PROFESSOR KELLAND ON THE THEORY OF WAVES. 



But -^ = the vertical velocity at the surface, which is a particular value of 



the quantity v, and as v is very nearly equal to when y + h-ax=Q, it follows 

 that k is very small, and for our present purpose may be neglected. 



We may state the results at which we have arrived, by saying that u t and v 

 do not represent the velocities parallel to x and y in any case, but approach nearer 

 and nearer to those velocities, as the points to which they correspond recede fur- 

 ther from the bottom of the fluid. The quantity 6 is always supposed to be less 

 than 2 v, a supposition which is required by the circumstance, that we are about 

 to remove it from beneath the circular sign. 



We suppose, then, that the above equations give the values of u and v at the 

 surface of the fluid, and shall apply the method of parameters to deduce from 

 them the variations of A, &c. The quantities \ b, &c. are now functions of x. 



Since ^ + ~ = 0, we get, calling a(z + h-ax) <f>; 



6 db _ 



cos 



. y, ... sin 6 db . - _ ffl / - d a d c\ 



f e f+e~ ? ) r- + (e f +e <f ) (x-ct- -- at ) 



b dx \ dx dx) 



da a 



,--*--" ) S m 0=0 (1) 



du dv 



And since -j- =-j-; 

 ay dx 



, a <f,cosdb - _ - / da t d c \ a 



(e v e p ) (e 9 e ^[xct- at \sin6 



b dx J \ dx dx) 



) = Q . . . . (2) 



dx 



dc 

 _ / 



dx 



By eliminating b, x-ct-^--at~smd-j-(z + Aax') a, successively, we obtain 



- . <-*) sin 0cos0=0 (3) 



dx dxj dx 



- 

 dx dor 



(4) 

 (5) 



Now, our object is not to obtain the variation of any of the quantities through 

 a small space, but through a large one. The variation of each of the quantities 

 will consist of two parts, the one depending on m, the other on a circular function 

 of x. With the former only we are concerned ; the latter goes through all its 

 values, and is as much additive as subtractive in the space occupied by one 

 wave. Confining our attention to the former variations only, we obtain, as the 

 non-periodic part of equation (3), 



