ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
207 
0(6 
'bt 
— (i) [v cox) — COV 
^ 4- w (w — (Olf) + 
dt 
dio 
ct 
ojU 
dx 
^!J 
(y'-pip), 
(V'-pIp), 
^C^'-pIp)> 
where V deiiotes the potential of the bodily forces acting’ on the licpiid, p the fluid 
pressure, and p the density. 
If now we put 
xjj = Y' — p/p d- ico'^ (x^ + ?/) -f const.(1), 
the above equations reduce to 
0(4 
di 
do 
dt 
dio 
dt 
— 2(ov = 
dp 
dx 
dp . 
dp ^ 
dp 
03 
while the incompressibility of the liquid is expressed by the additional equation 
dv d/'j 
dij 03 
= 0 
(3). 
These equations, originally given by Poincare,"^' sufiice, in conjunction with 
certain conditions which must hold at the boundary, for the determination of the 
four functions u, v, iv, p. They are perhaps the simplest equations of which to 
make use when dealing with the oscillations of a mass of liquid of finite extent in 
three dimensions, and, for this purpose, they were first solved by Poincare in a 
form adapted for satisfying boundary-conditions at an ellipsoidal surface, while 
additional applications have been considered by Bryan,! Love,| and myself § The 
possibility of solution in each of these cases however turns on the fact that it only 
required to satisfy boundary-conditions at a single ellipsoidal or spheroidad surface, 
whereas, in the problem presented b}^ the tides, it is necessary to satisfy conditions 
at two surfaces, namely, the ocean bed and the free surface of the ocean. 
There is, however, a feature attached to this problem which enables us to surmount 
* ‘Acta Mathercatica,’ vol. 7, ]). 356. 
t ‘ Phil. Trans.,’ 1889. 
X ‘ Proc. Lou. Math. Soc ,’ vol. 19. 
§ ‘Phil. Trans.,’ 1895. 
