542 PROFESSOR KELLAND ON THE THEORY OF WAVES. 



R = 6//t/^(^ ^ (e« y + e- « 2/) (1 + sin 0) 



Now X and t enter always together, and the limits x=p x=q correspond to 



t— 



+ r , — + r; therefore the two are equal, i. e. Q=R accurately. 



c c 



Consequently we deduce the following important conclusion : 

 " That all the fluid which was elevated above the statical level has passed 

 on with the wave, and no other fluid with it." 



Combining this with the fact that the velocity of the particles in all vertical 

 sections through points at the nearer extremity of the wave is zero, and that no 

 particle is moving backwards, it follows that a wave will not he followed hy another 

 resulting from the displacement which it has caused amongst the particles. Hence 

 the functions u, v, z are discontinuous ones, having the values assigned to them 



when the function x — ct lies between — j and + —^ , but being zero in all 



other cases. 



It is not necessary to determine what function this is : for, although the de- 

 rived functions of certain orders might differ widely from those deduced ft-om the 

 equivalent formul£e, yet the first derived functions of which alone we make use, 

 representing either a velocity or a force, cannot be very different in the wave as 

 expressed by its value from that expressed by the form ; they cannot, in fact, 

 differ from those obtained in ordinary undulations, provided we conceive all the 

 fluid in motion in the direction of the axis of x. We may conceive, however, a 

 difference of pressure, and hence probably arises the fact that p is not in the pre- 

 sent case a complete differential. 



44. The formulae we have deduced above will give us the velocity of trans- 

 mission, provided we know the length of the wave. 



Mr Russell has not yet published his results as to the length of the wave, 

 nor have I been able at present to deduce it to my satisfaction in terms of the 

 depth of the fluid and the height of the wave. It would appear at first sight 

 probable that the higher the wave for a given depth of fluid, the longer it will be. 

 Yet, from Mr Russell's conclusions, it appears the contrary to this is the case. I 

 have appended a few results deduced from the value of X, which Mr Russell has 

 favoured me with as the result of his present researches. I must, however, take 

 the liberty of observing, that, although the first part of it is just what I should 

 have expected, the latter part is by no means so. Had I chosen a formula em- 

 pirically, as that which the nature of the case seems to require, it would have 

 differed from Russell's; thus, for 2 tt A— 2 a:, I should have conceived 27rh + '7rx, or 



2 TT ('* + -|-) , as the probable form. But I forbear any further remarks until I shall 



