1905 - 6 .] Lord Kelvin on the G-rowth of a Train of Waves. 421 
velocity-potential at, and the vertical component displacement of, 
any point of the fluid at any time, become 
§ 136. The illustrations in figs. 36, 37, 38 are time-curves in 
which the ordinates have been calculated hy continuous quad- 
rature from one or other of the four formulas (189), (190). 
§ 137. The curves - in fig. 39, being space curves in which the 
ordinates are vertical component displacements of the water- 
surface, are therefore pictures of the water-surface (greatly 
exaggerated in respect to slopes of course), and may he shortly 
named water-surface curves. Their ordinates have been calculated 
by an analytical nrethod described in § 151 below. They cannot 
be calculated continuously for successive values of x by the 
method of continuous quadratures ; if that were the method 
employed, the value of the ordinate for each value of x would 
to the particular value of t for which the water-surface is repre- 
sented by the curve. The values of t chosen for fig. 39 are 
respectively it, (i + 1/S)t, (^ + 2/8)t, (z'+3/8)t, (^ + 4/8)r, where 
i is any very large integer, and r denotes 2 77-/00, the period of the 
In all our illustrations we have taken w = which makes 
r = 2 Jtt, and, with g == 4 as in § 105, makes the wave-length X — 8. 
§138. In figs. 36 and 37, all the curves Correspond to 
cos io(t - q) in ' the formulas. In fig. 38, all the curves corre- 
spond to sin o)(t — q) in the formulas. 
S = dq a>(t-q)<l>(x,z,q)-, 
need to be calculated by an independent quadrature ^ 
varying surface-pressure to which the fluid motion considered 
is due. 
