Lord Rayleigh on Waves. 27 S 



may be expressed by the series 



h = tu K u K (xy), ...... (D) 



the quantities a being constants with respect to space, but de- 

 pendent upon time. The potential energy of the displacement, 

 calculated on the hypothesis of a constant pressure on the surface, 

 will clearly be 



Y =9P #••£«*»« dx d V = ^h'dx Ay 



(E) 



by the conjugate property of the functions u. This is the po- 

 tential energy. 



The motion of the fluid throughout the interior depends, 

 according to a known theorem, only upon the motion of the 

 surface ; and the surface normal velocity 



j- =h=%'a li ii K , 



dz 



If I be the depth, the complete value of cj> is given by 



e <Z-l) _|_ q — K(Z-l) 



~ft =s tt(g-"-g"0 ' UKU ^ x y)- • • • ( F ) 



For, in the first place, this value of cf> satisfies Laplace's 

 equation, inasmuch as each term u K satisfies the equation 



d 2 u , d?u , 9 n 



Secondly, (F) satisfies the condition imposed by the rigid 



du 

 cylindrical boundary, since -~ = ; 



Thirdly, (F) makes ^=0 when z=l ; 



And fourthly, when z=0, —-^-=h. 



The kinetic energy T may now be readily calculated : 



2JJ dn 



Phil. Mag. S. 5. Vol. 1. No. 4. April 1876. U 



d *dS, 



