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BJLlIjl^JLlItalE, City EngiTteer, QUEBEC, M.S., F.R.S.O., &C., BBFORG THB MATHEMATICAL, PHYSICAL AND CHEMICAL SECTION OF THE EOYAI* SOCIETY OF CANADA. May ^^nd, J 384. OTTAWA : PfllNTBD BY MACLEAN, ROGER & Co.,' WELLINGTON STREET. 1884. 4 i i -1 I i TABLE OF CONTENTS. \ PAGE. INTEODUCTION 5 Experimental Enqairies. Appakatcs used — Mode of conducting Experiments 6 EXPERIMENTS: Coefficient of discharge through circular orifices in thin plates 8 Coefficients of contraction 11 Experiments on the flow of liquid through annular spaces, formed by introducing a rod or disk into a cylindrical orifice in a thin plate 19 Experiments on the stemming power of naturally contracted vertically dcHccnding veins passing through circular orifices in thin plates, from the reservoir of supply into a receiver, through a trumpet mouth shaped divergent tube, provided with a t-hort conoical con- vergent entrance 28 Experiments on the eiHux of water in the open atmosphere, through circular orifices in thin walls, whose sides form with the axes passing through the centre of the orifice perpendicularly to its plane, angles greater than a right angle on the inside of the reservoir 31 Theory: Discufwion of properties, formation, origin, etc., of contracted liquid veins, in general 32 Fundamental relations between coordinates, acceleration, velocity, time, etc., of theorical or horizontally projected veins abstracted fi'om gravity ; also of vertically descending and vertically ascend- ing Jets 35 Unsatisfactory character of coefficients of efflux for orifices in thin plates under different heads given by various authors, from a theoretical point of view 40 Variations in the discharge and contraction incident on changes in the inclination of the plane of the orifice to the adjoining walls of the reservoir, aflfecting the new theory 45 Applications of the New Theory : COMPARISON OF THEORETICAL COMPUTATIONS WITH EXPERIMENTAL RESULTS: 48 Horizontal jets 49 Yertioally descending veins 62 Vertically ascending jets 63 Discharge through cylindrical ajutages or tubes, viz. : 66 I. Application of the new theory- Example 1 (firom Venturis experiments) 68 do 2 (from Buffs experiments) 59 li II. PormiilfT for computing the velocity, etc., of tho stroams issuing from reservoirs through cihort cylindrical tubes, with and without entrance contraoted by adiaphragna... (From hydraulic table.s, coefficients and formulae, by John Neville, Bsq., Civil Rngiueer, M.R.J.A., etc., etc.) Remarks respecting tho theory, etc., given by Mr. Neville, etc. Diricharge throngh divergent ajutages or tubes, viz : Ist, Tubew applied directly to the wall of the reservoir without the intervention of conoidal mouth-pioco Example 2nd. Tubes applied to tho small base of a conoidal mouth- piece, having nearly tho form of the naturally con- tracted vein Example 1 Example 2, — Theoretical determination of ratio of velocity in small base of divergent tube with cycloidal mouth- piece, experimented with at Lowell, Mass., by Mr. J. B. Francis (1853)UV\«toY«J(i«r.xeIoaly:auA!taheai«U. [)i8chargo through conical convergent tubes On the flow of liquidrt through oblong orifices in thin plates « Liquid pressure motion, energy, etc Concluding romai'ks 59 62 64 64 66 67 68 69 75 77 79 80 :f Appendix. Pmsiro- Mathematical TuEoav of the Motion of Liquids IssorNO Pftosr Orifices in Resebvoiu, by Mb. Le Chevalier Lorgna 81 INTRODUCTION 81 Chapter I — Natural phenomena 83 Chapter II — liluquiry into the slate of over-flown liquids in reservoirs 87 Notes respecting contents of remaining chapters (III and IV) etc 91 ON THE CONTRACTED LIQUID VEIN AFFECTISC THE PRESENT THEORy OF THE SCIENCE OF HTDRiOLICS By R. STECKEL, Assistant EnoineeR; DEPARTMENT OF PUBLIC WORKS, CANADA. 1883-84. INTRODUCTION. It has boon proved in the most conclusive manner, a full century ago, by the celebrated Italian philosopher, Lorgna, founder of the "Societa Italiana," in the first chapter of his " Phisico Mathematical Theory of the Motion of Liquid? issuing from Orifices in Reservoirs," * and by other scientists, that the contracted fluid vein issuing from an orifice in the side or bottom of a reservoir constantly kept full of water, does not acquire its vis viva or living force by reason of the actual descent of the liquid particles, from the surface through the orifice. Yet, for the want of a sound theory, consistent with the results of experiment, respecting the formation of the liquid contracted vein, we are up to this day compelled, in the absence of any other alternative, to consider all liquid jeta or veins in the light of bodies falling, in each case, freely through a space equal to the height of the liquid surface above the centre of the orifice, according to the universally accepted law of gravitation. We are also forced, chiefly for this reason, to introduce into all hydraulic computations, empirical coefficients of velocity, coefficients of contraction and coefficients of efflux or discharge, in addition to a variety of coefficients of friction and other resistances. Some time ago I undertook a series of experiments, for the purpose of becoming practically acquainted with the leading hydraulic phenomena and thoroughly convinced of the truth of the commonly accepted laws by which the intricate and still imperfectly understood science of hydraulics is said to be governed. It is no more than might be expected, that such a prominent phenomenon as the contraction of the liquid vein at its exit from the orifice should attract a good share of attention on my part. I may state, however, that I was also incited to pursue deeply the investiga- tion of this particular part of hydranlios by the perusal of such passages of the literature on the subject as the following, viz.: — 1. " By applying the general laws of motion to the *' lateral fluid filaments of the stream which issues " through A B, it is found that they tend to describe a " curve which commences within the reservoir, for ex- " ample at A, and continues towaixis C S E. To deter- " mine the nature of this carve, it is requisite to know " and to combine together by calculation : the mutual con- « vergency of the fluid filaments in A B, the law of the '< lateral communication of motion between the filaments " themselves and their divergent progression from C to E. "These combinations and calculations are perhaps " beyond the utmost efforts of analysis. While the tube ja*.!. * Vol. IV., Mem. delU Societa Italiana. (See Appendix.) 6 " A r F B.posseBBCH a different figure from this natural curve, the results of oxpori- " mont will always differ more or less from the theory (1). 