Manchester Memoirs, Vol. I. (1906J, No. 8- 7 



On comparing with Stokes' results {loc. cit., p. 318), 

 and allowing for the change in the form of the coefficients 

 which I emplo}-, the results will be found to agree as far 

 the 4th order. 



If 2a stands for the height of the wave from trough to 

 crest, we get from (7) and (4) 



The next stage in the process is to apply the condi- 

 tion (2) to express the necessary relation between ii, a, b., 

 and /. 



This requires 



or 



/[{ I + 2y¥;-} + 2 {iT; + 2/r„y7;,+i}cos ku 



+ 2[H,,+ I,//„JI„+.,}cos 2ku + ...'\du^ct ^a. 

 This becomes 



{ I + ^H;)ku + f {/^i + 2Zf„/7„+i}sin ku 



+ %{H--v^H„H,,^.)%\x\ 2ku+ ... =k{ci + a). 



For convenience, I shall divide by the coefficient of 



ku, and write this 



^ + ^jSin '^ + -£'^sin 2i^ + ... =^ . . (8), 

 where 



(p = ku, 



H = k{a + a)j{i +I.ff„% 

 E=2{H^ + 2^,^.+,}/(i . { I + ^H,}]), 

 E.^=2{H.^ + ^H„H'„+.,]I{2 . { I + Si7„2}), etc. 



These expressions contain, though not in a very 

 convenient form, the connections which we are seeking, 

 and the solution is contained in (8) and the relations 



kx = <h + 'L — ^sin«0, 



H 



ky = kb -^ -- cos n<h. 



71 



