PROFESSOR KELLAND ON THE THEORY OF WAVES. 



COR. For the case in which the section is triangular g=Q. In this case equa- 

 tion (17) shews that 2 1 and 1-2 n 2 must have the same sign : and equation (18) 

 that 1 2w 2 and w 2 ( 2 + l) 1 must have the same sign. 



Therefore the three quantities a 2 -!, 1 2w 2 , and n 2 (a 2 + l) 1 are positive, 

 aero, and negative together. 



Now, the result obtained by an approximation was, that 1-2 2 =0. This re- 

 sult, then, corresponds to that triangular section for which =l. In other cases 

 we find that, on the hypothesis of continuity which we now adopt, 



. 11 



if a -=: 1 w ^ g -^ ^ + T ' 



; 11 



if a^-1 n ^ ^^r -5 T . 

 & a T i 



In the case for which we are furnished with experiments, viz. that given by 



Mr RUSSELL, the value of a was (Report of British Association for 1837, p. 442). 



114 

 For this case, then, we ought to have n^.^^- 5 ^- ^ 



4 + 



Mr RUSSELL himself is of opinion that n=\. I do not think that his expe- 

 riments warrant this opinion, and whilst I am not disinclined to admit that a 

 quantity a little less than | may suffice, I am still more confident in the truth of 

 results obtained from approximation, than in those obtained from the hypothesis 

 of continuity. 



64. With respect to the height of the wave, we determine its value from the 

 equation 



a dz 



but - Jfj .'A 



py+^vn^^a 



Hence the elevation of the wave is 



n. 

 ca 



COR. 1. If the form of the wave can be expressed by one function, 



This expression shews that the crest of the wave is higher at the sides of 

 the canal than in the middle. 



