400 
Proceedings of Royal Society of Edinburgh. [sess. 
and £, £ the velocity components, of any particle of the water 
whose undisturbed position is (x , z). We suppose the disturbance 
infinitesimal ; by which we mean that the change of distance 
between any two particles of water is infinitely small in com- 
parison with their undisturbed distance ; and the line joining 
them experiences changes of direction which are infinitely small 
in comparison with the radian. Water being assumed incom- 
pressible and frictionless, its motion, started primarily from rest 
by pressure applied to the free surface, is essentially irrotational. 
Hence we have 
dx dz dx dz 
where F(a? , z , t) y or F as we may write it for brevity when con- 
venient, is a function which may be called the displacement- 
potential; and F (x y z, t) is what is commonly called the velocity- 
potential. Thus a knowledge of the function F, for all values of 
x ,z,t , completely defines the displacement and the velocity of 
the fluid. And towards the determination of F we have, in virtue 
of the incompressibility of the fluid, 
d 2 F d 2 F 
dx 2 + dz 2 
(133). 
In virtue of this equation, the well-known primary theory of 
Gauss and Green shows that, if F is given for every point of the 
free surface of the water, and is zero at every point infinitely 
distant from it the value of F is determinate throughout the fluid. 
The motion being infinitesimal, and the density being taken as 
unity, an application of fundamental hydrokinetics gives 
p -n = g(z-h + 0- w =9(z-h) + g- rz - w . (134), 
where g denotes gravity ; n the uniform atmospheric pressure on 
the free surface ; and p the pressure at the point (x,z + £) within 
the fluid. 
§ 98. Suppose now that F(;r, z, tf) is a function which, besides 
satisfying (133), satisfies also the equation 
d¥_d 2 Y 
^ dz dt 2 
(135) ; 
