NOTE ON THE MOTION OF WAVES IN CANALS. 275 



Suppose a = length of the wave when t = ; then f (a) = 0, 

 except when a is between the limits and a. If therefore we 

 consider a point P before the wave has reached it, 



the whole volume of the fluid which would be required to fill 

 the hollow caused by the depression below the surface of equi- 

 librium when t = 0. Hence we get 



x = a + j 

 7 



x being the horizontal co-ordinate of P, before the wave 

 reaches P. 



Also, let x" be the value of this co-ordinate after the wave 

 has passed completely over P, then 



I dat, (a - 1 V#7) = 0, and x" = a. 



<>o 



If f were wholly negative, or the wave were elevated above 

 the surface of equilibrium, we should only have to write V 

 for F, and thus 



y 

 x a -- , and x" a. 



We see therefore, in this case, that the particles of the fluid 

 by the transit of the wave are transferred forwards in the direc- 

 tion of the wave's motion, and permanently deposited at rest in 

 a new place at some distance from their original position, and 

 that the extent of the transference is sensibly equal throughout 

 the whole depth. These waves are called by Mr Russel, positive 

 ones, and this result agrees with his experiments, vide p. 423. 

 If however f were positive, or the wave wholly depressed, it 

 follows from our formula, that the transit of the fluid particles 

 would be in the opposite direction. The experimental investi- 

 gation of those waves, called by Mr Russel, negative ones, has 

 not yet been completed, p. 445, and the last result cannot there- 

 fore be compared with experiment. 



182 