2. " Lorgna protends that 0'472 a (a being the head) is the height which would " produce, in any heavy body, the velocity of efflux in the orifice, and that the cou- " tracted vein is nothing else than the continuation of the Newtonian Cataract : ho " supports this proposition by computations deduced from the mutual action of the " particles of the fluid contained in the voasol. But after having seen the failures of " the greatest geometers on this very subject, wo ought to mistrust all these demon- " strations founded on mechanical principles very true in themselves, but "of which the application to an infinity of bodies, which move and are pressed in " every direction, becomes extremely difficult, if not impossible." (2) 3. "So long as we have no more accurate knowledge of the law of con- " traction of the stream, we can assume that the stream flowing through a circular " orifice, forms a solid of rotation whose surface is generated by the revolution of the "arc of a circle about the axis of the stream. (3) 4. " It has been latterly asserted in a Blue-Book that theoretically V,\ = ^i/2gh, " Fii denoting tho velocity in the plane of an orifice in a thin plate ; h, the head of " water on this orifice, and g, the acceleration produced by gravity, per second. It " is not necessary here to combat this error, which confounds the discharge with its " velocity, and a single practical fact, applicable only to a thin plate, with a theoretical "principle. The experimental discharge approximates to §|/2^A multiplied by the " area of the orifice; but the theoretical velocity y/2^/i always approximates to the " experimental velocity, or '974 \/2gh, obtained immediately outside the orifice, in " the vend contractd. it would be unnecessary to allude to this theory here, if it "were not supported and put forward by three engineers whose authority in prac- " tical questions may mislead others. Vide p. 4 of ' Brief Observations of Messrs. " * Bidder, Hawksley and Bazalgette on the answers of the Grovernment Referees on " ' the Metropolitan Main Drainage,' ordered by the House of Commons (London, " Bng.), to bo printed 13th July, 1858."* The first part of Lorgna's "Phisico Mathematical Theory of the Motion of Liquids issuing from Orifices in Reservoirs," especially, is well worth perusing. As the fourth volume of the memoirs of the " Italian Society," published in 1785, which contains this savant's original paper in extenso, is not easy of access for consultation, I have appended hereto a translation of the introduction and tho two first chapters. EXPERIMENTAL ENQUIRIES. APPARATUS USED— MODE OF CONDUCTING EXPERIMENTS. In order that the experimental data to which I shall have to refer hereafter, in support of theoretical deductions, may prove acceptable with some degree of confidence, it is indispensable that I should give a brief description of tho apparatus made use of, and of the modus operandi followed by me for their dotormination.f (1) and (2) See Tracts on Hydraulics, edited by Thomas Tredgold, London, 1836. Part 11 — Experimental enquiries concernini; tiie principle of lateral communication of motion in fluids applied to the explanation of rarious hydraulic phenomena, by Citizen Z. B. Venturi ; translated from the French by W. Nicholson. Pages 146 and 177. (3) See Weisbftch's Uechanics of Engineering, page 822, rol. I. Bnglteh translation by Ooxe. Von Nostrand, New York. • See Neville's Hydraulic Tables, coefficients and formula-, second eiition, p. 33. t The apparatus shown in Fig. 2, including mouth-pieces, orifices, tubes, hook gaui^es and fittings was constructed for me by Mr.;B. Obanteloup of lIontreal,wbo executed the work with bis accustomary ingenuity, precision and care. a I»L . I . [ Fiii.2. N ii^c Wx inar?inc| e.vpcaivuciify. i it Tho rosorvoir of supply A— n brass cylinder 12 inches in diamotor inside and some 3^ inches high in the clear, was mounted on two horizontal, parallel circular plates B, C, respectively 10 and 12 inches in diameter, connected by four ball and socket jointed levelling screws D, by moans of two guide rods E, and a feed screw P, about 3 feet in height, alt)ng which it could bo raised or lowered at pleasure to any desired height above the upper plate B. The orifice- plates O, mouth-pieces M, or tuboH T, were screwed from bolow into a threaded ring in the center of the horizontal circular bottom of this reservoir A, and a brass stand G, carrying a hook-gauge and scale S., provided with vernier furnishing readings to within ^J^ part of an inch, was screwed around the outer face of this interior ring-shaped projection, about ^ inch in height, provided on the bottom of ♦.he said reservoir. A cylindrical, vertical, perforated partition of sheet copper, some 9 inches in diameter, and 3^ inches high, was placed loosely in the centre of the reservoir A, for the purpose of counteracting such disturbanoes as might bo produced by any appreciable centrifugal or other motion which the water might still have had after passing into the reservoir of supply propci-, from a square tank situated in tho garret of the building, through an inch supply pipe I, connected with a f inch circular copper pipe laid on the bottom of the last mentioned reservoir, outside of the fuide rods and screw, which was pierced on tho outside by a number of small, round oles.* The water was first received into a light trough K, of sheet copper, held by hand or otherwise secured in position, so as to be easily removable, wheu aa experiment is liaished, from underneath the orifice, tube, or cock ; from this small .. r^h tho water ran into one or moro circular brass vessels L, which were weighed wh the experi- ment was over, on a scale Z, reading to half ounces. Tho time was furnished by a stop-watch, giving quarter secorf^s, and tho diam- otew of orifices, tubes, &c., &c., were determined by moans of ' ' oring iiheet-mctul gauges and so!' <'. coniral I "ass rods, measured with Brown and Shaij/s Vernier calipers, reading to 0"0(/i .nch. When th'> Uscharge took plaoe under water tho cylindrical h-.