PROPOSITION XXXVI. PROBLEM

To define the VIII. motion of water running out of a cylindrical vessel through a hole made at the bottom. Let AC D B be a cylindrical CD vessel, AB the mouth p

Q= EF a G the the horizon, of it, the bottom p the of middle circular hole in the bottom, irallel to GH the axis of the cylin- of the hole, and And suppose a horizon. the to perpendicular c-?ritre cler K j cylinder of ice APQ,B to be of the same breadth with the cavity of the vessel, and to have the same axis, and to descend perpetually with an uniform motion, and that its parts, as soon as they touch the and flow water, superficies AB, wn by their weight into the vessel, and in their dissolve into ( compose the cataract or column of water passing through the hole EF, and filling up the same exactly. Let the uniform velocity of the descending ice and of the contiguous water in the circle AB be that which the water would acquire by falling through and through lie in the same right line and let IH and the space IH fall ABNFEM, ; HG ;332 THE MATHEMATICAL PRINCIPLES the point I let there be and meeting the ice drawn the right [BOOK Jl KL parallel to the horizon K and L. Then the ve EF will be the same that line on both the sides thereof in locity of the water running out would acquire by falling from Galih cJ s Theorems, IG will be at the hole it through the space IG. Therefore, by to IH in the duplicate ratio of the velo I r city of the water that runs out at the hole to the velocity of the w ater in the circle AB, that is, in the duplicate ratio of the circle to the circle AB EF those circles being reciprocally as the velocities of the water which ; in the same time and in equal quantities passes severally through each of We are now considering the velo them, and completely fills them both. But the mo city with which the water tends to the plane of the horizon. tion parallel to the same, by which the parts of the falling water approach to each other, is not here taken notice of; since it is neither produced by gravity, nor at all changes the motion perpendicular to the horizon which the suppose, indeed, that the parts of the water cohere gravity produces. We that by their cohesion they may in falling approach to each othei with motions parallel to the horizon in order to form one single cataract, a little, prevent their being divided into several but the motion parallel to the horizon arising from this cohesion does not come under our present and to

consideration. CASE 1. Conceive now the w^hole cavity in the the falling water ABNFEM, w hich encompasses water may pass r vessel, to be full of ice, so that the through a funnel. Then if the water pass very near to the ice only, without touching it; or, which is the same tiling, if by rea son of the perfect smoothness of the surface of the ice, the water, though r touching it. glides over it w ith the utmost freedom, and without the le-ast through the ice as EF with the resistance; the water will run through the hole as before, and the whole weight of the column of water same velocity will be ABNFEM all taken up as before in forcing out the water, and the bottom of the vessel will sustain the weight of the ice encompassing that column. Let now the ice in the vessel dissolve into water yet will the efflux of ; It will not be the water remain, as to its velocity, the same as before. to descend it will not will endeavour dissolved the ice now because less, ; be greater, because the ice. now become water, cannot descend without hin The same force dering the descent of other water equal to its own descent. in the effluent water. the same to velocity generate ought always But the hole at the bottom of the vessel, by reason of the oblique mo tions of the particles of the effluent water, must be a little greater than before, For now the particles of the water do not all of them pass through the hole perpendicularly, but, flowing down on all parts from the sides of the it with oblique mo vessel, and converging towards the hole, pass through in tending downwards meet in a stream whose diameter is a little smaller below the hole than at the hole itself its diameter being to the tions

