The Classes of Progressive Long Waves. 213' 



various terms in (7). If the system be such that 



does not continually increase, we obtain, as in the case of 

 the virial theorem, 



2*»^|US(*Y+yX) (8) 



It would seem that this equation has application to the 

 molecular theory of the viscosity of gases, analogous to that 

 of the virial as applied to hydrostatic pressure. 



XVI. The Classes of Progressive Long Waves. 

 By R. F. Gwsther, M.A * 



ADOPTING the method employed by Lord Rayleigh in 

 his paper " On Waves " f, write 

 <fi 4- ity = F ( x 4- if) 

 where F is a real function, and therefore 



* = Fy.-F"'^+etc. l .... (I) 



while the condition expressing the uniformity of pressure 

 along the free surface, for which yjr= —ch, is 



u* + v* = c*-2gfy-h) (2) 



But now, instead of obtaining a differential equation 

 approximately related to the free surface, proceed to elimi- 

 nate y between (1) and (2), putting ^=— ch. 



The convergency of these expressions in the case of long 

 waves is clear, and we easily obtain 



F»-(F / F /// -F"*)^+ etc. 



= ^ + 2 9 h + 2g[^ + C ^ r/I + etc.j. . . (3) 



This mode o£ treatment has the merit of simplicity and also 

 of allowing the constants introduced in the integration by 

 approximate methods to be treated with a feeling of security. 



Write (3) to a first approximation 



(c>+ C ^)h*F>'' = ¥<*-(c> + 2gh)Yi-2gch. . (4) 



* Conirnimicated by the Author. 

 t Phil. Mag., April 1876. 



Phil. Mag. S. 5. Vol. 50. No. 303. Aug. 1900. Q 



