88 Ma GREEN, ON THE MOTION OF WAVES IN CANALS. 



In order to examine more particularly the way in which the 

 Primary Wave is propagated, let ns resume the formula?, 



{lit ; _ e±d w, _/■»-.) . 



gdt g \ •> s/gyl 



where we have neglected the function f, which relates to the wave 

 propagated in the direction of x negative. 



Suppose, for greater simplicity, that fi and y are constant, the origin 

 of x being taken at the point where the wave commences when 

 t = 0. Then we may, without altering in the slightest degree the 

 nature of our formulae, take the values, 



(1) <p = F{x-tVg^), 



But for all small oscillations of a fluid, if (a, b, c) are the co- 

 ordinates of any particle P in its primitive state, that of equilibrium 

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

 q> = f<pdt when (x, y, %) are changed into (a, b, c), we have (Vide 

 Mecanique Analytique, Tome 11. p. 313.) 



d<t> , d^> d<& 



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



$ = r= 'F(a- t y/gy), and x = a -7= F(a - t Vgy)- 



Vgy vgy 



r 



Neglecting (disturbance) 2 , we have 



£ = -\A F'{a-ty/gy), 



