280 NOTE ON THE MOTION OP WAVES IN CANALS. 



What immediately precedes is not given as new, but merely 

 on account of the extreme simplicity of the analysis employed. 

 We shall, moreover, be able thence to deduce a singular conse- 

 quence which has not before been noticed, that I am aware of. 



Let (a, I, c) be the co-ordinates of any particle P of the fluid 

 when in equilibrium. Then, since 



6 = jffif^' sin ? ('*-*); 



A< 

 -H\ -^ 27T, 



'=- ^os-(^-a), 



and the general formulse (2) give 



d H ^ . 27T , . 



x=*a+-j- = a -- re * sin -(vt a), 

 da v \ ^ 



d$> H -^ 27T,, 



z = c + -7- = c + A cos - (v t a). 

 ac v A. 



Hence, 



and therefore any particle P revolves continually in a circular 

 orbit, of which the radius is 



round the point which it would occupy in a state of equilibrium. 

 The radius of this circle, and consequently the agitation of the 

 fluid particles, decreases very rapidly as the depth c increases, 

 and much more rapidly for short than long waves, agreeably to 

 observation. 



Moreover, the direction of the rotation is such, that in the 

 upper part of the circle the point P moves in the direction of 

 the motion of the wave. Hence, as in the propagation of the 

 Great Primary Wave, the actual motion of the fluid particles is 

 direct where the surface of the fluid rises above that of equi- 

 librium, and retrograde in the contrary case. 



