Mr GREEN, ON THE MOTION OF WAVES IN CANALS. 95 



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 consequence which has 

 not before been noticed, that I am aware of. 



Let {a, b, c) be the co-ordinates of any particle P of the fluid when 

 in equilibrium. Then, since 



27T 



5 7T 



- HX -VL1 2tt 



<b = He \ sm — (v't-x); .-. <t> = — — — e ~ *• cos . (v't - a), 

 T X 2ttv' X ' 



and the general formulae (2) give 



x = a + -r~ = a — — e *• sin — - (v t - «), 

 da v X ' 



d<b H -!?« 2tt , . 



% = c + -j-=c-\ r e * cos — - w t — a). 



dc v X ' 



Hence, 





and therefore any particle P revolves continually in a circular orbit, of 

 which the radius is 



H «. 



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 equilibrium, and retrograde in the contrary 

 case. 