>^p ioservoir A, was connected with a square brass box H, 3 inches W'de by 2 inchoj high in the clear, and some 16 inches long, resting on the upper parallel plate B, by "leans of one or more brass tubes W, nearly 2J inches in diameter and 2J inches Iiigh, screwed together, the connections being rendered perfectly watertight by the interposition of ruober bands between the brass bearing surfaces. On top of the square brass receiving box H, and near one end thereof, stood a glass graduated tube N, open at both ends, of inch bore, some 50 inches high, hermetically connected with the square box by means of a stuffing box; this tube served the double purpose of indicating approximately the height of tho water or intensity of the presbure in the receiving reservoir and preventing any accumulation of air therein. The water that passed from the upper cylindrical reservoir A, through a submerged orifice or tube fitted into its bottom, was discharged through a |-inch gauged cock Y, inserted in a stuffing bo:: at the left end of the square receiving reservoir H, into the light conduit or trough of sheet copper K, already referred to, whence it ran finally into the brass vessel L, until the time allotted for each experiment, viz., usually from 100 to 300 seconds, was up, when the trough was quickly removed fi*om under the cock V, and the water allowed to go to waste ; everything, in other respects, remaining undisturbed, until it was settled whether or not it was desirable to repeat the experiment The square box or receiving reservoir H, was connected at the right end by moans of an India rubber tube P, f inch diameter inside, provided with brass couplings, with a cylindrical vessel Q of sheet copper, 6 to 8 inches in diameter, and some 3 inches high, supported on a movable bracket pushed tightly into one of the inter- stices, 1 inch high, left between every two of a tier ^f shelves let into two uprights, xj i>j| m m %! • The tank had an area of 36 feet, and was supplied from the water works of the City of Ottawa by means of aa inch service pipe, provided with a bib and ball cock, and its water surface stood, on an average, aay, 16 feet above the wat^r in the reservoir. 8 raised on a heavy base, the whole of wood, eo as to form a firm stand R ; by this means the water surface in the receiving reservoir Q, could be fixed at any elevation below that of the reservoir of supply A, that might be found desirable. A second hook>gange, with scale S^, and vernier, 5;ipported on a bracket similar to tlMJt^ just described, which was inserted into a compartment situated at a convenient height above the top of the reservoir Q, served to determine the actual difference of level between the water surface of this reservoir and that of the reservoir of supply A, to within -jJj-jf part of an inch. jtvo Prior to commencing a set of experiments, the^points 1 of the scales Si and S^, in connection with the respective reservoirs A and Q, were compared with each other, by taking the elevation of the water surface in both of them, while the liquid WA9 in a state of perfect equilibrium in the whole system of vessels and tubes, proper care being taken that no leakage or syphoning should take place anywhere, and sufBoient time allowed for the water to come to a perfect stand still in each case. When it was found requisite to use a greater head of water than that which could be directly furnished by the cylindrical reservoir A, viz., about 3 inches, the orifice plates or tubes experimented with wore screwed into the bottom of an auxiliary brass cylinder U, some 3 inches in diameter inside, and 8 inches high. This auxiliary ylinder U, itself, was then screwed into the bottom of the 12-inch reservoir A, in the place of the hook-gange stand G, and placed in communication with the iron 1 inch supply pipe, from the tank in the garret, by an intermediate f* inch rubber hose. The effective pressure on the orifice or tube was regulated by the iulet cock, its inlonsity being ascertained by observing to what height the water rose in a glass tube connected with the 3-inoh closed reservoir, at its highest point, by means of a flexible rubber tube X. EXPEEIMBNTS. OOBPFICIKNTS OP DISCHARGE THBOUGH CIRCULAR ORIFIOIS, IN THIN PLATES. It is generally conceded by all authorities in hydraulic matters, such as Micho- lotti, Bossut, Bytelwein, Venturi, D'Aubuisson, Weisbach, &c., that, for a circular orifice in a thin plate, the coefficient of velocity of efflux, corresponding to the plane of the orifice— that is to say, the ratio between the quantity of water actually dis- charged and the quantity which would be discharged fi-om the reservoir if the velocity in the plane of this orifice was equal to that acquired by a heavy body falling freely or in vacuo, through a space equal to the height of the water surface, above the centre of the orifice — varies between 0*60, or thereabonte, for large heads and small circular orifices, and 0*66 or 0-68 for small heads and large orifices, when the dis- charge takes place in the open atmosphere. I may remark, however, at the outset, that the experiments with small orifices, under large heads, on record, are not very numerous, so far as I have been able to find out, and to say the least, those that are available do not inspire unlimited con- fidence as to the accuracy of the results arrived at. Thus— while Michelotti found the coefficient of velocity of efflux to be 607, for an orifice 2-126 inches in diameter, under a head of 7-218 feet, and 0-697 for a circular orifice, the diameter of which was 3-189 inches, under ahead of 22.179 luet— Weisbach says that for an orifice of 1 centimeter, or about 0'394 inch in diameter, this coefficient is : 632 x 0'99=0-6256, under a head of 13-574 meters, or 44-536 feet, and 60x 994=0-5964, under a haad of 103,578 meters, or 339,839 feet ; these last two co-efficients appear to me to be much .aK) large, or else the two former are too small. The coefficients of velocity determined by myself for efflux, in air, through circular orifices in a thin plate, do not differ from those obtained by a number of others, before me, under similar circumstances, as may be seen by the following recapitulation of experiments, headed Table 1. ■HHSBHSS" \ >*- r ice If ■]?; ^mm~ Huilinp of JBt as per enlanje fMA/ffffb CIRCULAR ^ONTRACTL Ihraugha aanaidal arifin. AB\ O^^SSO - in of J 4i0 6,0 _ J PL.n \ psr Bnlarqei phaiographc rBDDrd_BBDxirBi. WNTR ACTED VQU/O \ VE/N praJBcbi ' hari^aniallij ?;tf'-*J-50 in did^meier iji a bra55 plah\, under, a h \ \ofi4 inches. ^ j I head 6\0 6io A H - TJO 7£ 8,0 9J0 I'O ^ ' L;^^a»- >^ 19 U 6*0 i rio 7^ 8i0 9*0 la 1 113 ,«aiW.*i^iau-isasKK3;Kii^3=BK-tR--.. Letter of Nun reference ezpei m A B D E F H I J K 1' L H , Letter of reference A B D E F O H I J E L H Number of ezperiment8 made. 3 3 2 II <( (I 8 6 6 5 14 3 4 Diameter of orifice in inches. 0-384 II i< II II II II 0-400 II 0-4186 0-420 0-482 0-484 TABLE I. Mean liead in inches. 