532 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



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



27r -J . r^db mbsaxd 



— -— . cos a H- sin a — = — 



A ax m (l—x) 



h + ; sin 6 



l—x 



[ dl . f. Inr d I n 



dx 2'Tr d cl ^ \ 



-— +^- -- COS0 I 



dt A l—x j 



Equating coefficients of like functions of x, we obtain approximately 



del 



bh= 



and 



db 



l-x 

 b 



dx l—x 



log6 = log-^— 



b _ I 

 ~y~ l-x 



b'l 



b= 



l-x 



h' being the value of h at the origin. 



du ds du „ 



Again, IT = -^d^^''di^^'' 



— -— 6 c . cos 6 — ^— b"^ sin 6 cos ^ + sin^ 6 — 

 A A dx 



2'Tr d ql ^ gal ./, . 



= ^: ^ cosO-^f — — sin 6+ &c. : 



A l—x {'—*) 



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



bc = 



But we have shewn that 



bh: 



l-x 

 del 



l-x 



h c 



or c^=9h. 



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 Mr 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. 



