386 Mr. J. R. Wilton on 



where X, c, (/>, and ^lr are' the wave-length, wave velocity, 

 velocity potential, and current function, respectively. 

 We assume that tj is a function of f , say, 



v=m)- ■ . ■ (i) 



The conditions which have to be satisfied are w=0 when 

 y = vo ; and, on the free surface, y = f (ct — x), 



where u and v are, respectively, the horizontal and vertical 

 velocities, and q is the resultant velocity. The axis of y is 

 vertically downwards, that of x horizontal. We shall also 

 take the origin at the trough of a wave when t=0. 



The first of the two surface conditions is satisfied whatever 

 the form of the function F, provided that f is real on the 

 surface, i. e. if 



ty~cy 

 on the surface. 



Let V = F'«') (2) 



be ihe equation obtained from (1) by changing the sign of 

 i throughout. We then have 



dw 



drj/dw \ 

 or c—u + iv=cl -ft 



and 



> — U — iV ~ c ' 



I ctg 



Also e =Ju|5 = g(*±i*+A 



2-7T dt dg\ c I 



i. e. 



o =- c +7^=-c«-«-), 



c 



therefore = — uc, 



so that the condition of constant pressure on the free surface 

 becomes 



