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•own altitude. The quantity of motion generated in the cylinder 

 in the time of the fall is equal to the fum of the quantities of 

 ■motion generated in that cylinder in all the little portions of time 

 into which the whole time is divided, taken feparately ; that is, 

 «qual to the fum of the quantities of motion generated by th^ 

 prefTure of that cylinder, in the fame little portions of time, in thp 

 fuccefllve plates of water (becaufe equal forces generate equal 

 quantities of motion in the feme time) j that is, equal to half 

 the quantity of motion in thefe plates as meafured by the ulti- 

 mate velocity continued uniform for thefe portions ,of time. 

 Whence it follows, that the quantity of motion generated in the 

 •incumbent cylinder, if it were fuppofed to fall freely through its 

 'height, is equal to half the quantity of motion of the cylinder 

 fuppofed to be difcharged in the fame time with a uniform velo- 



A x'F 

 city. That is, AFV= — —, whence V* = a' % and V = .v ; that 



is, the ultimate velocity with which the water is difcharged is 

 equal to that which a heavy body would acquire in falling through 

 the entire height of the water aboye the orifice. 



The mode of demonftration which Sir Ifaac purfues in the 

 fecond and third editions of his.Principia, and which has been 

 admitted and difcuffed at large by Juxin, Maclaurin, Robinfon, 

 •and other mathematicians, is this: Let MNCD be a cylindric 

 veflel filled with water to the height A B ; C D its bqttom parallel 

 tto the hcpzon ; EF a circular hole in the bottom, and IG a per- 

 pendicular to the horizon palling through the centre of the hole. 

 iNewton then fuppo^es water to be pouied in at the upper furface 

 AB as faft as .it fubfides by the efflux of the water through tiie 



aperture 



