6 GWYTHER, Range of PiOgressive Waves in Deep Water. 



where w is an integer, and /^ is a quantity taken as 

 defining the wave length and is a constant. 

 From this it follows that 



f\u + ib) = I + 2//„^-"'V"'" = I + :S^,«'"'" . (5), 

 where 



H^^h^fi-""' . . • • (6). 



Since the surface is, in the steady motion, the path of 

 a fluid particle, it is one of the lines for which /? = constant, 

 and we may most conveniently select this to be the line 

 from which b originates, so that <^ = defines the free 

 surface of the fluid. 



The condition necessary to secure the constancy of 

 pressure at the free surface is, from (3), that 



2gy - c^lf\u + ib)f\it - ib) 



shall be independent of u, when /; = 0. 

 This requires that 



2g^-h,f:o's,nku + ^(^(i + 2/^,/'"'")-\i + 2;//„^-""'"j-^ 



71 



shall be independent of ji. 



The determination of the resulting relations to any 

 desired degree of approximation, after the manner of 

 Stokes, is merely a matter of labour. I have carried the 

 approximation to the 6th order, in order to show rather 

 more of the character of the final solution of the Lagran- 

 gian equations. Stating the results only, I obtain 



^e = H^/lf (7) 



