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 dh 
dx ~ ^ dx 
dp dh 
dy~9^ dy 
Hence for the case of gravitational oscillations (1) become 
du 
"77 - 2<JiV = 
dt 
dv 
dt +2o>w = 
dh 
9 dx 
dh 
9 dy 
w 
From (1) or (4) we find by differentiation, &c. 
d fdv du\ (du dv 
dt\dx dy) w 
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 6 , y — r sin 0 
u = £ cos Or sin 9 , v — £ sin 6 + r cos 0 , 
become 
D£ d(DJ}_ d(Dr) 
r + dr + rdO 
dh 
dt 
^- 2wt = 
dt 
dr 0 
S + 3 “C“ 
dt 
d 
dt 
dh 
d£ 
dh 
rdO 
( 2 ') 
( 4 ') 
(t dr d£\ (l d£ dr\ . 
\r+dr~rdd) + (r + Jr + rdd) = () ' ( 5 ) 
In these equations D may he 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, 
