338 PHILOSOPHICAL TBANSACTIONS. [aNNO 17 18. 



Demonstration of the second part. — ^The bulk of water running out in a given 

 time, is to the cylinder under the hole and height of the water, as the length 

 of the space the effluent water shall run over with an equable velocity in that 

 given time, is to the height of the water. And since the velocity which is 

 ascribed to the bulk of the effluent water, is to the velocity of the cylinder, 

 reciprocally in the same ratio, the quantities of motion in both will be equal. 

 a. £. D. 



Cor. 1 . — Having given the height of the water, and the bulk of the effluent 

 water, the motion of the cataract is in the inverse ratio of the time in which 

 that bulk runs out. 



Cor. 2. — Having given the height and the time, the motion of the cataract 

 is as the bulk of the water that runs out in that time. 



Cor. 3. — Having given the height and the bulk of the effluent water, the 

 motion of the cataract will be in the ratio of the height. 



Cor. 4. — Having given the motion and height of the cataract, the bulk of 

 the effluent water is in the ratio of the time. 



Cor. 5. — Having given the motion of the cataract, and the bulk of the 

 effluent water, the height is as the time. 



Cor. 6. — Having given the time and motion of the cataract, the bulk of the 

 effluent water will be reciprocally as the height. 



Theor. 2. — If ab, as in fig. Ip, be taken so as to be to bd, as dg* is to dg* — 

 BC'' ; and the water running out at a round hole cc, in the middle of the bot- 

 tom of a given cylindrical vessel ggee, which is always full ; the motion of the 

 cataract of water towards the horizon, will be equal to the motion of the 

 cylinder under the hole and height ab, whose velocity is equal to that of the 

 water running out at the hole ; or it will be equal to the motion of the bulk of 

 water that runs out in any given time, and whose velocity is such, that a space 

 equal to the height ab, may be run over in the same given time. 



Demonstration of the first part. — Let As be drawn parallel to dg, and let the 

 Newtonian curve sgc be supposed to be drawn with the asymptotes as, ab, 

 through the points gc. To have the height of the water constant, the place of 

 the water that runs out, is to be supplied with a cylinder of water ^^gg, de- 

 scending with that uniform velocity, which is acquired by falling from a to d, 

 as the incomparable Newton demonstrates in the above-mentioned proposition. 

 The motion of the cataract ssgg, is equal to the motion of this cylinder by the 

 preceding theorem. Therefore the motion of the descending water, since it 

 is compounded of the motion of the cylinder of water ^g^GG, and of the mo- 

 tion of the cataract ggcc, is equal to the motion of the whole cataract sgccgs, 

 that is, by theor. 1, to the motion of the cylinder of water whose base is the 



