274 NOTE ON THE MOTION OF WAVES IN CANALS. 



where we have neglected the function /, which relates to the 

 wave propagated in the direction of x negative. 



Suppose, for greater simplicity, that ft and 7 are constant, 

 the origin of x being taken at the point where the wave com- 

 mences when t = 0. Then we may, without altering in the 

 slightest degree the nature of our formulas, take the values, 



(1), 



== 



gdt 



But for all small oscillations of a fluid, if (a, 5, c) are the 

 co-ordinates of any particle P in its primitive state, that of equi- 

 librium suppose ; (x, y, z) the co-ordinates of P at the end of the 

 time , and <&=f<j)dt when (x, y, z) are changed into (a, Z>, c), 

 we have (vide Mecanigue Analytique, Tome II. p. 313), 



Applying these general expressions to the formulae (1) we get 



and . . x = a j=F(a t *Jgi) 



Neglecting (disturbance) 2 , we have 



;;; - ^ 



and consequently, 



supposing for greater simplicity that the origin of the integral 

 is at a = 0. 



Hence the value of x becomes 



