PROFESSOR KELLAND ON THE THEORY OF WAVES. 121 



cept in one very simple case. If the canal have a triangular section such that the 

 breadth is double the depth, or if one side of the triangle be vertical, and the 

 breadth equals the depth ^ (*) =* and *' (*) = 1. In this case 



w=\/l w 2 = r and 



_ 



If aV2=a', |=*'; this gives 



v_ v 

 #,. e , ,~ 3^, which is the expression for c 2 in a rectangular channel of 



a e K z + e" 1 



which the depth is one-half that of the greatest depth in the triangular channel, 

 Art. 10. 



Hence we conclude, 1. That the velocity is that due to a rectangular channel 



of half the depth : 2. That the length of the wave x =v is to that in a rectan- 

 gular channel of half the depth as \/2 : 1, or to that of a wave in a rectangular 

 channel of the same depth as 1 : \/2. The former of these results is amply con- 

 firmed by experiment : the latter has not, so far as I know, been examined expe- 

 rimentally. 



63. Let us, in the next place, determine the relations which exist amongst 

 the functions for that particular case in which the functions v and w are conti- 

 nuous. We cannot suppose that this hypothesis is very greatly erroneous in a 

 channel of triangular section, but we hesitate to assert that it is strictly true even 

 in that case. Here we may put h for the statical depth of the surface h + 1 for the 

 depth at the time t, and expand the functions in terms of jj. 



Let us denote *+* by S n , e " n *-rf nh by D n , 



** by j)^. then substituting these values in equation 

 (13) we obtain 



and by (14) 



This last equation will give no results except those which arise from equat- 

 ing to each other parts independent of g, inasmuch as we obtain the approxima- 



VOL. XV. PART I. K k 



