THE TWO GREAT SOLITARY WAVES OF RUSSELL. 331 



.-. y = A cos (^nt - a) (10), 



and V = S,y = - n A sin (nt - a) (H)- 



V 



Also d,u = = OT tan (nt - a) (12). 



y 



If we knew the greatest and least values of y for any particle we should be able to deduce 

 results from these equations. Now for a particle in the surface, k and h are the greatest and 

 least values of y. If we call t^, #,, the values of t when the particle has these values for its y ; 

 then V = 0, when t = t,. and y = k ; 



.-. wi'j, - a = from (ll) (l3), 



and .-. k = A from (lO) ; 



.-. h = k cos {nt,^ - a) from (lO) 



= k cos («<,_ - nt,^ from (13); 



•■• '*~'^ = ;,'^''^"^- ^'*)- 



Since u is positive or negative according as a particle is in its ascending or descending phase, 

 it appears from (II) that nt is less than a as long as the vertex of a wave is behind a particle; and 

 equal to a when the vertex is passing it ; and greater than a when the vertex has passed it. 

 Hence the functions on the right-hand side of the equations (10) (ll) (I2) are to be treated 

 discontinuously, i.e. their variation is to be confined within certain limits; between these limits 

 however their variation is continuous. Since, from the nature of the case y cannot be zero 

 for a particle not originally at the bottom of the canal, it appears from (10) that nt ~ a must 



always be less than — . Equation (11) shews that the vertical velocity does not begin from 



zero ; but that it suddenly has a finite value, which gradually decreases till it is all lost ; at 

 which moment the particle begins to descend, gradually regaining the lost velocity, which being- 

 accomplished it is as suddenly lost, as it was suddenly generated. All this agrees exactly 

 with the recorded observations of Mr. Russell (see Report, p. 342). Equation (14) gives half 

 the time during which the vertical motion of any particle lasts. Consequently the time a wave 



. 2 _ fe 

 takes to pass a particle is - cos ' - (I5). The quantity n is unknown at the present 



stage of our investigation. 



We must now proceed to integrate the equation (8). For this purpose we must remember 

 that (4) gives us 



u = (p(ct - ,i), 



which being written in (8), using (p for (p {vt — ,v) for brevity, we have 



n^ = - cd^(p + (pdj'(j) - (rfj0)-, 



rf, (c - (p) _ d,(pd/(p 

 c - (p n' + {d,<p )- ■ 



from which by integration wc find 



c -(p = C v/n" + (d.(py ; 



U U 2 