r,r,d = OF NATURAL PHILOSOPHY. SEC. V1L! diameter of the hole as 5 to 6, or as 5^ to 6|, 333 very nearly, if I took the procured a very thin flat plate, hav a diameter of the circular hole being hole in the the middle, pierced ing an of of inch. that the stream And f parts running waters might not be accelerated in falling, and by that acceleration become narrower, I fixed measures of those diameters right. I this plate not to the bottom, but to the side of the vessel, so us to make the water go out in the direction of a line parallel to the horizon. Then, when the vessel was full of water, I opened the hole to let it run out and the ; diameter of the stream, measured with great accuracy at the distance of about half an inch from the hole, was f of an inch. Therefore the di J- ameter of this circular hole was to the diameter of the stream very nearly as 25 to 21. So that the water in passing through the hole converges on all sides, and, after it has run out of the vessel, becomes smaller by converg ing in that manner, and by becoming smaller is accelerated till it comes to the distance of half an inch from the hole, and at that distance flows in a smaller stream and with greater celerity than in the hole itself, and this 25 X 25 to 21 X 21, or 17 to 12, very nearly that is, in about the subdaplicate ratio of 2 to 1. Now it is certain from experiments, that the quantity of water running out in a given time through a circular in the ratio of ; made in the bottom of a vessel is equal to the quantity, which, flow with the aforesaid velocity, would run out in the same time through ing another circular hole, whose diameter is to the diameter of the former as hole 21 to 25. And therefore that running water in passing through the downwards equal to that which a heavy body hole itself has a velocity would acquire in falling through half the height of the stagnant water in the vessel, nearly. But, then, after it has run out, it is still accelerated by arrives at a distance from the hole that is nearly equal to and diameter, acquires a velocity greater than the other in about the ratio of 2 to 1 which velocity a heavy body would nearly subduplicate the whole height of the stagnant water in the acquire by falling through converging, till it its ; vessel. Therefore in what follows let the diameter of the stream be represented by that lesser hole which VW called EF. And imagine another plane above the hole EF, and parallel to the plane there we of, to be placed at a distance equal to the diame same hole, and to be pierced through ter of the with a greater hole ST, of such a magnitude that a stream which will exactly fill the lower hole EF may pass through it ; the diameter of which hole will therefore be to the diameter of the lower hole as this means the water 25 to 21, nearly. By run perpendicularly out at the lower hole and the quantity of the water running out will be, according to the magnitude will ;THE MATHEMATICAL PRINCIPLES 334 of this last hole, the same, very nearly, which the solution of the Problem The space included between the two planes and the falling stream be considered as the bottom of the vessel. But, to make the solution requires. may [BOOK 11 more simple and mathematical, it is better to take the lower plane alone bottom of the vessel, and to suppose that the water which flowed through the ice as through a funnel, and ran out of the vessel through the for the hole EF made in the lower plane, preserves its motion continually, and that let ST be the diame of a circular hole described from the centre Z, and let the stream run Therefore in what follows the ice continues at rest. ter out of the vessel through that hole, when the water in the vessel is all And let EP be the diameter of the hole, which the stream, in fall fluid. ing through, exactly fills up, whether the water runs out of the vessel by that upper hole ST, or flows through the middle of the ice in the vessel, ST And let through a funnel. the diameter of the upper hole be to the 25 to 21, and let the perpendicular di& tance between the planes of the holes be equal to the diameter of the lesser hole EF. Then the velocity of the water downwards, in running out of as diameter of the lower EF as about same that a body and the velocity of both the falling streams will be in the hole EF, the same which a body would acquire by falling from the Avhole height IG. CASE 2. If the hole EF be not in the middle of the bottom of the ves the vessel through the hole will be in that hole the ST, may acquire by falling from half the height IZ ; but in some other part thereof, the water will still run out with the same velocity as before, if the magnitude of the hole be the same. For though an heavy body takes a longer time in descending to the same depth, sel, by an oblique line, in its descent the than by a perpendicular same velocity ; line, yet in both cases it acquires as Galileo has demonstrated. CASE 3. The velocity of the water is the same when it runs out through a hole in the side of the vessel. For if the hole be small, so that the in terval between the superficies AB and KL may vanish ns to sense, and the stream of water horizontally issuing out may form a parabolic figure; from the latus rectum of this parabola may be collected, that the velocity of the is that which a body may acquire by falling the height IG the stagnant water in the vessel. For, by making an experi ment, I found that if the height of the stagnant water above the hole were 20 inches, and the height of the hole above a plane parallel to the horizon effluent or water HG of were also 20 inches, a stream of water springing out from thence w ould fall upon the plane, at the distance of 37 inches, very nearly, from a per For without resistance pendicular let fall upon that plane from the hole. r the stream would have fallen upon the plane at the distance of 40 inches, the latus rectum of the parabolic stream being 80 inches. CASE 4. If the effluent water tend upward, it will still issue forth with the same velocity. For the small stream of water springing upward, as-OF NATURAL PHILOSOPHY. SEC. V11.J 335 GH or GI, the height of the stagnant cends with a perpendicular motion to in far as its ascent is hindered a little by so water in the vessel excepting ; and therefore it springs out with the same ve it would acquire in falling from that height. that Every particle of locity the stagnant water is equally pressed on all sides (by Prop. XIX., Book II), and, yielding to the pressure, tends always with an equal force, whether it descends through the hole in the bottom of the vessel, or gushes out in an the resistance of the air

