Lord Kelvin on an Initiational Form. 3 



where g denotes gravity: II the uniform atmospheric pressure 

 on the "free surface ; and p the pressure at the point [x } c + f) 

 within the fluid. 



§ 98. Suppose now that F(.r, r, t) is a function which, 

 besides satisfying (133), satisfies also the equation 



r/F J*F -o-v 



'\L = clF (lo0); 



we see by (134) that the corresponding fluid motion of which 



F is the displacement-potential (132), has constant pressure 



oyer every surface (z + f) ; that is to say, every surface 



which was level when the water was undisturbed. Thus our 



problem of finding any possible infinitesimal irrotational 



motion of the fluid, in which the free surface is under any 



■constant pressure, is solved by finding solutions of (133) 



and (135). 



§ 09. Xow by differentiation we verify that, as found in 



§ 3 above, 



1 ~9 f2 



F_ eMz+a) .... (136) 



x : + ix 



satisfies (133) and (135). By changing t into — i, and by in- 

 tegrations or differentiations performed on (136), according to 



the svmbol t^i — ^t> where i, j. k are any integers positive 



or negative, we can derive from (136) any number of 

 imaginary solutions. And by addition of these, with constant 

 coefficients, we can find any number of realised solutions. 

 If, as in § 97, we regard any one of the formulas thus 



obtained as a displacement-potential, then by taking -,- of it 



we find J, the vertical component displacement, which we 

 shall take as the most convenient expression in each case for 

 the solutions with which we are concerned. Or we may, if 

 we please, take any solution of (135) as representing, not a 

 displacement-potential, but a velocity-potential, or a horizontal 

 component of displacement or velocity, or a vertical com- 

 ponent of displacement or velocity. 



§ 100. Thus it was that in § 12 we took 



-?={R.S}— ^-- e i(4W 

 Wz + vx 



., ^ n /gt 2 x 1 \ -** 



=*(*> - = V - p C0S (V ~ 2 x r 4pi 



where p= \S(s 2 + x 2> ), and ^=tan -1 {#/3:), 



(137), 



B2 



