432 Drs. Korteweg and de Vries on the 



When k=0 they determine the motion of the fluid for a 

 solitary wave. 



In the first place we now will endeavour to calculate the 

 velocity of propagation. For the solitary wave this is simple 

 enough. If we consider that the liquid at infinity is Drought 

 to rest when a uniform motion with a horizontal velocity 



- 9 =-^l(l+|) (23) 



is added to the motion expressed by (21) and (22), it is clear 

 that this velocity, with reversed sign, must be taken for the 

 velocity of propagation of the solitary wave. 



But for a train of oscillatory waves Sir G. Stokes has 

 shown* that various definitions of this velocity may be given, 

 leading at the higher order of approximation to different 

 values. It seemed to us most rational to define it as the 

 velocity of propagation of the w r ave-form when the horizontal 

 momentum of the liquid has been reduced to zero by the 

 addition of a uniform motion. This definition corresponds 

 to the second one of Sir G. Stokes. According to it, we have 

 to solve the equation 



dx\ tu-q)dy = 0, .... (24) 



where q denotes the velocity of propagation, and where 



X= 2K ^ (25) 



s/h + k ' 



is equal to the w 7 ave -length. 

 If, then, 



Jo v h+k i y 



= x{(/i + *)^® -k\ ... (26) 



denote the volume of a single wave reckoned from above its 

 lowest point, we get from (24), retaining only such terms as 

 are of the first order compared with r], A, and k: — 



* Math, and Phys. Papers, vol. i. p. 202. 