horizontal direction through a hole in the side, or passes into a canal, and springs up from thence through a little hole made in the upper part of the And it may not only be collected from reasoning, but is manifest from the well-known experiments just mentioned, that the velocity with which the water runs out is the very same that is assigned in this canal. also Proposition. CASE 5. The velocity of the effluent water is the same, whether the figure of the hole be circular, or square, or triangular, or any other figure- equal to the circular for the velocity of the effluent water does not depend upon the figure of the hole, but arises from its depth below the plane ; KL. ABDC CASE 6. If the lower part of the vessel be immersed into stagnant water, and the height of the stagnant water above the bottom of the ves B be sel is GR, the velocity with which the water that run out at the hole into EF in the vessel will the stagnant water will be the same which the water would acquire by falling from the height IR ; for the weight of all the water in the vessel below the superficies of the stagnant water will be sustained in equilibrio by the weight of the stagnant water, and therefore does riot at all accelerate the motion of the water in that is descending This case the vessel. in which COR. AK will also appear by experiments, the water will run out. 1. may Hence be to if CA measuring the times the depth of the water be produced to K, so that duplicate ratio of the area of a hole made in any CK in the part of the bottom to the area of the circle AB, the velocity of the effluent water will be equal to the velocity which the water would acquire by falling from the height KC. 2. And the force with which the whole motion of the effluent watei be generated is equal to the weight of a cylindric column of water whose base is the hole EF, and its altitude 2GI or 2CK. For the effluent COR. may r water, in the time by it its it becomes equal own weight from to this column, may acquire, by falling the height GI, a velocity equal to that with which runs out. COR. 3. The weigb t of all the water in the vessel ABDC is to that part
336 THE MATHEMATICAL PRINCIPLES of the weight which the circles and employed in forcing out the water as the sum is EF to AB II [BOOK twice the circle EF. For let IO be a of mean pro portional between IH and IG, and the water running out at the hole the time that a drop falling from I would describe the altitude IG, become equal to a cylinder whose base is the circle and its altitude EF will, in EF 2IG is ; that to a cylinder is, For 2IO. the circle IH whose base EF is the circle is to IG AB, and whose the circle AB altitude in the subduplicate ratio cf is, in the simple ratio of the mean Moreover, in the time that a drop falling from I can describe the altitude IH, the water that runs out will the altitude proportional IO to the altitude to the that ; altitude IG. hare become equal to a cylinder whose base is the circle AB, and its alti tude 2IH and in the time that a drop falling from I through to G de H ; the difference of the altitudes, the effluent water, that is, the will be equal to the difference water contained within the solid scribes HG, ABNFEM, of the cylinders, that is, to a cylinder whose base is AB, and its altitude 2HO. And therefore all the water contained in the vessel is to the ABDC whole falling water contained in the said solid that is, as HO + OG the water in the solid to 2HO, or IH ABNFEM and therefore the weight of all is

  • K ) to ABNFEM 2IH. employed as HG to2HO, But the weight of all in forcing out the water the water in the vessel is to ; that part of the weight that is employed in forcing out the water as IH + IO to 2IH, and therefore as the sum of the circles EF and AB to twice the circle EF. COR.

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