186 Lord Rayleigh : Uydrodynamical Notes. 



By successive approximation, determining yjr so as to make 

 the mean value of y equal to zero, we find as far as a 4 



y = (« -f- 1 a 3 ) cos X — (~ a 2 -4- f. a) COS 2x 



+ fa 3 COS 3 A' — 3 a 4 COS 4#, . . (13) 



or, if we write as before a for the coefficient of cos x, 



y = a cos x — (la 2 + ^a?) cos 2x 



+ § a 3 cos ?>x — I a 4 cos 4#, . . (11) 



in agreement with equation (20) of Stokes* Supplement. 

 Expressed in terms of a, (11) becomes 



g=l-a*-±a\ (15) 



or on restoration of k, c, 



g = kc i -k*a 2 c 2 -\k b a A c 2 (ltf) 



Thus the extension of (8) is 



c! = ^/yK(l-rFa 2 +f£4 a 4) 5 .... (17) 



which also agrees with Stokes' Supplement. 



If we pursue the approximation one stage further, we find 

 from (12) terms in a 5 , additional to those expressed in (13). 

 These are 



y 



, T373 243 Q 125 . ■> MO , 



= « j 7p>9 cos # + — cos dx + ^-T)~2 cos 0X \ • ( 18 ) 



It is of interest to compare the potential and kinetic 

 energies of waves that are not infinitely small. For the 

 stream-function of the waves regarded as progressive, we 

 have, as in (1), 



ifr = — ae~ y cos (x — ct) + terms in a 4 , 

 so that 



(d$/da:y + (dWdy)*=A*er* + tQTmB in a 5 . 



Thus the mean kinetic energy per length x measured in the 

 direction of propagation is 



where y is the ordinate of the surface. And by (3) 



