1905 - 6 .] Lord Kelvin on the Growth of a Train of Waves. 417 
time, we translate the formula (170) into the language of the 
integral calculus as follows : 
S (x,z,t)=f dqj(t-q)Y(x,z,q) . . . (171), 
*'o 
where f(t-q ) denotes an arbitrary function of (t - q), according to 
which the surface-pressure, arbitrarily applied at time (t — q) y 
is as follows : 
n(t-q)= -f(t-q)V{x,l,Q) . . . (172). 
Hence the pressure applied to the surface at time t , denoted by 
n(ir, 1, t), is as follows : 
n(a,i,*H* 1,0). 
(173). 
§ 129. The solution (170) or (171) gives the velocity-potential 
throughout the liquid which follows determinately from the 
dynamical data described in §§ 127, 128. From it, by differentia- 
tions with reference to x and z, and integrations with respect to t , 
we can find the displacement components £, £ of any particle of 
the liquid whose co-ordinates were x , 2 when the fluid was given 
at rest. But we can find them more directly, and with consider- 
ably less complication of integral signs, by direct application of 
the same plan of summing as that used in (170), (171). Thus 
if, instead of Y (x , z , q) in (171), we substitute -^-Y(x,z,q), 
ax 
and again — -Y(# , z , q), we find £ and £. And if we take 
az 
J d q^V(x,z,q) and ) dq^-V(x,z,q) . . (174) 
in place of Y(x y z,q) in (171), we find the two components £, £ 
of the displacement of any particle of the fluid. Confining our 
attention to vertical displacements, and using (179) below, we 
thus find 
£(x,z,t) = -[ dqf(t-q)~Y(x,z,q) ■ ■ (175). 
( JJo . d 1 
§ 130. To illustrate the meaning of the notation and analytical 
expressions in (171), (173), (175), take the simplest possible 
PKOC. ROY. SOC. EDIN. — VOL. XXVI. 27 
