THE TWO GREAT SOLITARY WAVES OF RUSSELL. 



Let now x,^, co^ be the values of x for a given particle at the times t^, t^. 

 Then (18) gives 



cth ~ xt, eh-xh 



2 tan {nt^ - a) = e ^ - e ^~ , 



elk -art, ctk - JCk 



and = 2 tan {nt,^ - a) = e c _ g c _ 

 The last line shews that ct^ = x^ ; and the preceding line gives, remembering that n - nt , 



and .-. ct^ - X,, = C \og^ tan ( ^ cos"'- 



• •■ x^ - X, 



But ctj^ — x^ = 0; 



ih - Q = Clog, tan (^^ - loos-'-) 



Now 2(,T?j - x,) is the distance through which a particle is horizontally transferred b3' the 

 transit of a wave ; as this is an observable element we will denote it by /3; 



Also the wave has travelled over the space c{f^-ti^ in tiie time t^^-t^. 



Now if \ be the length of a wave, the wave in the time 2(/j - t^ has travelled over the 

 space \ + fi. 



.-. X + /3 = 2c{f, - f,), 



h 

 k 



ch 2ch , Itt , ,h 



and .■. \ = 2 C log^ tan I — + ^ cos 



„ , ch 2cfi , (■jT , ,«\ 



Consequently 'J = ^^ = y^ l°ge (^"J + i '^"^ j^j (20). 



2c h 



Also \ + /3 = 2c(<t - 4) = — cos-' - ; 



71 K 



k h 

 - cos - 

 /3 h k ^ ^ 



••• X + ' = (-'^- 



1 /■"" 1 ,h 



log^ tan - + * cos '- 

 V4 k 



We may consider this equation as giving the value of the length of a wave ; and then 

 (20) gives the value of n in terms of c. 



If we expand the terms of equation (21) we find, 



" = cos-'- cos-'- + &c (22), 



X h 6h\ kl 12/t \ kl ^ ' 



which shews that as /( diminishes, /3 diminishes compared with X. 



We may now proceed to determine the velocity of transmission ; and the equation of a 

 wave surface. 



