418 
Proceedings of Royal Society of Edinburgh. [sess. 
example, f(t — q) = 1. This makes II the same for all values of t ; 
and (173) becomes 
n= -Y(x, 1 , 0) (176); 
and by integration (175) becomes 
£(^M) = y[v(a,M)- V(a,z,°)] . . (177). 
Putting now in this z — 1, and using (176), we find 
1 _ 
9 L 
V(*,M)-V(*,1,0) 
1 
J y 
Y(aj,l,#) + n 
(178). 
The interpretation of this, as t increases from 0 to oo , is that 
the sudden application and continued maintenance of a pressure 
— V(a:,l,0) over the whole fluid surface, initially plane and 
level, produces a depression, £, which gradually increases from 
0, at t = 0, to its hydrostatic value II jg t at t= oo . The gradual 
subsidence of the difference from the static condition, as time 
advances from 0 to go , is illustrated by the diagrams of fig. 34, 
for the case in which we choose for Y(x , 1 , 0) the i f/(x , 1 , 0) of 
.§§ 100-104 above. 
§ 131. To understand thoroughly the meaning of Y(x,z, q) as 
defined in § 127 ; remark first that it is the velocity-potential of a 
possible motion of water, under the influence of gravity, with no 
surface-pressure, or with merely a pressure uniform over its infinite 
free surface. This is equivalent to saying that Y(x,z,q) fulfils 
the equations 
d 2 Y d 2 Y ^ , dY d 2 Y 
dx 2 + dz 2 ’ an ^ dz dq 2 
. . (179). 
Secondly, remark that at the instant q = 0, there is no surface 
displacement; hence Y(x , z , q) is the velocity-potential at time q, 
due to an instantaneous impulsive pressure, — Y(x, 1 , 0), applied 
to the surface of the fluid at rest and in equilibrium, at time ^ = 0. 
iSow, allowing negative values of q, think of a state of motion from 
which our actual condition of no displacement, and of velocity- 
potential equal to Y(x , z , 0), would be reached and passed through 
when q passes from negative to positive. It is clear that the 
