1905-6.] Lord Kelvin on the Growth of a Train of Waves. 429- 
be described as components of the first and third, according to the 
following formula : 
£(x , 1 , t) = P sin wt - Q cos wt . . . (193), 
where P= - A cos (194), 
and Q is a continuous transcendental function of x, having equal 
values for ±.r, expressed by (195) for positive or negative values 
of x, exceeding a wave-length. 
For x positive, Q = - A sin 2-irx/X ; for x negative, Q = + A sin 2ttx/X (195), 
where A denotes the semi-amplitude of the vibration, at any time 
long enough after the beginning, and place far enough from the 
middle of the disturbance, to have very approximately sinusoidal 
motion. The determination of the transcendental function Q, and 
the calculation of A, for both P and Q, will be virtually worked 
out in § 151 below. 
§ 146. We have now an exceedingly interesting and suggestive 
analysis of the circumstances represented in fig. 39. Consider 
separately the two motions corresponding to P sin wt alone, and to 
- Q cos wt alone. The motion P sin wt , if at any instant given 
from x = • — oc to x — 4- oo , would continue for ever, as an infinite 
series of standing waves, without any surface-pressure. Hence 
our application of surface-pressure is only required for the Q- 
motion : and if this motion be at any instant given from x= - oc 
to x = + oc , it will go on for ever, provided the pressure 
— cos wt , 1 , 0) is applied and kept applied to the surface. 
§147. The plan of § 1 46 may be generalised as follows: — 
Displace the water according to the formula (193) with P omitted, 
and with Q any arbitrary function of x for moderately great 
positive or negative values of x, gradually changing into the 
formula (195) for positive and negative values outside any 
arbitrarily chosen length MON (MO not necessarily equal to 0 H). 
Find mathematically the sinusoidally varying surface-pressure, 
F(.e) cos wt , required to cause the motion to continue according to 
this law. Superimpose, upon the motion thus guided by surface- 
pressure, the motion -A cos 27 t^/A sin (o^, which needs no surface- 
pressure. In the motion thus compounded, we have equal 