51 44 35 29 19 12-10 3-08 2-97 2-92 3 03 3-07 3-00 2-81 C (onf.) Average value of coeffioint of velocity of efflux, in air, at plane of orifice. (1) 06210 0-^263 0-6269 0.6277 0-6268 0-6281 0-6544 0-6702 0-6727 0-6802 0-6776 0-6803 0-6844 Remarks. The diameter of each orifice was obtained bv measur- ing, with Brown k Sharpe'a vernier calipers, readinir to 001 inch, a slighUy co- nical brass mandrel intro- duced into the hole, at the point where it filled the same, the largest dimen- sions being assumed to be nearest the true one. l/2ff was taken at 27-78 in inches ; 1 ounce was taken equal to 1-7316 cubbic inches. ri,-l m In order to establish coefficients of efflux for very small beads and large orifices, I made experiments with submerged orifices. A synopsia of the results arrived at is given in— TABLE II. Letter of reference. Number of experiments made. Diameter of ori6ce in inches. Mean head in inches. ^.1 (orif.) Average value of coefficient of velocity of efflux, under water, at plane of orifice. Rtfmarka. A 7 0-484 0-12 6616 B 7 0-13 0-6564 4 0-23 0-6640 D B 3 7 0-38 '•60 0- 6531 0-6528 Temperature of water, 52° to 65° Fahrenheit. from F 3 1-42 0-6632 2 2-60 0-6503 H 10 1031 0-040 0-6698 I II 0063 0-6684 J II 0-103 0-6676 K II 0-165 6619 L II 0-206 0-6639 On comparing the above coefficients for discharge under water, with corres- ponding ones for otllax in air, given in Table I, it is found that from 4^ to 5 per cent must be subtracted from the coefficients of efflux in air, to convert them into co efficients of efflux under water, instead of only 1^ per cent, obtained by Dr.Weisbach for ordinary heads of water I suppose, * indicating a difference of over 3 per cent., which, although comparatively large, may still properly be considered to be due, in a great measure, to the very small heads which I used exclusively. The coefficients to be used for efflux under water through circular orifice in thin plates, which are given by Mr. J. B. Francis, in his " Lowell Experiments," differ very materially from those obtained by myself, as recorded above, in Table II., and still more from those established according to Dr. Weisbach's rule, just re* ferred to (1). Mr. Francis entertains, apparently, no doubt but that the coefficient of efflux, through a circular submerged orifice, 0*1017 foot = 1*2204 inch in diameter, should not exceed 0*57 under small heads of from 1 to 5 inches, for at page 225 of his work (1), he says : " It is the general result of the great numberof experiments, on record " on the flow of water through orifices in a thin plate, discharging freely into air, that " the coefficient of discharge (which in simple orifices is the same thing as the ratio -' of the velocity at the smallest section of the orifice to the velocity due to the head) '' !Q groatest for very small heads. In these results where the discharge takes place " !;;kj Weisbach's Uechaaics of Eagineering and of the Oonstraction of Machines.— English Trans- latioii, by Goz, page 8^6. " Lowell H „ _ ^ trand.NiY.— Table XXVII— Izperimenta 93 to 101 (1). See '* Lowell Hydraulic Experiments by J. B. Francis.' '—Third edition, 1871— D. Yan Nos- a, NiY.— T( ■'" - ■»'-•' 11 srifices, ed at is (yater, from 3nheit. th corres- j per cent m into CO Weiabach per cent., be due, in ice in thin Its," differ II., and juat re- t of efflux, er, should hia work , on record tto air, that the ratio the head) ;akes place igUsh Traas- -D. Van Nos- " under water, the coefficient of discharge is least with the very am '11 heads. This " result is so marked and uniform that there oau be no doubt of the fact." Nevertheless, my fifty experiments, H, I, J, K, L, Table II, indioate unmlstake- ably that even under the very small heads, varying between ^^ and /„ of an inch, the coefficient in question is at least as high as 66, for an orifice of 1.U31 inch in diameter. The only diatinctive feature that I can see in Mr. Francia' experiments on sub- merged circular orifices in a thin plate, as compared to my own, is that hia orifice, of 1*22 inch, was in a vertical plane, while my orifice of l'03l inch in diametr was in a plane parallel to the horizon . I may be allowed to observe, in regard to the discrepancies faund to exist be- tween Mr. Francis' coefficients and those of other experimenters, for efflux under very small heads, that his mode of establishing the quantity of liquid flowing in a given time, through a circular orifice in a thin plate, 1*2204 inch in diameter under small heads, varying from say 1 to 6 inches by means of the measured depths of the con- tracted stream passing over the sharp crest of a weir TB inches long, placed in the wall at the far end of a rectangular reservoir 111- feet long and 3'0 feet wide, viz. : 6^ feet beyond the plane of the discharging orifice, does not appear to me to be one calculated to lead to unquestionable results. I do not see that it is possible to determine, with unerring certainty, the dis- charging power of an orifice in a reservoir, otherwise than by weighing the quantity of water which actually flows out of it into a receiving vessel in a given time and un- der a constant head, and I consider this to be more especially the case when the heads used are small and the reservoirs comparatively large. I cannot help thinking that had Mr. Francis bored small holes in the wall wherein the weir was placed, at a depth of 1 footer so below the level of the crest, and weighed the water that would have flowed in a fixed space of time, out of the openings, taking one after another or aa many together as would have been convenient, he would very probably have ar- rived at a different conclusion respecting the value of the coefficients of efflux which are applicable when submergea circular orifices, in thin plates, are used. On the whole, I think we can admit with confidence that the coefficient of efflux, in air, through my orifice of 1 031 inch in diameter, would be, under the very small head of about ^% inch — if such a vein could be produced in its complete state — in air, no less than 0'b68 -t~ 0032 additional for discharge in air, instead of water, viz. : 0*70 ; even this value is perhaps yet slightly smaller than it woul 1 be if ordinary river water was a perfect fluid in every respect. The Chevalier liorgna contends that the reduced velocity of the liquid in the plane of the orifice, as compared with the ordinary theoretical velocity ; F=|/2y// 1 due to the head, H, of water in the reservoir, is due to the simultaneous pressure of : the whole liquid mass around the orifice, which, he says, prevents the free efflux from the reservoir ; and he computes the theoretical velocity in the plane of the orifice to be : FoHf. =(/2/i^\3\|/2^__j/.472^[27x2^==0-68'7ll5*i/2jflr [Mr. H. Eesal