188 WAVES IN LIQUIDS. [CHAP. VII. 



It appears that the velocity of propagation is not independent of 

 the wave-length, but that it increases continuously with the value 



of the ratio -, being Jgh (as in Art. 148) when this ratio is in- 

 A/ 



finitesimal, and A / l | when it is infinite. To any given value of 

 c there corresponds only one value of X, and vice versa. 



157. Let us examine the nature of the motion of the 

 individual particles of fluid as a system of waves of the above 

 kind passes over them. If f, 77 be the component displace 

 ments at time t of the particle whose mean position is (#, y) t 

 we have 



We ought, in strictness, to have on the right-hand side of these 

 equations x + f for x, and y + tj for y, but the resulting correction 

 would be of the order a 2 which we have agreed to neglect. Inte 

 grating then the above equations on the supposition that x, y are 

 constant, and remembering the formulaB (28), (31) for c and a, we 

 find 



,. 



COS L ( X - C $&amp;gt; 



.(32). 



yar^zs-- ***(*- d)-\ 



Each particle therefore describes an ellipse whose major axis is 

 horizontal ; the law of description being the same as for a particle 

 attracted to a fixed point by a force varying as the distance. The 

 ratio of the minor to the major axis of the ellipse, viz. 



+ e -k(u+h) &amp;gt; 



diminishes from the surface to the bottom, where it vanishes. 



