Classes of Progressive Long Waves. 311 



From this the form of/ is easily found ; but the point of 

 most interest is the relation between m and the parameters 

 c and h given by (17). 



The method of comparison which T am now about to make 

 was very kindly suggested to me by Sir George Stokes, and 

 is based upon the article communicated by him to Professor 

 Lamb, which appears on p. 421 of the latter's ' Hydro- 

 dynamics/ . n 7 



In that article the formula c 2 = glt — ^— - — ; is shown by a 



J 2mh J 



simple method to be the necessary relation between m and the 

 parameters in the outskirts of a solitary wave. This result 

 follows from assuming 



<p = Ae- 2m( - x - ct) cos 2my 



as the form of the velocity-potential when x is large. 



Sir George Stokes has now pointed out to me that this is 

 an exact relation provided we admit (1) the possibility of 

 the uniform propagation of the disturbance as a solitaiy wave, 

 and (2) that in such a wave, at a great distance from the crest, 

 the coefficient of disturbance varies in geometric progression 

 as the distance from the crest increases in arithmetical 

 progression. 



If then we admit that c =gh tan 2mh/2mh is the exact 

 formula for the wave, being found without assuming any 

 relation between c and gh ; and if also the conditions of surface 

 pressure can only be everywhere satisfied provided 



$<?-gh)(<?-gh) "I 



approximately, the approximate identity of the two formulae 

 forms the condition for the existence of a Solitary Wave of a 

 certain amplitude. And as under proper restrictions we 

 know that such a wave is capable of propagation, the com- 

 parison of the relations under these circumstances is the 

 proper test of the justness of this mode of treatment. 



So long as mh is small, we can easily make the comparison 

 by writing the exact formula in the expanded form 



gh _ 1 4mV 



16, 



and substituting in the approximate expression. The expres- 

 sions are then found to agree identically as far as these three 

 terms in the expansion are concerned. 



