214 ' MR.. G. H. BRYAN ON THE WAVES ON A ROTATING 
The height of the forced tides is therefore the same as we should get by the 
“ equilibrium theory,” ie., by neglecting the small relative motions of the fluid 
particles entirely. In fact, these relative motions have no effect on the height of the 
tides. It does not follow that they do not exist; in fact, it is evident, on the contrary, 
since the tides move relatively to the liquid, that they must exist. But on referring 
to equations (34) we see that, when k = 1 , the small relative velocity components 
U, V, W may be finite, even though i// vanishes. 
The zonal oscillations are, however, given by 
.(95), 
since = 1, and therefore = 1, T>Vn{v) =■ ^n{n 1). For these oscillations 
xjj does not vanish. 
24. Another interesting application is to determine the height of the permanent 
corrugations produced by disturbing forces which remain constant and fixed relatively 
to the rotating liquid. We now have to take k=0; therefore, v = 0 and v/k:= cos a. 
If s is different from zero, then, whether n — s be odd or even, equation (90) 
gives us 
3(My/c)a/K,/(0 = W,/.. . (94), 
and (92) gives A,/T,/(i^) = 0, whence [xfj] = 0 ; and therefore xfj is everywhere zero. 
If 5 = 0 and n is odd, then P„ (0) = 0, DP,, (0) is finite, and, as before, we find 
3(My/c)aK4C) = W„ .(94 a), 
and 
i// = 0. 
Lastly, let s = 0 and let n be even. This is the case of a harmonic disturbance 
which is symmetrical both about the axis of the spheroid and also with respect to 
its equatorial plane. Then, as in §16, 4i^P„ (i^)/DP,, (i^) approaches the finite limit 
— 4/{n(?i + 1)}, when v is diminished indefinitely, and therefore 
c 
a 
K«(^) 
4g.(r) 1 
n {n + 1)J 
= w. 
(96), 
12 
M7 
n {n + 1) 
(97). 
In the first two cases the height of the corrugations is given by the “ equilibrium 
theory,” and, since i/; = 0, it follows from equations (34) that U, V, W are all zero. 
Thus, the fluid continues to rotate as if rigid in a form differing slightly from the 
