186 
Proceedings of Royal Society of Edinburgh. [sess. 
surface suffices to determine its value throughout the water ; in 
virtue of the equation 
d 2 cf) d 2 cf> _ q 
dx 2 dz 2 
( 2 ). 
The motion being infinitesimal, and the density being taken as 
unity, another application of the fundamental hydrokinetics shows 
that, as found by Cauchy and Poisson, 
p-H = g(z-h + Z)- 
d 2 <f> 
dP 
g(z 
( 3 ); 
where g denotes gravity ; II the uniform atmospheric pressure on 
the free surface ; and p the pressure at the point (x, z + £) within 
the fluid. 
§ 3. To apply (3) to the wave-surface, put in it, z = h; it gives 
and therefore if we could find a solution of this equation for all 
values of z, with (2) satisfied, we should have a solution of our 
present problem. Now we can find such a solution ; by a curi- 
ously altered application of Fourier’s celebrated solution 
[ 
(t + &y 
for 
dv 
dt 
~ k dx 2 ’ J 
his equation for the linear conduction of heat. Change t + c, x , k , 
into z + ix,t,g~ l respectively: — we have (4), and we see that a 
solution of it is 
1 -g* 2 
p 4(z+t*) 
J(z + tx) 
( 5 ); 
which also satisfies (2) because any function of z + tx satisfies (2) 
if t denotes J-l. Hence if {RS} denotes a realisation* by 
taking half sum of what is written after it with ± we have, as 
a real solution of (4) for our problem 
I -gt* 
1 4>(x,z,t) = {nS}~^r ) e^ .... ( 6 ), 
* A very easy way of effecting the realisations in (6) and (9) is by aid of 
De Moivre’s theorem with, for one angle concerned in it, x = tan-kc/z ; and 
another angle = gt 2 x/ 4(« 2 + x 2 ). 
