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 



so = a + r e kb sin k (a + ct\ 



= b Te kb cos k (a + ct) 



ox 



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

 a, b are two parameters serving to identify a particle, and #, 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 



_ kb 

 d(a,b)~ 



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 



-j- (- + = e sn 



, 



%r ( - + 



db \p 

 whence 



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

 P I * ) 



- c 2 e kb cos k(a + ct) + $c*e* kb ...... (4). 



For a particle on the free surface the pressure must be 

 constant ; this requires 



&amp;lt;? = g/k ............................ (5); 



cf. Art. 218. This makes 



= const. - gb + |c 2 e 2W&amp;gt; ................... (6). 