proves (see article 268, page 288, second volume of his "Traite de ^ Mecanique Gen^rale " — Paris— Gauthier Vi liars, 1874) that the coefficient of discharge through an orifice, in a thin plate, can never be less than ^ or 5. coefficients' of oonteaction. It has been usual to take for granted that the coefficient of contraction of the I circular vein projected from an orifice in a thin plate, becomes a minimum at a dis- j tance from the orifice, equal, on an average, to once or twice its radius. At or near \ this point, the diameter of the vein has been measured repeatedly by means of four j pointed set screws, mounted on a circular diaphragm, these screws being directed, by the eye, as nearly as possible, towards the centre of the vein, until the points touched 12 its aarfaoe. The mean of the two distanoes, between opposite points, has been invariably held to be the true diameter of the vein at its greatest oon traction ; this diameter was found to be, on an average, 0-8 of that of the orifice. From the manner just described, in which these coefficients of contraction aro commonly obtained, it is manifest that although they are, as a general thing, suffi- ciently accurate for practical purposes, for the objects of theoretical research they are not equally serviceable. In onier to arrive at something more reliable, in my opinion, I measured two vertically descending veins, projected through circular orifices, in thin plates of 0-4 inch and 0*482 inch in diameter, respectively, under a constant pressure of some 3 inches. For this purpose, the position of the cylindrical reservoir of supply A (See fig. 2), into the bottom of which the orifice plates were screwed, was adjusted by means of the four levelling screws D, so as to render the plane of the orifice truly horizontal in every case. The diameter of the vein was measured at various points bv means of pointed screws, mounted opposite each other on a circular diaphragm d, secured with a screw c, to a vertical cylindrical brass standard r, along which it could be moved up or down, by slidiug. The foot of this upright brass rod r, was ground to fit closely into each of three long vertical tapering sockets s, united by three radiating bars to a central ring, so as to form a kind of tripod, which was placed concentrically under the falling liquid vein. The rod r, together with the diaphragm d, was turned round in each of these sockets, until I succeeded in adjusting the positions of the screws, so that their points would describe, about the centre or axis of the rod or socket, circular arcs tangent to the liquid vein at both sides. The distance between the points of the screws was then ascertainid, by measuring, at the proper place, with the vernier callipers, already described, the diameter of a conical manduil introduced between them. ' l>.-: 18 The dimensions and coefficients of contraction found are given in Tables III and IV, which here follow :— ^ *»wiwi xxx TABLE III. Liquid Contracted Vein, falling vertically under a head of 2-99 inches, through a circular oriBce, in a ihin horizontal plate, 0-4 inch in diameter. Letter for reference. A B D E F G Q I J K & H P Q R S T U X, Abcissa, or distance from the plane of the orifice, down to the measured section. Inches. 0-000 ly-eoo 1-000 1-635 2-535 3 635 4-635 S-535 6-635 7-535 8-036 8 536 8-800 9-635 10-635 11 635 12 635 13-635 14635 16-635 16-930 2y = d. Diameter of the vein. Depth of the measured section below the water surface. Inches. 2-990 3-790 3-990 4 625 6-525 6-626 7-626 8-525 9 625 10626 11025 11-626 11-790 12-626 13-626 14-626 15 -525 16-526 17 625 18-625 19 920 Mean value of (?„« = saj 0-813, whence (T-cont = 0-813= 0-661. Coefficient of velocity of efflux, ^ vd. s = 0-6662, whence ^'i n = Coefficient of velocitj at section of greatest contraction = ^'^^"^ 0-6610 C cont. '; '"lv'2-99j"* Ooeflicient of contraction, abstraction being made of the acceleration produced by gravity, outside of the reservoir? 4«e fooV note n^rf bugt) ■ 1-0000 0*8197 0-8143 0-8207 0-82 0-8203 0-8063 0-80fi6 0-8083 08116 0-8082 0-8089 0-8070 0-8118 08166 8165 0-8160 08127 08129 0-8165 0-8086 0-44382. — 1-0078. 2 14 TABLE IV. Liquid Contracted Vein, falling vertically under a head of 300 inches, tbroagh a circular orifice, in a thin horizontal plate, 0'482 inches in diameter. Letter for reference. X, Abcissa, or distance from the plane of the orifice, down to the measured section. 2y = d. Diameter of the vein. h. Depth of the measured section below the water surface. ^h d. ""'• 1 1^3-00 •'•sa Ooeificient of contraction, abstraction being made of the acceleration, produced by grayity outside of the reservoir.* Inches. Inches. Inches. A 0-000 0-482 3-000 1-0000 B 0-925 0-380 3-926 0-8431 1-926 368 4-926 0-8407 2 926 0-341 6-925 0-8387 B 3925 327 6925 0-8366 F 4 926 0'316 7-925 0-8353 6-925 O-306 8 925 0-8337 H 7-636 0-289 10-535 0-8205 I 10-636 0279 13-635 0-8436 J 13-536 0-260 16-636 0-8263 ♦ At ft distance of one to two diameters below the plane of the oriffce, the vein-form is here sup- posed to be governed only by the ordinary laws of the descent of heavy bodies subjected to the forec of gravity. Mean value of Ccont. = say 0-835, whence C^cont = 836^ = 0-6972 C C^ Coeffioientof velocity of efflux /-vei, \ = 0'6803, whence (y<^\.\= 0-46281. ■^ ^orif. ) Wif. / Coefficient of velocity at section of greatest contraction = gTgS^o ^ '^'''^8- m through ontraction . g made of , produced lide of the In order lo gain, at loaat, an approximate knowledge of the rate of variation of the coefficients of contraction applicable to liquid veins in general, I made the experiments under various heads, which are recapitulated in Table V, with a polished brass mouth-piece, having nearly the form of the contracted vein projected through a circular orifice in a thin plate, of 4 inch diameter, under a head of say between 1 and 2 feet Ei§. 2^ too 131 107 J87 368 363 337 206 136 263 is here sup- the forec of This mouth-piece or artificial contracted vein, shown full size in Fig 21-, is 1 inch long, the diameter of the bore at the small end being 0313 inch, white at the junction with the reservoir its cross-section may be considered to be infinitely great us compared to that of the small end. The coefficients of contraction, C emu. given below in Table V, wore coraputed on the supposition that inasmuch as the form of the month-piece coincided nearly with the true cono'idal form which the naturally contracted vein would assume, in each case, the fluctuations of the coefficients of discharge, C j{,eh. were entirely due to defficiency of the waterway afforded by the mouth^piece in comparison to the areas of the respective corresponding cross-sections of the natural contracted veins pro- jected under equal heads through an orifice of 4 inch in diameter. As the actual amount of acceleration produced by gravity during the passage of the liquid downward, from the large to the small base of the mouth-piece, in addition to that due to the hydrostatic pressure in the reservoir, cannot be computed with uneri'ing certainty, when the efiiux takes place in air, I preferred to have the dis- charge take place under water, running the risk of having to apply, for efflux in air, approximate connections to the coefficients as found for discharge under water. 