94 
Proceedings of the Royal Society 
constant, and that the difference of pressures at any two points in 
the fluid is equal to g x difference of levels, we have 
dp dli 
dx~y^ dx 
dp dli 
dy~9^ dy 
Hence for the case of gravitational oscillations (1) become 
du dh 
dt v ~ ® dx 
(3) 
dv 
dt 
+ 2 u>u 
dh 
g dy 
W 
From (1) or (4) we find by differentiation, &c. 
d /dv du\ /du dv\ 
dt\dx dy) + \dx + dy) ~ 
( 5 ) 
which is the equation of vortex motion in the circumstances. 
These equations reduced to polar coordinates, with the following 
notation,— 
x = r cos 0 , y = r sin 0 
u = £ cos 6 t sin 6 , v = £ sin 6 + r cos 0 , 
become 
DC rf(D£) d(Dr) 
r + dr + rdO 
dh 
dt 
dt 
dt 
_ dh 
2mt= 
dr n 
dt + 2 “t= 
dh_ 
rdd 
( 2 ') 
(*) 
d 
dt 
(r dr d£\ /£ d£ dr\ „ /r ,, x 
Vr + dr~7de) +2 “ \V + dr + 7de) = 0 ' < 5 > 
In these equations D may be any function of the coordinates. Cases 
of special interest in connection with Laplace’s tidal equations are 
had by supposing D to be a function of r alone. For the present, 
