Chap. XVII.

THEORY OF THE MOTION OF FLUIDS.

1127

A, r, which shows that the effort made by the water of the vessel to occupy

Eli, c its circumference, and A the height EF of the water. Thus we shall have

for the pressure which the surface sustains, and for its weight; whence we have

h k c r he

2 _ ’ “T"

the place of the column, is to the inclination of this column to descend, as its height is tothe radius of its base. Hence we may conclude that when the height of the columnexceeds its semi-diameter, the portions of the water which compose it can never escapealtogether by an opening equal to its base, because the force with which it will tend todescend will be less than that of the water which seeks to replace it: thus showing thatduring the time of flowing the column will always have the same weight, since the partswhich escape will instantly be replaced by others.

It follows that when water in a vessel is continually maintained at the same level, thatwhich escapes by an orifice made in the bottom will always have the same velocity, sinceit will be driven by the whole weight of the column, which presses it, and which we mayregard as a constant force acting uniformly on the whole extent of the orifice.

The same will not take place with water in a straight tube, which voids itself by anopening equal to the base, because it falls in one piece, like a cylinder of glass, that is tosay, that in issuing it has at first a very slight velocity, which increases like that of heavybodies, from the instant of its fall. As the water of the column is not replaced eitherfrom the top or sides, its surface answering immediately to that of the vessel, it is in theposition of all heavy bodies, and consequently follows the law of their accelerations, notmeeting any circumstance which can cause a change in its fall. The time which such atube would take to empty itself totally will be equal to that required by a body indescending freely through the same space which the upper surface of the water goes overin the tube.

Since we can always render a retarded or acceleratedvelocity uniform, by taking half the great velocity, wemay apply the rule when we wish to compare the ex-penditure of a tube such as the preceding with that ofanother always maintained full.

Fig. 1709. If the surface of the water contained ina tube or reservoir, after having remained during acertain time at the same level B C, was then at the levelF G, notwithstanding the expenditure through theorifice E H in the bottom of the vessel: its uniformvelocity in the first case would be to its uniform velocityin the second, as the square root of the height I E is tothe square root of the height K E.

Fig. 1710. When in two reservoirs of different heights,the orifices 0,0 are unequal in superficies; the smallprisms of water which issue therefrom in the same instantmay be expressed by O V and o u, because they can onlyhave for base that of the columns which impel them,and for height the space which a film detachedfrom the same columns may go over with auniform motion in this instant. As these spaceswill be in the ratio of the velocities exercisedin the same instant, the velocities may holdthe place of spaces, since in this case thequestion is only of the ratio of these prisms.

As the columns of water which the orificeswould expend in two different times T, t, willcontain as many times the small prisms OV,ou y as the number of instants that elapse in

the duration of time T,^ we shall have-^ = OV,

and — =ou: whence we havet

M mT’ 7 :

O V,on,

which gives

Mt

iOV

M o t u =

OT'i'o [|1

Fig. 1710.

T t

m O T V, or a general formula comprising themasses or quantities of water, their velocities,the time of their flowing, and the size of the

orifices ; that is to say, all the circumstances which enter into the measurement of the water.

4 c 4