Manchester Memoirs, Vol. xlvi. (1902), No. 15. 5 



Consideration of the actual problem proposed. 



In the case of progressive surface waves in the 



question which we are to consider, I shall write 



F^ a\F{ U+ iV) + F{U- iV)] + b[F\ U+ iV) - F\ U- iVj]li 



/i = ai[F{u + iv) + F{ti - ivy\ + b^F\u + iv) - F\u - iv)\\i . (3) 



etc., where 6^+ z'F stands for 



{x - a')cosa + (jv - )'')sina - ^/sech/3 + /2tanh/3, 

 and u-\-iv for 



{x - .T')cosa + (7 -y)sina - -^/sechy + /ztanhy. 

 I shall take it to be an essential condition that the 

 amplitude shall diminish asymptotically to zero as the 

 depth increases. 



The condition, generally, that the stresses on the free 

 face shall be null, requires that 



and 

 R^\(x. .,)^, . x(g, + ^1)]^+ .,|,(/. +/, -/.) . (4) 



shall all vanish, when ^ = 0. 



In order to satisfy any such conditions, it will be 

 requisite, in the first place, that 



>^sech/3 = /2sechy = <7 (5), 



where ^ is the common rate of propagation of the com- 

 pounded waves. 



There may, of course, be solutions into which the 

 F functions only enter, and solutions into which the 

 / functions only enter, as well as those with which we 

 are mainly concerned in which both functions occur, and 

 in which the presence of both sets of functions is necessary 

 to satisfy the surface conditions. 



