2 Lord Kelvin on an Initiational Form. 



liquid, (called water for brevity,) in a straight canal, in- 

 finitely long and infinitely deep, with vertical sides. Let it 

 be disturbed from its level by any change of pressure on the 

 surface, uniform in every line perpendicular to the plane 

 sides, and let it be left to itself under constant air pressure. 

 It is required to find the displacement and velocity of every 

 particle of water at any future time. Our initial condition 

 will be fully specified by a given normal component of 

 velocity, and a given normal component of displacement, at 

 every point of the surface. 



Taking 0, any point at a distance h above the undisturbed 

 water level, draw X parallel to the length of the canal, 

 and Z vertically downwards. Let f, f be the displacement 

 components, and f, f the velocity components, of any particle 

 of the water whose undisturbed position is (,r, -)• We 

 suppose the disturbance infinitesimal ; by which we mean 

 that the change of distance between any two particles of 

 water is infinitely small in comparison 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 incompressible and friction- 

 less, its motion, started primarily from rest by pressure 

 applied to the free surface, is essentially irrotational. Hence 

 we have 



where F(a% z, £), or F as we may write it for brevity when 

 convenient, is a function which may be called the displace- 

 ment-potential ; and F(.r, s, 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 o£ the incompressibility of the fluid, 



d^ + d?-° (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 



^n«rt*-A+B--J£=ite-A)+yg -^ (134), 



