428 Drs. Korteweg and de Vries on the 



It is obvious that this solution coincides with the one usually 

 given for the case of long waves of arbitrary shape made 

 stationary by attributing to the fluid a velocity 7 equal and 

 opposite to that of the waves, on the assumption that the 

 velocity in a vertical direction may be neglected and that 

 the horizontal velocity may be considered uniform across each 

 section of the canal. 



But, if we wish to proceed to a second approximation, we 

 have to put 



/= ?0 -^(,+ a+y ) ...... ■ <?) 



where 7 is small compared with rj and a. On substituting 

 this in (6) and (7) and on writing out the result, rejecting 

 all terms* which are small compared with any one of the 

 remaining terms, we find respectively : — 



?& + 'g-f<'t-?S-(*!Hr)S-* - (10) 



and 



goto S?_f(2 v + a )p +)P 9 p d =0. . .(11) 



In eliminating ^ from these equations, we have at last 



At ~ 21 ^ • • [iZ) 



where 



-v-S < u > 



This very important equation, to which we shall have fre- 

 quently to revert in the course of this paper, indicates the 

 deformation of a system of waves of arbitrary shape, but 

 moving in one direction only. Before applying it, we may 

 point out the close connexion between the constant a, which 

 may still be chosen arbitrarily, and the uniform velocity 

 given to the fluid. Indeed it is easy to see from (1) and (9) 

 how a variation Sot of the constant a corresponds to a change 



in com- 



* The terms for instance with 2^« ^-? and (— ) are rejected i 

 parison with rj ~t, which is retained in the equations, those with ^J and 



da ,2 d£ dt 



