434 Drs. Korteweg and de Vries on the 



fore we have dx=(u' — g)dt y or to a first approximation 



dt = dx= 7= dx : but then 



q ^gl 





2 2K.« , 



en — r — dx* 



Or, according to a well-known formula *, 

 g= -,(^[z-( 2K ^^-«)) -z(^)]. . (29) 

 At the same time we have 



-^WW\ (30) 



Of course, as all fluid particles with the same y describe 

 congruent paths, these formulae may be simplified by sup- 

 posing ,2* =0. 



IV. Deformation of Non- Stationary Waves. 



In order to study the deformation of non-stationary waves, 

 we will now apply our formula (12) to various types of waves. 



Solitary Waves. — As a first example we choose a solitary 

 wave whose surface is given by 



7)=hsech 2 jiM (31) 



According to (12), the deformation of this wave is expressed 



by 



+ 8(W-A) ] 8eoh8 i w ' tanh l >a! - • • ( 32 ) 



But before we are able to draw any conclusion from this 



dvi 

 expression, it is necessary to separate the two parts of -~, of 



* Z(«) = u 1 1 - ^ ] - M 2 1 sn 2 u . du. Compare, for instance, Cayley, 

 'An Elementary Treatise on Elliptic Functions,' 1876, ch. vi. § 187. 