16 H M 1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 18 17 18 19 20 21 22 23 24 2& 26 27 28 29 30 31 32 33 34 36 36 37 A" at 1*3 . H inches. 66-000 66 '000 68-600 68 600 61-600 61-600 43-800 43-800 38-000 38 000 32-400 32-400 24-200 24-200 19-700 19-700 3-114 3 078 3 078 3-080 3-082 3 074 3-072 3-110 3 104 3-066 3-084 3-088 3-092 3-090 3-088 3-068 3-082 3-080 3-072 3-072 3-064 .H S) ■3 o III inches. 8 000 8 000 8 000 8 000 8 000 8 000 8 000 8-000 8 000 8000 8 000 8 000 8-000 8 000 8000 8 000 — 2-680 — 2154 — 2 166 — 1-230 — 0-732 — 640 0-536 0-100 0-100 0-620 0-652 1-220 1-210 1964 1990 2 '600 2-496 2516 2-874 2-900 2-878 ^ o ■a 2 •So ig-5 o inches. 68000 68 000 60-600 60-600 43-600 43-600 35 800 36-800 30-000 30000 21-400 24 400 16 200 16-200 11-700 11-700 5-800 5 232 5-234 4-310 3-814 3-614 3 608 3010 3 004 2 446 2-432 1-868 1-882 1126 1098 0-568 0-586 0664 0198 0172 0-186 h Mean e fleet! vo head. inches. 68-000 60-600 43-600 36-800 30-000 24-400 16200 11-700 6-800 6-233 4-310 3 814 3-611 3007 2,439 1-875 1-112 0-573 0186 000 3 a a M sec 60 CO 60 60 60 50 60 60 60 6U 100 100 100 100 100 100 200 200 200 '200 200 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 a .2 * a 4) O V, Vo V, Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo Vo K. V, "a v.. 1 Vo Vo •S 4) ™ .. ° a 0) ''as 53 o 4jja lbs. oz 34 34 32 32 30 30 27 27 26 26 42 42 I 35 35 30 30 40 38 38 35 33 46 46 41 42 38 38 33 .33 26 i ^''' I 19 : 19 10 10 10 9' d ji ^S o 6 6 6 6 6 6 14 14 1 4 6 4 5 9 9 lOA 14i 14 10} 13 4 6 8 7 14 14 13} 14 13 13 6 6 6 5 6 4i ounces. 469 427 395 366 32fJ 686} 475 .^90 6G0i 632^ 480} 451 651 689} 632 45lj 339 10 c a bo TAm,B 11 V. cub. in. 16 8961 14 7870 13 8788 12 3283 11-3067 10 1663 8-2246 6-9087 4 8625 4 6079 4-1600 3-9016 3-7673 3-4024 3-0706 2-60735 1-9586 220 I 1-2697 109-5 0-6320 k4 O «, •3 I a inches. 206-4308 192-0390 177-6474 100-1076 146-8402 131-8863 -106-8126 89-7232 63-0198 69 8436 64-0250 60-7082 48-7968 44-1870 39-8770 33-8617 26-4103 16 4906 8-2078 a s 00 Taulb 1 I 1 bo =i a & a m a *» « It 0. :i.-i a s "«" s II bl e IS. w 0/ 13 p< .a >* j:] 2 •3 ;a o a •s > 89-7232 630198 59 8436 640250 60-7082 48-7968 44-1870 39-8770 33-8617 26-4103 16 4906 8-2078 • n V. 12 13 14 15 16 17 ! tc a 1.4 '3 a .S-w 2 _ a ♦^ t-..2 9 S< - ■ !J , ^ uV S r- s. o_, -• -.-c i 2gh = 27,78 V h cal velocity due to the head A. 13 a a ^^ - .a II •s a 1 1 1 TO ^ 1 II •w^ .a ® o .2 + -? ., « .2 " "3 V. o ♦J a T3 K MoJ i c> a a ^ ■= a o o c — ® O 2 a 2S" ? Ill Si II § 9 « M - 1 °^ |s II •<• J 1 Hrmarki. 1 . orrectii of disc to red charge oefficie efficier measu: jected 0-4 in. ', H o U u O (' inches. \ 211-638t. 0-9751 00132 9883 0-79914 0-40845 107-5991 09718 0137 9855 0-80057 41077 183-4321 9684 00147 0-9831 0-80165 0-41278 . t 163 2163 9632 0160 0-9792 0-E0314 0-41607 ; 1521511 0-9651 0174 0-9825 0-80180 41328 . . 1 '■ , • y t 137-2230 9611 0-0189 O930O 80281 0-41540 ' ■ . -■ ■ '■ -" 111-8145 9552 00225 9777 80376 0-41736 95-0222 0-9412 0-0?- 1 9696 0-80711 42436 66-9031 0-9141 0327 9763 080418 0-41812 63-5488 0-9417 0-0337 9754 80470 0-41932 576727 9368 00355 9723 80.599 0-42200 54-2543 0-9346 0-0367 9713 08U610 42287 62-7805 0-9241 0368 0-9612 81003 43180 48-1723 9173 0-0383 0-9556 0-81300 0-43688 43 2851 0-9191 0403 9594 0-81139 43343 38 0393 8902 0412 0-9314 0-82349 45988 . It is probable that, 29-2945 86741 0136 09110 83266 0-48070 owing to the very small heads used in experiments No 28 to 21-0285 0-78419 00422 8264 87424 0-68416 87, the coefficiente of discbarge and con- traction are sensibly 119816 0-6850 0389 0-7239 0-93409 76130 affected by friction. 0-6647 0-0424 7071 1-COOO 1-0000 supposed supposed supposed supposed s'pposcd limiting limiting limiting limiting limiting value. value. value value. value. ■ L i '.1 1 18 ; , , . The following are the results obtained by Miohelotti, the younger, with large jets, under great heads. He refers the curve assumed by the longitudinal profile of the contracted vein to a cycloid, and in one of his experiments with a cycloidal tube, he found the coefficient of velocity at the section of maximum contraction to be 0-981:: TABLE VI. » Head above the orifice in teet. Diameter In inches. Ccont. Coefficient of contraction or ratio between the diameters. Distance from orifice to contraction, in inches. Ratio of the distance to the contracted diameter. C*caat- At the orifice. At the contraction. 6890 12008 7-349 12-502 22-179 6-394 6-394 3197 3-197 3197 6 047 5039 2-611 2-604 2-413 790 0-788 0786 0-783 0-765 2-620 2-620 1-260 1-210 1-181 601 0-500 0-600 0-492 0-497 3895 0-3866 03817 3769 3249 Mr. H. E^sal says, at page 290, vol. ii., of his ** TraitS de meoanique generalo " (Paris, Gauthier Yillars, 18*74), that the results of experiment respecting the con- traction of a liquid vein through a circular orifice in a thin plate, show that fur any head less than 6-80 met = 22*3088 feet = 2677038 inches, the co efficient of con- traction is equal to v^-62i or '7874 — for all orifices the diameter of which is less than '"16 =. 6 299 inches, and greater than "> 02 = -78737 inches. EXPERIMENTS ON THE PLOW OF LIQUID THKOUGH ANNULAR SPACES FORMED BY INTRODUCING A CYLINDRICAL ROD OR DISK, INTO A CIRCULAR ORIFICE, PIERCED IN A THIN PLATE. 20 EXPERIMENTS on the flow op liquid Tiiaouau annular spaces formed bv intro- I, Tho discharge took place, in air, under a uniforna head, an orifice in a thin plato, 0*4 inch diameter, and the surface axis, J K L, through the centre of the orifice, to points at Area, a, of the annular opening, A B C I G II, = 0*09980 AreaABCGHI 0-098800 j» Scclion, =0- •78622. Area AhO -125064 Ratio of breadth, A G, of ring-shaped opening to its mean Table VII. I'M 3 a K o a to r di 1 2 3 4 6 6 7 8 9 10 11 ri2 13 u 16 16 17 18 19 (20 f 21 22 (.23 f24 25 I 26 f27 28 29 fSO 31 (32 33 "o to > 3 oTo u ,03 O ^ u- "> 5 ■♦-» cS - ^< <4-> ~ .2 "^ oj »•-" inches. 