1887 .] Sir W. Thomson on Procession of Waves. 39 
surface, the other to the bottom. The latter, for our present case 
of infinitely deep water, is simply 
P •= 0 when y = oo (5). 
To find the former, or upper-surface kinematical equation, at time 
i, let it be y = 0 at time 0, and let j) be the height at time t above 
the level y = 0, of the upper-surface particle whose coordinates at 
time 0 are (x, 0). Remembering that f y positive was taken as 
dowmvards, we have by (1), 
The most general upper-surface dynamical condition which can 
be imposed is 
P(y = 0) == f( X ’ t) 
where / denotes an arbitrary function of the two independent 
variables. 
Suppose now the water to be at rest at time 0. It is clear from 
dynamical considerations that the solution of (4), subject to the 
conditions (5), (7), (3), is fully determinate : and when it is found, 
(1) gives the position at time t of the fluid-particle which at time 0 
was in any position ( x , y) ; and so completes the solution of the 
problem. 
The particular solution which we are now going to work out to 
represent a uniform procession of waves commencing at time 0, and 
produced and maintained by the application of changing pressure to 
the surface in the neighbourhood of the zero of x, must, as its 
appropriate form of (7), fulfil the condition 
i ? (y=o) == $( x ) sin tot + F(a?) cos oit .... (8), 
where g ( x ) and F ( x ) denote functions which vanish for all large 
positive or negative values of x. 
If we wish to make only a single procession, in the direction of 
x positive for example, we may take 
g(*) = I (9)- 
A perfectly general formula is easily (by the Fourier-Poisson- 
Oauchy method) written down to express the value of P ; and so, 
