1887.] Sir W. Thomson on Procession of Waves. 
41 
Remembering Cauchy and Poisson’s discovery that every surface 
of particles which are in a horizontal plane when undisturbed 
fulfils the condition of a free upper-surface (so that if all the water 
above it were annulled the motion of the water remaining below it 
would be undisturbed), in the case of free waves of infinitely deep 
water; we see that when P(y=o) = const., we have also, in our 
notation, p = const., for every constant value of y. Hence, looking 
to (3) above, we must find, in the case of free waves. 
9f P 
dy 
d* P 
dt 2 
( 13 ); 
for every value of y, and not only at the upper surface y = 0. 
Thanking Cauchy and Poisson for this as a suggestion, hut not 
assuming it without the proof of it which we immediately find ; 
and borrowing now from Fourier* his celebrated “ instantaneous 
dv d^v 
plane-source ” f solution of his equation ^ f° r thermal con- 
duction, assume, as an imaginary type-solution of (4) and (13) for 
free waves, 
(U), 
(b + y + ixf 
Mb+y+ix) 
where t denotes J — 1. Whence, as a real solution by adding the 
values of (14) for i and - 1 , and dividing by J 2, 
gte 1 
4r 2 f H 15 )- 
1 f 
gt 2 x 
<£(*) = - | (r + y + bf eos'-^- + (r-y-bf sin 
where 
r — Y(y + b ) 2 4- a; 2 ]- 
Curves representing calculated results of this solution for free 
waves were shown at the meeting of the British Association (Section 
A) at Birmingham in September, and at the last meeting (December 
20) of the Royal Society of Edinburgh. To build up of it a solu- 
tion for a uniformly maintained procession of waves (a double 
procession it shall be, of equal and similar waves travelling in the 
two directions from x = 0) take 
P(0= /dtcf>(t) (16); 
J o 
* TMorie Analytique de la Chaleur. 
t Sir W. Thomson’s Collected Papers, vol. ii. p. 46. 
