532 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



By substituting these values in equation (1), we obtain 



27r, ^ . ndb mbsm.6 



— — — 6 . cos y + sin y — - = — - 



A. ax m{l—x) 



dx Inr del 



1 { d I . n 217 dl 



dl . 

 'I + z — sm 

 l—x 



{d I . n Stt dl n d^ 'Zirdcl n \ 

 Tjsmt) — ^ cost) -_-+—- cost) \ 

 [l—xf A l—x dl A l—x J 



Equating coefficients of like functions of ^r, we obtain approximately 



and 



b' being the value of b at the origin. 



du ds du 



— " — b c . cos 6 — r— b^ sin 6 cos 6 + sin' 6 b — 

 ■ ■ A A dx 



27r d g I ^ gal . n c 



= — i- cos0-7f — — sm0+ Sic. : 



A l—x {l—^T 



or equating only those coefficients which belong to the large terms, 



dgl 

 bc= " 



But we have shewn that 



or c'=ffk, 



Thus it appears, that the velocity is not altered, whilst the height of the 

 wave increases in harmonic progression. This result does not agree even roughly 

 with Ml- Russell's experiments ; the reason for which is, that his waves were of 

 considerable length, so that the variation of the channel through the length of a 

 single wave cannot be neglected. 



To attempt the solution of the more general problem, would lead us into 

 very complex analysis. It must consequently be reserved for another memoir. 



