72
HYDROSTATICS.
it will How to the greatest distance. For DE is the greatest per-pendicular. Also, since the perpendiculars are equal at equaldistances above and below D,the iluidllowing from those points,as from F and G, will reach to the same horizontal distance
NK.
Prop. IS. The quantities of a fluid proceeding in the sametime through different apertures, at equal depths, (supposing thatthe vessels are kept equally full,) are to each other as the productof the areas of the apertures by the square root of the heights.For the quantities discharged are as the areas and velocity, andthe velocity is as the square root of the depth, (Prop. 11,) there-fore the quantities are as the areas and square root of the depth.Thus it has been proved by experiment, that a circular apertureof one inch diameter, in a thin vessel, gives in one minute of time,the water being four feet high, 5436 cubic inches of water. Toknow what will be furnished in the same time by an aperture twoinches in diameter, the altitude of the water nine feet (Frenchmeasure,) use the following proportion (observing that the aper-ture of two inches is four times as large as that of one, because theareas of circles are as the squaies of the diameters) : as I X ^4 isto 4 X \/9, so is 3436 to x: or, as 2 is to 12, so is 5436 to 32,616cubic inches of water, the quantity that will he furnished by anaperture of two inches diameter from a reservoir whose surface isalways kept at nine feet from (lie aperture. If you till with watera prioinalic vessel, and let the water run out by an aperture in thebottom, observing the time employed by the water in runningout; and then fill the vessel again, keeping the surface of the wa-ter at the same height; you will find in this last case, that in thesame interval of time that the water was running out of the vesselin the first instance, nearly double the quantity of water is ex-pended in the second. In practice the water often issues fromlateral openings, which, although but small in comparison with thesize of the reservoirs, cannot be considered as having all theirpoints at an equal distance from the surface of the fluid. In thesecases, the usual method of determining the quantity of water flow-ing through the aperture depends on the following principles:Imagine the whole to be stopped by a plate, and this plate to bepierced with a great number of holes through which the waterescapes; now, considering each of these holes as a single insulatedaperture, the velocity for each will be according to the correspon-dent height of the fluid. If the number of these holes be infinite-ly augmented, or, what comes to the same thing, if the plate betaken away, the velocity of each point of the given aperture willbe as the height corresponding thereto; and in determining thequantity of effluent water, regard must be had to this 'inequality©f velocity. This mode of reasoning, however, is not conclusive ;for though it may be just as far as relates to the number of insu-lated holes, it does not appear that the water will flow exactly inthe same manner when the threads thereof are united, as whenthey proceed from small separate apertures. As the results oftheory, however, upon this plan differ little from experiments, itmay be useful to adhere to it till some better method is discover- |ed. The quantity of water flowing through holes in a given timeis not so great as might he expected, because the water does notflow in a compact parallel stream, but contracts in diameter oncoming out of the aperture, and this contraction extends to a dis-tance nearly equal to half the diameter of the aperture. The dia-meter of the contracted stream is to the diameter of the apertureas 3 to 4, or as 3~ to 4, or as Iff to 24, so that its area to that ofthe aperture is as 10 to 16: it is nearly the same thing when thewater flows from lateral apertures. This contracted stream is aproof that wilhin-side the vessel the lateral particles are directedtowards the hole, with different degrees of obliquity, which ob-liquity may be decomposed into two forces, one parallel to theplane of the hole, which contracts the fluid ; the other perpendi-cular to the same plane, which occasions the efflux. This con-traction takes place also when water passes through tubes, and thecontraction is at the entrance of the water into the tube, not at itsgoing out, where it preserves its cvlindric form. This contractionsensibly diminishes the quantity of water (iiat should be furnishedbv the tubes. To ascertain these facts, M. Bossut made a greatnumber of experiments, the results of which are as follows. Theapertures for the efflux of the water were all pierced perpendicu-larly in plates about half a line thick, and the time of each experi-ment was reduced to one minute.
Constant Height of the Water, Eleven Feet N a 'of cubic
Eight Inches Ten Lines from the Centre of inch. disc,each Aperture. in 1 min.
Exp.
1. With a circular horizontal aperture, six lines
diameter. 2311
2. With ditto, one inch diameter. ff281
3. With ditto, two inches diameter........ 37203
4. With a rectangular horizontal aperture, one
inch by three lines. 2933
5. With a square horizontal aperture, the side
one inch. 11817
6. With two ditto, the sides two inches. 47361
Constant Height Mine Feet.
7. Lateral circular aperture, six lines diame-
ter. 201S
8. Ditto, one inch diameter. 8135
Constant Height Lour Feet.
9. Lateral circular aperture, six lines diame-
ter . 1353
10. Ditto, one inch diameter. 5436
Constant. Height Seven Lines.
11. Lateral circular aperture, one inch diame-
ter . 628
From the above experiments M. Bossut draws the following de-ductions :
1. ‘ The quantities of fluid discharged in eciual times from dif-ferent sized apertures, tiie altitude of the fluids being the same,are nearly to each other as the areas of the apertures.’ Thus inthe second and third experiments the areas of the apertures are as1 to 4, and the watts'discharged 9,281 cubic indies; 37,203 i*nearly in the same ratio.
2. ‘ The quantities of water discharged, in equal times, by thesame aperture, with different altitudes of the reservoir, are nearlyas the square roots of the corresponding altilude'of the water in th°reservoir above the centre of the aperture.’ Compare the eighthand tenth experiments, in which the respective altitudes of the re-servoir were 9 and 4 feet, of which the square roots are 3 and 2 *and we find the water disdiarged by the first was 8135 cubi‘Jinches, the second 5436 cubic inches ; nearly in the proportion o'3 to 2.
3. ‘ That in general, the quantities of water discharged in thfsame time, by different apertures, and under unequal altitudes o'the reservoirs, are to each other in a compound ratio of the an’"’of the apertures and the square roots of the altitudes.’
4. ‘ That on account of the friction, the smallest apertures di* -charge less water than those that are larger and of a similar fignrf>the water in the respective reservoirs being at the same height.’
5. ‘ That of several apertures whose areas are equal, that whichhas the smallest circumference will discharge more water than the’others, the water in the reservoirs being at the same altitude,’ a"”this because there is less friction. Hence circular apertures a ,emost advantageous, as they have less rubbing surface under d' esame area.
The quantities of water, we find, expended in the foregoing efperiments are not nearly so much as thev ought to be, eonsidern'Sthe size of the apertures and the altitude of the reservoirs. I wjquantity discharged is diminished considerably by the friction, an ,by the contraction of the stream ; and probably on account also®the circular motion of the fluid: for the velocity which depenf.on the altitude of the reservoir is not sensibly altered. The d> £ference in the discharge of water, supposing, 1. 'That the area ^the stream is (lie same with that of the aperture; 2. That tl>>stream is contracted ; is as H) to 10: in other words, by supP<Ling the area of the orifice to be diminished in the proportion ot ^to 10, we may determine with sufficient exactness the efflux efluids from vessels where the surfaces are maintained at the san>height. If the water, instead of flowing through an aped." jpierced in a thin substance, passes through the end of a verb®tube of the same diameter as the aperture, there is a much g' ca . edischarge of water, because the contracted stream is greater in 1first instance than in the second. In the following experimen^