Change of Form of Long Waves. 433 



I dx I dy \ {l + rj)dx 



h Jo Jo 



,-./, , £ + /» A + AE(K)\ .... 



r^(l+.-gj FTr) < 27 > 



On subtracting this velocity from that expressed by equa- 

 tion (21), we obtain 



u ' =u - q ^-\f£^+k-(k+h)^-y=-^(v- ^y, (28) 



and it is obvious at once that in this manner we have annulled 

 the velocity of the particles for which 



V 



v= T 



This last equation has a simple geometrical meaning. It 

 designates those particles E (fig. 1) whose height above the 

 bottom of the channel is equal to the height where the surface 

 of the liquid would stand when the waves were flattened. 

 Therefore for a first approximation we may say that the 

 various particles of the fluid change the direction of their 

 horizontal motion at the very moment when one of these 

 points E is passing over them. 



We now proceed to the calculation of the path of a single 

 particle of fluid. Let # , y denote the coordinates of such a 

 particle at the origin of time, and a} — x + %^ y l z=y -\-^ 1 its 

 coordinates at the time t, u' and v l its horizontal and vertical 

 velocity at that time, l + rf its elevation above the bottom, then 

 we have , 



Here rf is equal to the value of rj for x=id +qt\ and there- 



