Change of Form of Long Waves. 431 



which is the equation of a train of periodic waves whose 

 wave-length increases when h decreases. 



For & = this length becomes infinite, and the equation may 

 be shown to coincide with (17). 



The following figure (fig. 1) represents such a train of 

 stationary waves for the case in which k=-f$7i, M=0*8. 



Fiar. 1. 



III. Stationary Periodic Waves (Cnoidal Waves). 



Proceeding now to a further investigation of the waves 

 determined by equation (20), we calculate from (10) and (11) 

 the value of y. From these equations we get 



dj_ 1 B5. /1 /2 __T\^ 

 dx~ 2r.3.r \3 'fypfW 

 or by integration, 



1 2 , /1 72 T \B 2 t, 



where the constant of integration is rejected because its 

 retention would onlv have had the effect of augmenting in 

 equation (9) the value of the arbitrary constant a. 



On substituting, then, /from (9) in (1) and (2), observing 

 that in virtue of (14) 



§5 =-^(3^ + 4^ + 6^) = - ~(S^-2(h-k) V -U), 

 these equations are replaced by 



.= ^ v /f{^K*-*>-£+(J+'£)[C*-U.i 



+ 



i^h-iv 2 ]} 



+ 



2a- 



2gpt 

 £{(h-k) V +ihk-&?}f + .... (21) 



v=*/ MVi-v)(k + v) t2/m . ........ (22) 



V la- 



