a Free Procession of Waves in Deep Water. 117 



From this and the second form of (17), we find 



whence, by (18), 



d? d*P (r + y + bf . /00 . 



'^riBi- - r mm • • ■ (22): 



and therefore finally, by (3) above, we have, for the surface 

 pressure, 



lV=o) = C + | V gq^g J «pb< . . (23), 



as promised in (12) above. 



To work out our solution, remember that dV/dt is the 

 velocity-potential of the motion : and calling this <E>, we find, 



by (19), 



®= -Vdt' sin co(t-t')cj>{t f ) . . * . (24); 

 and by (22), (3), and (2) we find 



1 jd® (r+,y + &)* . X t9to 



r)=z—<-j- + ± Z '—$m(ot> . . . (25). 



g{dt r ) 



What we chiefly want to know is the surface-value of rj, 

 which we have denoted by — -(j ; and we shall work this out 

 for the case b = 0. But it is to be remarked that the assump- 

 tion of b — does not diminish the generality of our problem, 

 because the motion at any depth, c, below the upper surface 

 with 6 = 0, is the same as the motion at the surface, with b = c. 



Put now b = Q and y = in (15) : we find 



Using this in (24), and putting a 2 =zgt' 2 j4x, we find 

 = -2A/-f ^^sin^-2a>A/-^sin^ 2 + ^y . (27), 



=x /y; v4 *{~[('-Vi)*-i-»f] 

 -"•[('■^VS'-i-'+i]}- <28> 



