a Free JProcession of Waves in Deep Water. 119 



Now, remembering that cay (oo ) = say (oo ) = 1 / ttj we see 



that if co \/ - is large, and if t \f 4- <*> \J ~ * s ^ ar g e 



positive, we have 



R=0, Q=V?r ? 6=0 .... (36); 

 and therefore 



*^\/T oos (f- (0t ) • • • ■ (37); 



whence, and by (25) with 5 = 0, 



or, since co*/(x\g) is very large, 



,. co /2tt . /co 2 x \ /onN 



b=-Wj sm \7~ ) • • ■ (39X 



This represents a uniform procession of free waves, of which 

 the wave-length, X, and the wave-velocity, U, are as follows: — 



\ = 2tt 9 /co 2 , JJ=g/co (40). 



To explain the meaning of " very large " as we have just 

 now used it, let 



,# = tt\, which makes co\f- = ^lirn^ and —\f -J^— = l/£7r s /n (41). 



Hence the term of (38) omitted in (39) is ljAir^n of that 

 retained. And the value of the R, omitted by (36) in (37), 

 is of the order 1/2 ^2n of the Q which is retained, because 



/ \ / fa — \- sm (27rn) 



cay (co) -cay ( V 2irn)= — - — ' > 



Is/ lirn 



and say (co)— say (\/2™)= C ° S ^ Z 7I ^ I . . . (42), 



when n is very large. 



In (36) and its consequence (31), we supposed t so large 



that t\/ ~ — co\/- is large positive : let us next suppose 



t so small that it is large negative ; that is to say, let 



t=2<oxlg- m/s J^ (43), 



