610 Lord Kelvin on Deep- Water Two-dimensional 
convenient, is a function of the variables which may be called 
the displacement-potential; and $(2, z, ¢) is what is commonly 
called the velocity-potential. Thus a knowledge of the 
function @, for all values of x, z, t, completely defines the 
displacement and the velocity of the fluid. And, by the 
fundamentals of hydrokinetics, a knowledge of @ for every 
point of the free surface suffices to determine its value 
throughout the water; in virtue of the equation 
Ue 8929 
da dz 


The motion being infinitesimal, and the density being taken 
as unity, another application of the fundamental hydrokineties 
shows that, as found by Cauchy and Poisson, 
f2 d 1A : 
p—W=g(e—h+ 9 — SP =g(2—) ty — SE. 8): 
where g denotes gravity; I the uniform atmospheric pressure 
on the free surface ; and p the pressure at the point (z, 2+ ¢) 
within the fluid. 
§3. To apply (3) to the wave-surface, put in it, z=h; 
it gives 
dd ad 
( == fre ae Ae 4 3 
a( dz z=h oe jy ( ) 
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 curiously altered application of Fourier’s celebrated 
solution 


—7Z 
fr lv dv 
\ SE teeHe). | for = ae 
| (e+e) FeO, for rel 


his equation for the linear conduction of heat. Change 
t+c, x, k, into z+.u2, t, g~! respectively:—we have (4), and 
we see that a solution of it is 
T° —gt? 5) 
Ben oe ee eietez) é. A . . A 9) . 
a/ (2 +42) ! ( 
which also satisfies (2) because any function of «+ satisies 
(2) if c denotes ,/—1. Hence if {RS} denotes a realization * 
* A very easy way of effecting the realizations in (6) and (9) is by aid 
of De Moivre’s theorem with, for one angle concerned in it, x= tan heyz; 
and another angle =gt?2'/4(z*+-2°). 
