38 
Proceedings of Royal Society of Edinburgh. [jan. 7, 
Friday, 7th January 1887. 
Sin DOUGLAS MACLAGAX, Vice-President, in the Chair. 
At the request of the Council, Mr J. J. Coleman gave an 
Address on Processes of Refrigeration. 
The following Papers were laid on the Table : — 
1. On the Front and Rear of a Free Procession of Waves in 
Deep Water. By Sir William Thomson, F.R.S. 
Preliminary. 
General Problem of Deep-Sea Wave-Motion in two dimensions , 
(. Infinitesimal Motion.) 
Taking x horizontal, and y vertically downwards; let ( x + £, y + f) 
he, at time t , the position of the particle whose position at time 0 is 
(x, y) ; let $ denote the velocity-potential at (x, y, t) ; and let P 
denote its time-integral^^ dt&. 
We have 
J o dx 
dP 
dx 
; and 
V 
=f\uf 
Jo dy 
dP 
dy 
( 1 ). 
Let p he the pressure at (a + £, y + rj). (The motion being infinite- 
simal,) we have . 
or, in virtue of (1), 
P = G + g(y + rj)- 
dt 
n , . dP 
p = C+ffy + g- 
d 2 P 
dt 2 
The kinematical conditions are, the equation of continuity, 
dffP dfP 
dx 1 + dy 2 
and the boundary equation, in two parts — one relating to the upper 
