231-232] GERSTNER'S WAVES. 413 



in these waves is not irrotational detracts somewhat from the 

 physical interest of the results. 



If the axis of x be horizontal, and that of y be drawn vertically 

 upwards, the formulae in question may be written 



x = a -f T e kb sin k (a + ct), 



y = bj-e kb cos k (a + ct) 



K 



where the specification is on the Lagrangian plan (Art. 16), viz., 

 a, b are two parameters serving to identify a particle, and x, y are 

 the coordinates of this particle at time t. The constant k deter- 

 mines the wave-length, and c is the velocity of the waves, which 

 are travelling in the direction of ^-negative. 



To verify this solution, and to determine the value of c, we 

 remark, in the first place, that 



d(x,y)_ 



3<M)~ 



so that the Lagrangian equation of continuity (Art. 16 (2)) is 

 satisfied. Again, substituting from (1) in the equations of motion 

 (Art. 13), we find 



(* + )- kc^^k(a + et), 

 db\p 



(3); 



^P 



\ 



= kc 2 e^ b cos k (a 

 dl)\p "V 



whence 



- = const. g ] b j e kb cos k (a + ct) [ 

 P ( * 



- c 2 e* b cos k (a + c) + Jc 2 e 2fc6 (4). 



For a particle on the free surface the pressure must be 

 constant ; this requires 



cf. Art. 218. This makes 



= const. - gb + Jc 2 ^ 6 ................... (6). 



