132 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



motion =^/g(z + K), but that every successive wave is transmitted with the same 

 velocity. 



Also the coefficient of the velocity -^ varies inversely as VTh, that is the 



(velocity) 2 a inversely as depth, therefore mass moved x velocity 2 or vis viva is 

 constant. 



Another, and, for our present purpose, a still more important result appears 

 from the form of the function ; viz. that the initial conditions which we have 

 assumed to exist must of necessity give rise to a wave transmitted in the negative 

 as well as in the positive direction. Thus the hypothesis belongs only to a canal 

 open in both directions. And farther, since the other hypothesis, that when '=0, 



j?=0, and (j)=a constant, will give the sum of sines as the function correspond- 



ing to the sum of cosines in the present case, it is clear that no hypothesis of this 

 nature can apply to the case of motion in a closed canal. 



We must therefore look for some other process when we come to that case, 

 and proceed at present with the problem before us, admitting it to be restricted 

 to an open canal. 



71. Let us now expand the different quantities which enter into the expression 

 for</>. 



!_,*(+*) 



5 

 3 



+ *)*) 



sin c^ t = sin fJL Vff (z + A) tcosfj. Vg (z + h) 't . ^- (z + h) 2 V ' g (z + h) 



2 * 2 - 9 A 2 



-& c . 



8 



