214 Mr. R. F. Gwyther on the 



As we are seeking solutions which we have reason to think 

 rapidly convergent, put 



F'=-c+/', (5) 



where we consider f small compared with c. On this 

 assumption (4) becomes 



(o 2 -^ )h 2 f'"=fi*-3cf> 2 + 2(c 2 -gh)f. . (6) 



Whenever the form of the function/' is found, the form of 

 the free surface is given by (1) or (2). 



Under the conditions stipulated we may neglect /' 3 com- 

 pared with 3c/ 2 , but we can only neglect 3c/ 2 compared with 

 2(e 2 - gfyf, when / is small compared with 2(c 2 —gh)/'6c. 



Taking this to be the case, we obtain 



, . /2{gh-c i ) x 



If X is the wave-length, we have 



4,ir 2 h 2 _ 2(gh-c 2 ) 



3 



subject to the condition that the height of the wave shall be 

 small compared with 2{gh—c i ')/?>g. 



If, for example, h : \=1 : 10, we get approximately 

 gh:c 2 = 9 : 8, and the height of the wave must be small 

 compared with 2h/27. 



This wave can therefore be only of small amplitude, and 

 cannot be the general case of the long wave. It has, on the 

 usual theory, a group-velocity given by 



3 C/>2_ /7 7^2 



c*-ghy 

 gh 2cgh ' 



which is slightly less than the wave-velocity. 



Proceeding to the more general case, we have from (6), 

 neglecting/' 3 and integrating, 



(c*-^ W /2 =-2c/' 3 + 2(c 2 -^)/" 2 + constant ; (8) 



from this it follows that f can be found in terms of x as an 

 elliptic function. I do not propose to consider this further 

 here, but it is doubtless connected with the stationary cnoidal 

 wave of Korteweg and de Vries *. 



* Phil, Mag-,, May 1895. 



