Ill 
i 889-90.] Prof. Tait on Ripples in a Viscous Liquid. 
where e is the density of the liquid, and P the potential energy of 
unit mass at p ; and the double integral is taken over the surface 
of the element. This is a perfectly general equation, so we must 
proceed to the necessary limitations. 
First. Let the displacements be so small that their squares may 
be neglected. Then we may write d for 8. 
Second. Let the liquid be incompressible ; then 
SV^ = 0 (2). 
With these, the equation of motion becomes 
e W= -V(p + eP)-y.Vl- (3). 
Third. Let the motion be parallel to one plane, and we have 
S&cr = 0 (4). 
From (2) and (4) we have at once 
cr = \/w.k ( 5 ) 
where w is a scalar function of Vkp. 
Operate on (3) by V. V? and. substitute from (5), and we have 
( e ^ + f*V 2 ) v2 «’ = ° (6). 
Fourth. Limit w to disturbances which diminish rapidly with 
depth. Here the problem has so far lost its generality that it is 
advisable to employ Cartesian coordinates, the axis of x ( i ) being in 
the direction of wave motion, and that of y (j) vertically upwards. 
Then it is clear that a particular integral of (6) is 
w = (Kt ry + Bs S2/ ) s (rx+nt)l (7) 
where i denotes J -l. The only conditions imposed on r , s , and 
n, are that the real parts of r and s, in so far as they multiply y, 
must be positive ; and by (6) 
p,(s 2 -r 2 ) = em (8). 
The speed of vertical displacement of the surface is found from 
V= -(S;V) 0 = (S^)o= -n(A + B)6^)‘, . . (9). 