3-P80 3 936 3 980 3-960 3-958 3-958 3- 3- 3- 3- 3- 3- 3 3 -996 -980 ■970 •932 •904 -982 982 950 3 980 3-980 3 980 3-982 3-980 3-€J0 3-970 3-970 3-970 3-966 3-966 3-964 3-954 3-954 3-964 3-964 3-966 3-966 3-966 03 .2 S t3 3 11 1^ inches. 3-982 3 958 3-996 3 980 970 932 904 982 982 950 980 980 980 3 980 3 970 3 966 3^954 3 956 3-966 a " «3o 03 - . Oj '^ O •2-«i " ■- ^ « fco 3 O »' _ .2 B.'i* 03 .S O (»3 O inches. 1-016 1-016 o •c o a o "S 2119 2032 4457 4457 4457 232-5 232 6 4 0267 "4" 0257" a o 21 BY INTRO- forra head, be surface > points at := 0-09980 DUOINQ A CTLINDBIOAL BOD OR DISK, INTO A OIRCULAB ORIFICE, PIERCED IN A THIN PLATE. throngh the horizontal annular space left between the the circa tnfcronco, A B C, of G H f of a cylindrical rod, M N O P, 0- 185 inch in diameter, lot down along the various disiances, K L, above and below the plane of the said orifice, square inch. Area of orifice A B = 0*125661 square inch. Hence o its moan length DBF, measured in the centre __ 0-918918 _ g.'s ~ 0-107600 ~" Table vir. 10 3 a ad fH I cub.inchs. 3'4197 3-2812 2466 2379 2335 2206 2033 2U>o 2119 3-2032 3-4457 3 4457 3-4457 . ■6 .... •5 40267 5 4 0257 II a •xsi* ioches. 34-6126 33-2105 32-8600 32-7723 32-8040 32-6971 32-4913 82-6094 32.60! 32-421 34-8763 34-8':o>^ 34 4763 12 inclies. 47 8371 47-6490 47 9557 47 Si68 -74i!0 'Kiao •8*/.9 ^269 ?68 4". '■7 * !i7- 47 32-0SS>^ 4/-6328 32-0SS6 I 47-6328 13 to o 0-7256 6970 6852 6852 6865 6871 6868 6796 6796 0-6814 0-72Si 0-7292 0-7J92 0-6726 "o'em 14 o o .. a> • J-S.2 2* I " 3 J- a .- * o t» ^ " 9. ^"3 + wis inches. 0000 000 0-000 — 050 — 0-050 — 0-050 T 100 100 100 100 100 200 200 200 006 006 006 020 020 020 060 060 060 100 100 0-100 0-aoo 0-200 0-200 0-300 0-300 0-300 Remarks. Thelniisc vessel, Vi, weiehcd 55-6 ounces. The vein appeared troubled by air carried along with the water and at a short distance below the cylinder, the space in the centre ot the rinf:; disaitpeared, tlie cross section changing invariably from an annular to a circular one. Vein appeared still troubled by theprcsenceoi'airwithinit. The vein continues troubled by air. The rein always a little troubled by air, but not so much as in preceding experiments. Air mixed with water, apparently. The base of the cylinder O-005 inch iiliove the plane of the orifice. The vein yet slightly troubled by air. Vein appears perfectly clear and transparent ; no air pre- sent in it. The plane where the presence of the cylinder ceases to effect the discharge, is apparently from 0-25 inch to 030 inch above the plane of the orifice. The cylinder was altogether removed, the vein being per- fectly transparent. 'M: ■m 22 EXPERIMENTS on the plow of liquid, throcoh annclae spaces formed by intbo II. — The discharge took place freely in air, under a uniform an orifice in a thin plate, 4 inch diameter and the surface orifice tangent to its circumference, to points various distances Area A B C H G B Area ABO "^ JC'/a/t. Table 1 2 3 4 5 C , 8 9 10 <« a, ,' oj a "2 t'ja DO a o a o O -3 a u g o t-> •** •3| aj •- go . o :.2 fe; ?iS '^ , P.M^ Cm c 0) 52i o9 m^ 1 73 r g'S- -I-' J, o .2 a be =5|. a I " i ^ B o .2 o- Is g-s.a £;2 <= a-E g 3 Q m a hi * O M 3 o p. & .a o m ^ £ 00 pi) 1' 1 ° inches. o " * Eh Q inches. inches. inches. seconds lbs. ozs. ounces. cub ins. f5 3 -960 3-958 • 3-958 1016 •2-942 100 II Vi Vi 15 16 6} 6 a • 190-75 3-3028 3 3-966 K II Vi 15 6; ••••••••• •■(■•• r 4 6 3 960 3-980 • 3970 II II 2-954 II II Vi Vi 14 16 16 b 1840 3-1869 I 6 ■ 7 8 3 960 3-964 3-966 ■ 3-964 II II II [2-948 K ll l( Vi Vi Vi 14 14 14 16i isl 13} ^ e ■ 182-0 3-1613 I 9 rio 3-964 3-998 3-988 tl 11 J 2-982 ll II Vi Vi 14 14 13} 13 181-6 8-1427 d n 4-010 4-010 II 2-994 II Vi 14 13 181-6 3-1427 12 3-970 3-970 II 2-954 II Vi 14 124 181-0 3-1341 ri3 14 IB 16 17 IR 3 980 3-980 3-932 3-966 3 964 3-964 3-980 3-964 II II II II II II 2-964 ■2-948 II II II II II II Vi Vi Vi Vi Vi 16 16 16 17 16 16 3 3 3 16* f • 203-6 9 200-26 ri9 3-960 II II Vi 17 \Q ••■••■•• •*•••• h 20 .21 22 23 26 26 3-960 3 960 3-950 3-948 3-916 3-964 3-960 • 3-960 3-948 ■ 3-966 II II .. II II II II 2-944 2-932 2 940 II II II II II II II Vi Vi Vi Vi Vi Vi Vi 17 17 17 17 17 18 18 9 10 14} 14 14 226-66 i 231-0 i „„ •••••• ?'■ 232-6 4-0276 f.1 3-966 3 966 3 966 II << 2-940 II II Vi 18 18 28 232-6 4-0276 •a a o u 6 ». I*. JO S7 INTBO L uniform B surface iistances HG_B_ to ~~ DPCINa A CTLINDRIOAE ROD OR DISK INTO A CIRCDIiAR ORIFICE PIERCED IN A THIN PLATE. head, through a horizontal lunular space loft between the circumference A B 0, of G H B, of a cylindrical rod M N O P, 0-185 inch diameter, let down through this K L, above and below its plane Q R. Fig. 4. 0-098800_ 0-125664~^ "78622 Table VIII. 10 m r^ < 03 5 cub ins. 3-3028 3-1869 3-1513 3-1427 3-1427 3-1341 4-0276 "i'om 11 12 13 14 ®J3 So ^^ g s" .a > II Ol > f.* Distance K I the cylindri orifice+abo inches. inches. inches. 0-000 0-000 0-000 -0-050 —0-050 —0-050 —0-100 —0-100 —0-100 —0-200 —0-200 The vein is twisted and troubled by air mixed with water. 33-4295 47 6490 0-7016 33 2466 47 7137 0-6758 '"6-"6687 Veins twisted and still apparently slightly troubled by air 31-8961 47 6975" 31-8084 31-8084 47-9718 48-0682 6631 0-6617 Vein twisted but almost perfectly transparent. 31-7208 47-7460 0-6643 -0-200 I-0-020 -0 020 f-0-020 Vein twisted and troubled by air. 1-0-050 -0-050 -0-050 -0-100 -0-100 -0-100 -0-200 -0-200 hO-200 M«*««ttt •■■•( ■•■•••'• ■■(•( Vein appears to be perfectly transparent. "32- 0356" "47-6328" "'""6*6726" T 1 T^T^ 8000 WWW 000 000 32-03&6 47-6328 0-6726 The cylinder remorcd altogether. ff' h:-:,} Ji, 84 pi: \\\ % Is i ; l^::i; Pi: if m iS It! h ■■!'■ EXPERIMENTS on tub flow op liquid TnaouoH annulae spaces, formed by intro III. The discharge took place freely, in air, under a A B C, of an orifice in a thin plate, 0*482 inch in diameter, thick, fastened to the point of a conical needle, as shown Fig. this orifice to points at various distances K L, above or below Area of the annular opening A B C I G H = 0083487 A rea A BCIGH _ 0-08348T _ ^^ ^j. Aroa ABO 0-18246'7 Batio of breadth A G, of ring-shaped opening to its Table »i~~e — y Oeixiart.. o ■a d C3 CQ /(IJ 13 14 15 JVoTt. 03 « 2 o o a o inches. 1 4 036 2 II 3 ti 4 II 5 4-032 6 4 040 7 4-036 8 4 036 9 4 036 10 4 038 4-038 4 034 4*036 26 16 17 18 19 20 21 22 23 24 25 00 2 .S » u •c OS o . inches. 4-036 II o a to Vi Vi Vi Vi Vi Vi Vt Vi Vi Vi Vi Vi Vi Vo Vo Vi Vi Vi Vi Vo Vo Vo Vo •3 '^ JS'^ d ^■S.H ■gS lbs. 15 15 16 15 14 14 14 14 16 15 15 15 17 24 24 27 27 16 16 16 17 22 24 26 27 27 ozs 4 4 1 1 13} 14j 15 IS I 8J 2 2 ■a 13 11 13 12 9 2 9 03 •3 9 m rj tJ ^3 d second 1-7316Z) a s a if 10 .S o ounces. 