426 Drs. Korteweg and de Vries on the 

 expressed by ^ — = : — 



w o# ?i(n— 1) ^^ 2 



In this manner we obtain the following set of equations : — 



U J 2 y -da? + 24 y d# 4 W 



v - y a« + 6 y B«» 120^ a^ + '-- • ( 2 > 



and, moreover, if </> be the velocity potential and ty the stream- 

 function : — 



which set of equations satisfies for the interior of the fluid all 

 the conditions of the problem, whilst at the same time it is 

 easy to see that for long waves these series are rapidly con- 

 vergent. Indeed, for such waves the state of motion changes 

 slowly with oc, and therefore the successive differential- 

 quotients with respect to this variable of all functions re- 

 ferring, as / does, to the state of motion, must rapidly 

 decrease. 



Passing now to the conditions at the boundary, let p 1 (a 

 constant) be the atmospheric pressure, y{ the pressure at a 

 point below the surface where the capillary forces cease to act 

 and T the surface-tension. We then have, distinguishing here 

 and elsewhere by the suffix ( a ) those quantities which 5 refer 

 to the surface, 



but, according to a well-known equation of hydrodynamics 



f =xW-^ i -i(V+^)-^ I , 



therefore 



*W+W+".' + £|£. . (5) 



