1905 - 6 .] Lord Kelvin on the Growth of a Train of Waves. 419 
values of Y (x , z , q) are equal for equal positive and negative 
values of q. Hence, 
when q = 0, we have f-Y(x , z > 0) = 0 
dq 
. (180). 
§ 132. Consideration of the V (x , z , q), defined in § 127, which 
allows Y(x, 1 , 0) to he any arbitrary function of x, but requires 
dY/dq to be zero when ^ = 0, suggests an allied hydrokinetic 
problem: — to find W fulfilling (179) with W in place of Y ; and, 
at time q = 0, having W = 0 and dW/dq any arbitrary function 
of x. We assume, as is convenient for our present purpose, that 
for large values of x 
Y(x , z , 0) '= 0, and W(cr , z , 0) = 0 . . . (181). 
This implies that for all values of x and z, large or small, 
but for large values of q, 
Y(x , z , q) = 0, and W(a; , z , q) = 0 . . . (182), 
§ 133. In the Y-problem the initiational condition is: — displace- 
ment zero and initiational velocity virtually given throughout the 
fluid as the determinate result of an arbitrarily distributed im- 
pulsive pressure on the surface. 
In the W-problem the initiational condition is : — the fluid held 
at rest with its surface kept to any arbitrarily prescribed shape by 
fluid pressure, and then left free by sudden and permanent annul- 
ment of this pressure. 
Without going into the question of a complete solution of this 
(Y, W) problem for any arbitrary initiational data, we find 
a class of thoroughly convenient solutions in a formula origin- 
ally given in the Proceedings of the Royal Society of Edinburgh , 
January 1887 ; republished in the Phil. Mag., February 1887 ; 
and used in § 3 and § 99 above. We may now write that 
formula in the following comprehensive realised expression 
for Y or W :— 
{KS} or {RD}yT h ’ + * 
-gv 
4 ( 2 + 1 *) 
dfdx^z* J(z + lx) 
= Y(x,z, t ), when i is even ; 
= W (x,z , t), when i is odd. 
(183). 
