lo GWYTHER, On the Propagation of a Solitary Wave. 



values for w and therefore for //i, and the wave repre- 

 sented is not a wave of depression. 



Although I have found no explicit equation to the 

 free surface, it is more easy to draw the forms of series of 

 such surfaces from the equations 



^ = - + //, sinh 2!n I Ian mj^y cosh — - + cos 2w/j 1- 

 y = fi + hi{i + cos 2mjl)li cosh — - + cos 2/«/3 |. 



than it would be to work from the explicit equation. 



Leaving the discussion of the mathematical expres- 

 sions, the physical ideas which they are intended to 

 express are there. In the experiments of Scott Russell, 

 the wave, by whatever method it was formed, in a short 

 distance cleared itself at its base from extraneous disturb- 

 ances and masses of water, and then travelled in its 

 definite form, breaking, if it did break, near the crest. It 

 has therefore been made an essential point in the mathe- 

 matical discussion that the surface-conditions are satisfied 

 with great nicety in the outskirts of the wave, and so the 

 slopes of the wave serve as buttresses to it, allowing of 

 any defect of pressure near the crest to be rectified by a 

 slight local readjustment of the fluid particles. This could 

 only be the case so long as the graph of uih in relation to 

 ?;//3 (Fig. i) has a slight gradient. When the gradient 

 becomes steep, the rearrangement would no longer be 

 small and local, but a change in the circumstances of the 

 wave would demand an inordinate change in the elevation. 

 The point at which this would happen cannot be fixed, 

 and is perhaps not very definite, but there is evidently a 

 very definite difference of the circumstances represented 

 at the two extremities of the graph. 



The elevation of the waves experimented upon 



