TRACT 38.

problems:

353

same or an equal time, are equal. And as the force in thiscase, is the weight of the superincumbent column of thefluid-over the hole. Let the one body to be moved, be thatcolumn itself, expressed by ah , where a denotes the altitudeab, and h the area of the hole ; and the other body is thecolumn of the fluid that runs out uniformly in one secondsuppose, with the middle or medium velocity of that intervalof time, which is \hv, if v be the whole velocity required.Then the mass ihv, with the velocity v, gives the quantityof motion ±hv x v, or ^hv 1 , generated in 1 second, in thespouting water: also 2 g or 32-| feet, is the velocity gene-rated in the massa/t, during the same interval of one second;consequently ah x 2g, or 2 ahg, is the motion generated inthe column ah in the same time of one second. But asthese two momenta must be equal, this gives i/zw 1 — 2ahg :hence then -a 1 = 4ag, and v = 2*/ag, for the value of thevelocity sought; which therefore is exactly the same as thevelocity generated by the gravity in falling through thespace a, or the whole height of the fluid.

For example, if the fluid were air, of the whole height ofthe atmosphere, supposed uniform, which is about 5A miles,or 27720 feet = a. Then 2Vag — 2\/27720 x 16 T r ^ =1335 feet =: v the velocity, that is, the velocity with whichcommon air would rush into a vacuum.

Corol. 1. The velocity, and quantity run out, at differentdepths, are as the square roots of the depths. For the velo-city acquired in falling through ab, is as ^/ab.

Corol. 2. The fluid spouts out with the same velocity,whether it be downward or upward, or sideways; becausethe pressure of fluids is the same in all directions, at thesame depth. And therefore, if an adjutage be turned up-ward, the jet will ascend to the height of the surface of thewater in the vessel. And this is con tinned by experience,by which it is found that jets really ascend nearly to theheight of the reservoir, abating a small quantity only, for theYOL. HI. A A