ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
247 
{u — oiy) = 0 , 
— X^y + ww — 0. 
These equations will be satisfied if X = 0, = — — = const. It follows that the 
O) 
system is capable of a small free steady motion relative to the rotating axes, defined 
by 
y ~ const, u — — m^yjo). 
If both m^y are zero, the equations for the free motions become 
— X% — oiiky = 0, 
— \^y fi- o}i\x = 0 ; 
both of which are satisfied by supposing that x, y are small arbitrary constants, and 
therefore X = 0. 
In the latter case the equilibrium-state defined by a; = 0, y = 0 is not the only 
condition of relative equilibrium, but any other configuration of the system in the 
neighbourhood of this one will also form a configuration of relative equilibrium. 
Let us now consider the nature of the limiting forms of the forced oscillations 
when the period is indefinitely prolonged. In the former case we must suppose that 
the disturbing forces are such that they do no work when the coordinate x is varied, 
so that X = 0, as otherwise the stability of the system will be destroyed ; the 
equations of motion for the forced oscillations then become 
iku — Oiiky — 0, 
— X^y -j- (Oil -b wry = Y ; 
II z= coy = 0 , / . 2 , 
a-" + ( III — X ) 
_ Yo) _ Y 
-t- 'lli^ + co" 
The velocity-component tt will therefore always remain of the same order as the 
disturbing force Y, while the amplitude of vibration ol‘ the coordinate x will tend to 
increase without limit. 
In the latter case the equations of disturbed motion may be written 
whence 
and in the limit 
iku — oiv = X. 
iXv -f- oiU = Y 
