320 OF THE MOTIONS OF 1- LUIDS. 



being d -^ gzn--^ g^ making dar constant, it may be sup- 



posed to be increased, with reference to the acceleration of 

 the upper surface of the fluid, in the ratio of the synchronous 

 variations Sdx and Sy, or that of da: to y, and it will then be- 

 come — . -~- cr=~-^crv» which will be the measure of the 

 djr dx ^ &x-^^ 



acceleration of the surface, and the surface will ascend or 



descend precisely as if immediately subjected to the opera- 

 tion of such a force. We may therefore inquire what 

 must be the velocity of a body moving along the curved 

 surface, or what must be the horizontal velocity of a similar 

 surface moving along through the body, in order that the 

 vertical motion should represent the efi*ect of the force 



-j^gy- Now in the common expression of the magnitude 



of a force acting in the direction ofy, we say/bi — ^; we 



^ ^x r I tld?/ dd?/ dix" - dx 



must thereiore make —-^z=.-—^qy, or — — nz ov, and -— 



zz s/ (gg) : consequently if x flow with the constant velocity 



dx 

 v=— — :=. s/igy)* the second fluxion of y will always repre- 

 sent the actual acceleration of the surface of the fluid, the 

 part of the curve corresponding to the time t always repre- 

 senting the actual position of the particle, as well as its mo- 

 tion. But s/{gy) is the velocity Required by a body in falHng 

 through i y, since in general v-—2gs, (232) and v— s/{2gs), 

 or =■ s/{2gM). In this simple manner we attain a strict de- 

 monstration, on the premised supposition respecting tlie 

 nature of the fluid, that the velocity of the surface will be 

 represented by that of the surface of a wave advancing with 



