4 GWVTUEK, Oil the Propagatiivi of a Solitary Wave. 



term is not gravely altered. It seems more than probable 

 that if a series of these terms were included we could obtain 

 a good representation of the wave b)' ensuring that the 

 surface conditions are satisfied for a series of values of 0, 

 say = /{v/3, when k represents a series of suitably selected 

 numbers. This method of treatment being too laborious, 

 we take only two of the terms and in order to ensure the 

 fulfilment of the conditions over a wide stretch of the wave, 

 we apply conditions to cover all cases when ^ is consider- 

 able, and also when (/> = o at the crest of the wave. 



The general expression for the pressure at any point 

 in the fluid is 



when C is an absolute constant. Hence the defect of 

 pressure at any point of the free surface below that at an 

 infinite distance along that surface is 



r.{y-\^)—Jf--j^j^y-^^.i.^)\ ■ ■ (6)- 



Tliis defect of pressure we arc unable to make rigidly null 

 at all points ; we first make it very small for considerable 

 values of ^j. For this purpose we write it as a fraction 

 having the terms arranged in powers of 



(2W0 -, \ , j^ 



cosh—' + cos2;///> I, or i/y>', 



and by making the coefficients of the lower powers of 

 \\D in this fraction null, the defect of pressure over the 

 outskirts of the wave will be made negligible, and we thus 

 learn the circumstances under which the wave must 

 travel if, as in the solitary wave, the conditions seem most 

 completely satisfied at a distance from the crest. 



