120 PROFESSOR KELLANB ON THE THEORY OF WAVES. 



=. .9*1 ....?_ =< ,,_L. 



m J ^ m 



n m n 1 + - 



a s + --az n 



n 



But area of vertical section 



ion = C*dzaz n 

 Jo 



area of vertical section 



1 m 



n az n 



and c 2 =ff . 



area of vertical section 

 breadth at surface 



area of vertical section 

 breadth at surface 



This is the result obtained in Art. 28, Part I. It may be doubted whether our 

 present result is more approximate even in other cases than this. 



61. One remarkable circumstance is pointed out by the form of the result, 

 viz. that the velocity of transmission is independent of y, provided one value of n 

 is sufficient to satisfy all the conditions. The ridge of the wave is, consequently, 

 a line extending directly across the channel, and not curved so as to be more 

 advanced in the middle or deepest part of the channel than at the sides. This 

 circumstance I supposed to be altogether in opposition to fact, from the form 

 which Mr WHEWELL gives the tide curves in his excellent essay on the Approxi- 

 mate Determination of the Cotidal Lines (Phil. Trans. 1832). I attributed this 

 to the approximation which had been used, and in part to the fact, that in all 

 probability the form of the wave is not the most simple possible, and cannot be 

 satisfied by one value of n. On mentioning the circumstance to Mr RUSSELL, he 

 informed me that, in experimenting on perfect triangular channels, he had found 

 that the wave is not retarded, and stated that this fact astonished him very much 

 on making the experiment. As far, therefore, as simple cases are concerned, it 

 appears that theory and experiment are perfectly at one. 



We may remark that the result of the approximation is the more to be de- 

 pended on, inasmuch as the results obtained by applying the same process of 

 approximation to the simple case of a perfectly uniform canal, produces results 

 remarkably conformable with experiment, and such as have been long recognised 

 as expressing the laws of the phenomena. 



62. Equations (13) and (14) cannot be reduced without approximation, ex- 



