rOL. XLI.] PHILOSOPHICAL TRAN8A.CTIOK8. 283 



Therefore the adequate motion of the water, flowing out of the hole, is tlie 

 pressure or weight of the column of water over the hole. But a given force, 

 howsoever applied, generates a given quantity of motion in a given time, to- 

 wards those parts whither the force tends. Therefore the weight of the in- 

 cumbent column, generates a like quantity of motion, in a given time, in the 

 effluent water, as it could generate in the same time in the column itself, fall- 

 ing freely through a vacuum. 



Now because, by the hypothesis, the particles of water find no resistance 

 for want of lubricity, and all those particles, which are just going out in the 

 very hole, are urged by an equal pressure of the superincumbent water, it is 

 plain that the velocity of all these is equal. 



Let V be that common velocity; a the height, in falling from which in 

 vacuo that velocity would be acquired ; A the height of the water above the 

 hole; V the velocity acquired by falling in vacuo from the height a; t the time 

 of falling from the same height; f the area of the hole; and let, the water flow 

 out of the hole in the time t. Now, because in the time t, with the velocity v, 

 the space 2a will be run over, the space — will be run over in the same time 

 with the velocity v. Therefore this will be the length of the column of water, 

 which flows out of the hole in the time t; and the magnitude of this column, 

 or the measure of the water flowing out in the time t, will be , and the 



motion of the same will be _1£L. But the motion which can be generated in 

 the column of water over the hole, in the same time, x, if carried by its own 

 weight through a vacuum, is thus. Its velocity will be v, and as its magnitude 

 is AF, its motion will be afv. But that motion, from what has been said above, 

 is equal to the motion of the column of water flowing out in the time t, or 

 afv = . Hence v = vy' ■ . 



V 



Also, the measure above assigned, of the water running out in the time x, 

 or = AFv/2. a. E, I. 



V 



Carol. 1. — Since a : a :: v^ : v% therefore a = ^ = ^A. Therefore the 

 height a, which the effluent water can reach, by turning the motion upwards, 

 is half the height of the water in the vessel above the hole, which is the very 

 height determined by Sir I. Newton, Princip. ed. 3, lib. 2, pr. 36. 



Carol. 2, — If we ascribe to the effluent water, that velocity which is acquired 

 by falling from the whole height of the water above the hole, that is, if we 



suppose t; := V, then the above determined motion of the water , is := 



2afv, or double that motion which can be generated by the column over the 



oo 2 



