PROFESSOR KELLAND ON THE THEORY OF WAVES. 53I 



50.52. „ ^ 

 and c-=g . — ^ in feet 



32.2.(50.52) (32.2.) 50.5 



=— lii ^=144 '"P^^^^ 



. 40.3 ..„ 



Experiment gives it 3.08. 



The discrepancy is due entirely to our not knowing in any case the exact 

 height (Ji) which gives c^=gh for a square formed canal. 



This discrepancy is not much greater than that between two waves of the 

 same height in a rectangular channel. 



By computing the velocity of the next waves given by Mr Russell, we ob- 

 tain c=3.45. 



By observation c is equal to 3.50. 



For the next and last set we obtain 3.56, which by observation is 3.86. 



30. I propose, in the last place, to deduce a first approximation to motion in 

 a channel of variable breadth, taking only that case for which the channel dimi- 

 nishes very slowly and uniformly. 



Let z be the breadth at the point whose other variables are x and t. 



z z= m (l—x) 



I being the whole length of the channel from the origin. 

 Adopting all the previous notation, we obtain 



zsa=-s;-\-Oz s+os a + 6a 



dz ds da. - 



or a*- t-a^- — \- zs -^— — 



dt dt dt 



Ida 1. dz Ids 



a. dt z d t s d t 



du 1 dx \ds ,-,, 



— = -m (1) 



dx z dt sdt 



31. Now, we may find the variation in the height of the wave, by supposing 

 its length to remain constant, an hypothesis which must be considered as merely 

 approximative. 



The volume of the wave will vary as 





'^ r^" . 2'7r , , , , 



asm-^— (ct—x)dxdz 

 A 



o 



or as aXz + C 



hence, if a' be the value of a at the origin, 



a' \ . I = a\ {I— X) 



. I 

 a = a . 



l—x 



1 d I . ^ 

 s = h + - — sm V 

 l—x 



u = b sind . 



