frequency-equation for the free modes of vibration, from which it follows that 
(a) either free steady motions relatively to the rotating system are possible, or 
(yS) that the configuration of relative equilibrium defined by x = 0, y = 0 is not 
isolated. The two conditions (a), (yS) may both be expressed by stating that the 
steady motion defined hy x = 0, y = 0 is not the only form of steady motion of which 
the system is capable. 
The two cases are both illustrated by our problem. For if we supjDose the waters 
of the ocean displaced horizontally in such a manner that the form of the surface is 
unaltered, we shall evidently obtain a new configuration of relative equilibrium, v’hile, 
as we shall see in the next section, if we suppose that the fluid is in relative motion 
in such a manner that the fluid particles are moving along parallels of latitude, it is 
possible by a proper adjustment of the free surface to ensure that such a motion should 
be permanent. 
The coordinates which depend on the horizontal displacements alone are analogous 
to the coordinate x in the former of the illustrations we have given above, and to the 
coordinates x, y in the latter. They do not a 2 :)pear explicitly in the equations of 
motion, but only through the corresponding velocity-components. We conclude, then, 
that the horizontal velocities will be of the order of the disturbing forces, whereas the 
horizontal excursions of the fluid particles will tend to increase without limit as the 
period is prolonged. 
By way of explaining how these circumstances may arise physically, let us suppose 
for the moment that X is actually zero, and consequently that the disturbing force is 
constant. In the case of a system oscillating about a position of equilibrium, the 
introduction of a constant disturbing force will have the effect of slightly changing 
the configuration about which oscillations corresponding with the free modes of 
vibration take place. Suppose now that a disturbing force, such as that v/hich gives 
rise to the long-period tides, tending to increase the surface-ellipticity of the ocean, 
is suddenly applied to our rotating system when in a configuration of relative equi¬ 
librium. It will immediately set up oscillations, the initial motion being such that 
each particle will tend towards the position in which it would be in equilibrium under 
the new disturbing influence. The new position of equilibrium is such that in it there 
will be more Avater in equatorial regions, and less water in polar regions, than in the 
old. Thus the initial motion involves a flow of water directed from the poles towards 
