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in -vacuo from that height V, and the time of falling T; and if the water flows out with this conftant velocity V, in the time T, then the length of the column of water, which flows out in that time will be 2 A, and the meafure of it will be 2 AF. But if we calculate from Poleni's a accurate experiments, we (hall find the quantity of water which flows out in that time to be no more than about T s -oVo °f tms meafure 2AF.— [ 8 Pcleiz. de CafteMs, art. 35, 38, 39, 42,
43-5
Polcni alfo found, that the quantity of water flowing out of a vefTel through a cylindrical tube far exceeded that, which flowed through a circular hole made in a thin lamina, the tube and hole being of equal diameter, and the height of the water above both being alfo equal; and he found it to be fo when the tube was inferted, not into the bottom, which others had obferved before, but into the fide of the veflel.
6. Since the meafure of the water running out in the time T, is 2 AF X T sV<5> the l en S tn of the column of water which runs out in that time, is 2 A X i^%* Therefore if each of the particles of water, which are in the hole in the fame fpace of time, panes with equal velocity, it is plain that the common velocity of them all is that with which the fpace 2 A x T VoV would be gone over in the time T, or the velocity V x T bVd- But tn * s * s the ve l°city with which water could fpring in vacuo to near 4 of the height of the water above the hole.
7. But when the motion of water is turned upwards, as in fountains, thefe are feen to rife almoft to the entire height of the water in the cittern.. Therefore the water, or at leafl fome portion of the water, fpouts from the hole with almoft the whole velocity V, and certainly with a much greater velocity than V x -rc^rc*
S. Hence it is evident, that the particles of water, which are in the hole in the fame point of time, do- not all burn- out with the fame velocity, or have no common velocity; though fome mathematicians have hitherto taken the con- trary to be certain.
9. At a fmall diilance from the hole, the diameter of the vein of water is much lefs than that of the hole. For in- ftance, if the diameter of the hole be 1, the diameter of the vein of water will be i\, or 0.84, according to Sir Ifaac Newton's meafure, who firft obferved this phenomenon;
and according to Poleni's meafure — , or — rS tnat ' s > taking
20 20
the mean diameter 0.78, nearly.
As to the manner of accounting for thefe phenomena we have already obferved that authors are not agreed; and it would be far beyond our defign to ftate their different theo- ries, we muft therefore refer to the originals above quoted. Neither are authors agreed as to the force with which a vein of water, fpouting from a round hole in the fide of a veflel, prefies upon a plane directly oppofed to the motion of the vein. Moil authors agree that the preflure of this vein, flowing uniformly, is equal to the weight of a cylinder of . water, thebafis of which is the hole through which the i&ater flows, and the height of which is equal' to the height of the water in the veflel above the hole. The experiments made by Mariotte and others feems to countenance this opinion. But Mr. Daniel Bernoulli rejects it, and eflimates this pref- fure b by the weight of a cylinder, the diameter of which is equal to the contracted vein, (according to Sir Ifaac Newton's obfervation above mentioned) and the height of which is equal to twice the height of the water above the hole, or, more accurately, to twice the altitude correfpond- ing to the real velocity of the fpouting water: and this preffure is alfo equal to the force of repulfion c , arifing from the reaction of the fpouting water upon the veflel. For he fays that he can demonftrate, that this force of repulflon is equal to a preffure exerted by a vein of fpouting water upon a plane directly oppofed to its motion, if the whole vein of water ftrikes perpendicularly againfl: the plane. From whence it would follow, that the preflure or force of the vein will be greater in proportion, as its contraction is lefs; and this contraction vanifhing, as it does when the water fpouts thro' a fhort tube, and the vein being at the fame time fuppofed to have the whole velocity it can acquire by theory, the fpouting water will then exert a preflure double to what is commonly fuppofed. But the actual velocity of the water being always fomething lefs than it ought to be by theory, and the vein of ivater being not uncommonly contracted to almoft one half, experiments have led authors to think that the preflure, exerted by fpouting water, was equal to the weight of a cylinder of the fame diameter with the vein, and of the height of the water above the hole. The ingenious author remarks, that he fpeaks only of finglc veins of water, the whole of which are received by the planes upon which they prefs : for as to the preffures exerted by fluids furround- ing the bodies they prefs upon, as the wind, or a river, the cafe is different, though confounded with the former by writers on this fubject. — [ b Hydrodynamics, feet. 13. p. 289. c lb. p. 279.]
