Sir G, G. Stokes on the Theory of the Solitary Wave. 315 



real, is (at least for my purpose) admissible. It is not true 

 that a non-periodic function of x cannot be expressed by 

 means of periodic functions ; for example, the non-periodic 

 function e~ x * may be expanded in the definite integral 



_2f 



e a2 cos 2&,x . dx. 



each element of which is periodic. On the other hand, the 

 form of expansion proposed by Mr. McCowan is (at least for 

 my purpose) inadmissible, on account of the discontinuity of 

 the expression. 



I will now mention more particularly the step which led 

 me to a wrong conclusion. It is easily shown that in a very 

 long wave propagated in water the depth of which is small 

 compared with the length of the wave, the horizontal velocity 

 is nearly the same from the bottom to the surface for any 

 vertical section of the wave made by a plane perpendicular to 

 the direction of propagation. For a given depth of water and 

 maximum height of wave, this is so much the more nearly 

 true as the length of the wave is greater. The horizontal 

 velocity tends indefinitely towards constancy from top to 

 bottom as the length of the wave increases indefinitely. Now 

 Sir George Airy has shown that for a wave in which we may 

 suppose the particles in a vertical plane to remain always in a 

 vertical plane, as they must do if the horizontal velocity is the 

 same from top to bottom, the form of the wave must gradually 

 change as it progresses. It might seem therefore that, how- 

 ever small we take the height of a wave at the highest point, 

 we have only to make the wave long enough and Airy's inves- 

 tigation will apply, and the wave will change its form in time 

 as it travels along. Now in the solitary wave of Ilussell, the 

 lower the wave, the longer it is ; and therefore it might seem 

 as if we had only to make the wave low enough and long 

 enough and the length would be so great that Airy's investi- 

 gation would apply, and the form would change, though 

 slowly. 



The answer to this is that, however small we take the 

 height, we are not at liberty to increase the length indefinitely. 

 There is in fact a relation between the height and the length 

 in a solitary wave which can be propagated uniformly, which, 

 though it is of such a nature that the length becomes infinite 

 when the height becomes infinitely small, nevertheless forbids 

 us for a given height, however small, to increase the length 

 indefinitely. 



The possibility of the existence of a solitary wave of uniform 

 propagation is so far established by my investigation relating 

 to oscillatory waves, as that it is made to depend on the prin- 

 ciple that the infinite series by which the circumstances of the 



