Manchester Memoirs , Vol. xliv. (1900), No. t>. 3 



the case of cither small or large values of (/>, the equation 

 connecting x and y can be formed and is found to 

 give relations in the form found by Boussinesq and 

 Lord Rayleigh. 



To show this relation wo may conveniently write the 

 exqM'essions 



W(/j / lIKh ., -, . , 



jc = t + 2//tanh - / ( I + tanh- tan-w/5) sin 2w/J, 



J' = /3 + /^ sech-//^a- / ( I + ~. X ) ; 



V sin 2mnJ 



y -ft + h seclr— ^ / (i + tanh- — ? tan^w/i) , 

 c c 



whence we obtain, when ^ is small, 



znih 



and, when ^ is large, 



7 = /3 + h cos^ VI ft sech^ mx . 

 This function would then appear to give a wave surface 

 having something of the character of that of the solitary 

 wave. It will be seen later to what extent this will be 

 found insufficient to satisfy the conditions which it is 

 proposed to apply. 



In working out the analytical conditions, I take 

 another term in the suggested series, and attempt to find 

 whether the expressions 



X 4- I'y =/{<p + i\p) 



- X — T 4. x^ tanh pi ^- ^ + Xs tanh^';;; - — - . . (4), 



(where the free surface is given by \p = c(5, and 



/?! = Ai tan m3, h^= - Xg tan" w/3, 



and ]i = Jh-\-Jh is the greatest elevation of the wave), 

 can be made to give a good approximation to the 

 known characteristics of the wave, subject to the 

 condition that Jh : h\ is a small quantity so that the 

 general shape of the free surface indicated by the first 