Mr. Bernoulli endeavours to confirm his theory by a difierta- tion in the eighth volume of the Acta Petropolitana d; where
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he ob&rves, that the experiments formerly c made before the Academy of Sciences at Paris, to eftablifh the quantity of the preflure exerted by a vein of fpouting zvater, are very far from proving the truth of the rule they are brought to eftablifh. For inftance, in one of thofe experiments, the height of the water in the veflel above the hole from whence the vein fpouted was two feet Paris meafure, the diameter of the circular hole which was cut in the horizontal bottom of the veflel was four lines, and the force of the vein of water was obferved to be pne ounce and three quarters. But the weight of a cylinder ot water of the diameter of the hole, and of the height of the water in the vcffel, is fcarce equal to one ounce and three eighths. The difference therefore is at leuft |- of an ounce, which is about ^ 3 - r of the whole weight of the beforementioned cylinder of water. So that it is furprifmg, that this difference fhould have been afcribed to the removal of the plane, receiving the impulfc, to fome diflance from the hole; for this cauie, fuppofing the plane removed to the diftance of two inches, could not produce an increafe of T ^ of an ounce. It appears there- fore, that the common opinion is rather overturned, than confirmed by experience. — [ d P. 114. c Du Hame} % Hift. Acad. Paris, An. 1679. feet. 3. cap. 5.] Mr. Bernoulli, on the other hand, thinks his own theory fufficiently eftablifhed by the experiments he relates; for the particulars of which, we refer to the Acta Petropolitana, Vol. cit. p. 122, feq.
This ingenious author thinks, that his theory of the quan- tity of the force of repulflon, exerted by a vein of fpouting water, might be ufefully applied to move fliips by pumping; and he thinks the motion produced by this repiillive force would fall little, if at all, inert of that produced by rowing. He has given his reafons and computations at length in his Hydrodynamica, p. 293 to 302.
The fciencepf the preflures exerted by water, or other fluids in motion, Is what Mr. Bernoulli calls hydratiluo-jlatica. This fcience differs from hydroftatics, which confiders only the preflure of water and other fluids at reft; but hydraulico- ftatics confiders the preflure of water in motion. Thus the preflure exerted by water moving through pipes, upon the fides of thofe pipes is an hydraulico-ftatical confideration, and has been erroneoufly determined by many, who have given no other rules in thefe cafes, but fuch as are applica- ble only to the preflure of fluids at reft. See Hydrodynam. feet. 12. p. 256, feq. Ralfmg of Water, in hydraulics. The great ufe of raifing water by engines for the various purpofes of life are well known. Machines have in all ages been contrived with this view, a detail of the beft of which, with the reafons of their conftructions, would be very curious and inftructive. Monfieur Balidor has executed this in part in his Architec- ture Hydrauhque, and it is to be hoped that he will foou finifh that ufeful undertaking. Dr. Defaguliers has alfo given us a defcription of feveral engines to raife water, in his Courfe of Experimental Philofophy, Vol. II. not to menti- on the compilers of treatifes under the title of Theatres of Machines, iffc.
Dr. Defaguliers f has fettled the maximum of engines for raifing water thus : a man with the beft water-engine can- not raife above one hogfhead of water in a minute, ten feet high, to hold it all day; but he can do almoft twice as much for a minute or two. [ f Experim. Philof. Vol. II, p. 498.]
Dr. Defaguliers obferves E , that when we come to examine the beft engines, and thofe that are moil cried up, if we look narrowly into them, and meafure the water they deliver, and at what height they deliver it; or bring to calculation the beft attefted relations concerning them, we (hall find that they do not exceed this maximum, though they may far outdo fome very bad engine that they are compaied with. — [s Ibid.]
The famous machine at Marli, though it has many ingeni- ous contrivances, muft yet be reckoned a bad one, if we confider the vail lofs of force in it. This lofs, according to Mr. Daniel Bernoulli's h computation, is not lefs than -|-§ of theabfolute force of the machine. — [ h Hydrodynam. p. 182.] Mr. Daniel Bernoulli ' thinks that a man may, with the moil perfect machine, raife one cubic foot of water, in one fecond of time, to the height of one foot. But by an ex- periment he made at Geneva with a good pump, he found that a man could not raife above four fifths of that quantity. — [* Hydrodyn. p. 199.J
Yet the greatcft quantity, mentioned by Mr. Bernoulli, falls fhort of Dr. Defaguliers's maximum : for according to the Doctor's rule, one hogfhead being equivalent to eight cubic feet, and the railing one hogfhead of water to the height of 10 feet being equivalent to the raifing 10 hogfheads of water, or 80 cubic feet, to the height of one foot, it fol- lows, that a man may raife 80 cubic feet of water, in one minute, to the height of a foot. But by Mr. Bernoulli's rule, he can only raife 60 cubic feet of water to that height in the fame time.
Engines for raifing water are either fuch as throw it up with a great velocity, as in jets j or fuch as raife it from one place
