430 Proceedings of Royal Society of Edinburgh, [sesb, 
sinusoidal waves travelling outwards in the two directions beyond 
M N (semi-amplitude A) : and, in the space M H, we have a 
varying water-surface found by superimposing on the motion 
P sin c ot an arbitrary shape of surface, varying sinusoidally 
according to the formula — Q cos a >t. 
§ 148. A curiously interesting dynamical consideration is now 
forced upon us, The P-component of motion needs, as we have 
seen, no surface-pressure. The Q-component of motion is kept 
correct by the surface-pressure F(a;) cos cot , which, in a period, does 
no total of work on the Q-motion ; but work must be done to supply 
energy for the two trains of waves travelling outwards in the two 
directions. Hence this work is done by the activity of the surface- 
pressure upon the P-component of the motion. 
§ 149. Another curious question is forced upon us. Our 
solution of §§ 135-145 has given us determinately and unambigu- 
ously, in every variety of the cases considered, the motion of 
every particle of the water throughout the space occupied. The 
synthetic method of quadratures which we have used could lead 
to no other motion at any instant due to the applied surface- 
pressure ; but now, in § 147, we have considered a Q-motion 
alone, kept correct by the applied surface-pressure. Would this 
motion be unstable? and, if unstable, would it in a sufficiently 
long time subside into the motion expressed in the determinate 
solution of §§ 135-145? The answer is Yes and Ho. At any 
instant, say at t = 0 , let the whole motion be the Q-component 
alone of § 148. Let now the surface-pressure, F(a?) cos c ot , be 
suddenly commenced and continued for ever after. It will, 
according to §§ 135-145, produce determinately a . certain 
compound motion (P , Q) which will be superimposed upon the 
motion existing at time t = 0 ; and this last-mentioned motion, 
given with its infinite amount of energy distributed from x='—cq 
to x = + oo , and left with no surface-pressure, would clearly never 
come approximately to quiescence, through any range of distance 
from 0 on the two sides. Thus we see that, though the Q- 
motion alone of § 148 is essentially unstable, the condition of the 
fluid does not subside into the determinate solution of §§ 135-145. 
It would so subside, if it were given initially only through any 
finite space however great, on each side of 0. In fact, any given 