188-5 185-6 182-6 183 5 186-0 187-6 228-0 291-0 302-6 344-0 I960 204-0 212-6 227 276 306 338 344 346-0 cubic in. 3-2639 3-2119 3 1600 3-1773 6*2033 3-2466 3 9478 6 0906 6-2378 5-9563 IX. D BT INTRO *, under a diameter, hown Fig. e 01- below = 0-083487 M DCCINQ A CYLINDRICAL ROD OR DISK INTO A CIRCULAR ORIFICB, PIERCED IN A THIN PLATE. aniform head, through the horizontal annular space left between the oircamferonce and the eurface G H I, of a cylindrical disk, 0*355 inch in diameter, and 0048 inch 5, and let down in the water along the vertical asia J K L, through the centre of its plane Q R. square inch. Area of orifice A £ C = 0'18246(r sq^uare inch. Hence iug to its 9 mean length D E F, measured io the centre := 1'3U1 0-0t)35 20'70 Table IX. 10 • S s ja o a u J3 :3 o n ■^-sS iS &i " ;h u » a. o bo u oi ja u .SS Q 1. cubic in. •( 3-2639 5 3-2119 6 3-1600 5 3-1773 6-2033 5 3-2466 3 9478 6 09 06 6-2378 6-9563 11 12 13 14 15 Velocity per second = - y 8 1' .1 o 1 ■a Distance K T, between the upper base R S, of the disk and the plane Q R, of the orifice (+ above and — below it). Distance K L, between the lower base M N, of the disk and the plane Q R, of orifice (-f- above and — below it). Remarks. inches. 39-0941 38-4722 37-8601 38-0676 38-3686 17-9786 inches. 48-2765 (1 li II i< II II 9ii •••••••••••*•• ...••*••• ..■••* At this elevation the presence of the disk ceases apparently to inflo^tice the discharge eensibly The disk removed altogether. 26 EXPERIMENTS on the flow of liquid, tbrouoh annular spaoks formed bt intro lY. The discbarge took place freely, in air, under a aniform bead, tbrougb tbe 0-384 iucb in diameter, and the Burface of a ovlindrical disk, 355 inch in diameter, ceding page, in case III, and let down along the vertical passing tbrougb tbe centre Area a, of the annular passage = 0-016832 square inch — Area o, of the complete a _ 0016832 _ T- 0115812 -^'^^^3. Batio of breadth of ring to its mean length measured in the centre = * ^ 0-0144 Table ii « 1 2 3 4 5 6 7 8 9 a"a ^ *'M O •^1 »o a 11 1 g| 11 o .a d 1 J 1 S 1' Si .9 s o a -c S. "S 00 It 5.g So § "go 1 '^2 1-9 .3 m •■3 1 a 8 S ^ o S5 •3 B • o O m 0-S.S. a a S, « It a Q 1 5 a •3 s * 1 111 ■■3 S » » iQ H -« inches. inches. inches. seconds. lbs. OS. ounces. cubic in. 1 3-942 0-832 3-110 300 Vii 10 14 127-26 0-73444 2 3-942 00 M II < 1 •3 u o •s o -3 g e : II e i s Distance E T, between the upper base R S, of the disk, and the plane Q R of the orifice (+ above, and — below it) Distance, K L, between the lower base H N, of the disk, and the plane Q R of the orifice (-|- above, — below it) Remarks. inches. 43 6338 43 6338 37-1187 36-7759 36 4330 37-1187 41 6764 43 1195 44-4911 44-4911 45-0054 24-0353 inches. 48 9906 II II II 48 8327 48 9906 II II 48-9117 II 48-9906 ■ 1 II 8907 0-8907 0-7577 7507 0-7461 0-7577 8487 8803 0-9096 9096 9187 0-4906 0-6371 - - inches. h 0-048 - 048 - 0-036 - 032 - 0-032 - 0028 - 008 - 003 0- 0- 0- - nnoR inches. 000 0-000 - 0-012 - 016 - 0016 - 020 - 0-040 - 046 -0 048 - 048 - 048 The under side of the disk is in the plane of the orifice. Vein troubled by air mixed with flowing water. Vein apparently still somewhat troubled by air— in experiments Noa. 3, 4, 6, 6, but not so ) much as in experiments Nos. I and 2. In all the experiments from No. 1 to No. 12, the liquid fillets meet, in the axis passing through the centre of the orifice approximately at a distance of from |^ to J inch below the orifice. Vein rendered somewhat opaque by air carried along by water, in experiments Nos. 8, 9, 10, 11, about to the same extent as in experiments Nos. 1 and 2. \ 31 2123 — 184 - h 008 . 028 - 0'096 - 0-192 - 0-244 - 3 198 Vein much clearer, apparently, than in any experiment between Nos. 1 and 12. ^ 33-3688 . 49 0693 II P Q T "i's'-oo' "6-646 ••••••••a H, "e'se" 7-60 "o-'is ••••■•••• io-85" 12-16 riD yiRTi ressnre of ches loDg, TABLE 9 in V, project- ah orifice 0, ;ii diam., for BoefiBcient of of ei&ttz = Hi' 0*632 0-672 0*689 0*695 0*686 "6*676 •••••••• ••a ••■•• 0*6*46 ■•••#■•• 29 OALLY DMOENDINa VEIN V, PASSINQ THBOUOII A SlMrLE ORIFICE IN A THIN PLATE, 3 inches = M N through a trumpet mouth shaped diverfjent tube, hhown full size and provided with a short conoidal convergent entrance E. ( .<^«« ^6) XI. 10 11 12 13 No. 3— Vein V, projected through ori- fice 0.400 in. diam.. for wliicli coeffi- cient of velocity ofefflu.v = 0.670. H, laches 1*35 180 3-25 4-80 5-30 5*65 6 90 6*15 6-85 "f*60 9*26 10*85 12*16 Inches 0*370 0*433 631 0671 0*693 0*693 0*682 0*672 675 0*672 0*663 0*632 0-603 0317 0*301 0-287 0-262 0-258 0-252 0-248 0-244 0*239 0-234 0-222 0-210 0-204 Sq. in. 0-07892 0-0711G 0- 06469 a 0-07360 005391 0-05228 0' 04987 0-04830 0-04076 0-04486 0-04300 003871 03464 0*03268 l-08n2 9739 0-8854 C-7379 0-7155 0-682S 0-6612 0-6400 06140 0-5886 0-5298 0-4741 -4474 16 16 No. 4— Vein V, pro jected through ori- fice 0348 in. diam., for which cocfl' velocity ot efflux = 0661. H, Indies. H, 645 o-ib" 0-669 "o'eeV 6-60 0-650 7-30 0-654 8 65 0-611 10-00 11-20 0-493 0-602 0-611 0-634 0-643 0-654 0-683 556 17 A B O I) B G H I J Iv L M N P Q R S T Remarks. "2 Ratio — max for veins Nos. 3 and 4 Ratio - — a maximum for vein No. 2 Hi Ratio — a maximum for vein No. 1 Hi ■* u li T?iL6% is «a S ?5-S' S » •- a .^ •• H'lr a KJ at M ^ l^ a ^ bfi Oi K^ lii o 4> •a Ps (0 H4 1 .d O Fc ^ $ »? 50 M d «r B J3 31 ♦J a a. • (-4 3 M • ^ <1 S ua ia a e il '%. 9 % f) u ^ .1-^ V ^ M « bt o fl •X3 o •r" ^N O*- re t the 2^ 4-a •3 *-■ rt So ca n> b S. 5 s o o ja »- Z*' a ja«»- '^ ♦J o 2 a ^ "is V! S,« o -Jd M *^n 1 00 V •RJJ at St bX} ^ ^ a ■*-) 'B s JQ " a <-• *i o ..a d bf) ti -M ••-< R "C *" Z ^ rt gs C4 o^ W <-i ■»-• e* « H pa !:=§ t * B II S *I J •J 2 Kb '2 5 O 3 ^ «-• *-* I- o 'II J3 4-* ^ V 9 '■c? a a O >>^ itlM ■■ •w oa !j u 73 o vjafi «) u, belt (-. °ja .4'^ *-" «, i* ^ a a 2 o .-• ^ .M t« * • J30 ■sg w ^ gd w 4-1 .Z. m Is? aJ t '*sl o _ U -a L. It O a^ II Y^J/l = """' I : 9})j«l(»fip JO a'iI30]3A ,|0 )D8|»D).1C ) O o « CO CO u) OS o 09 o> V/1 9111 =■ v^ y /I o « 00 C-I 0> c^l a 5! oi o o GO 0> ^ rl <0 t- "f cu m ■ o> e>» s (O Cfl m « D _ pU033B .19(1 -CipoioA 6BaA JO uopvuSisaQ >■ >• o (> > > t- •B^uarauadxa jo noijwnQ J, see's 100 100 o o o o o S M M e^ •g V aoijijo IB)uozuoq 9qi uo a9}VJ& jo puaq nva]^ — 'y d o CO •aiBas aSnsS ijooq jo o aAoqw 'ay 83'JMO 9qi JO 9U«[d oq) jo aoi^BAaig o o CO CO o o o OS OS O) o o o jjooq JO 8A0(i« 'v 'iiddiiB (o jioa -J999.I UI 'H O '33BJJII6 JajVAl JO UOIJVAai^ o o 00 —I CO ^ ■* ■o eo "10 J ao.i0 3 8l3uv ™ O St, g?' &* S' CO CO lO r^ rt ^ rt rt r4 1 « 'goguo JO a V JojarautQ inches 0-405 0-405 0-416 0-416 0-416 •jnatajaadxg jo -oij •1 w ?r 32 THEORY. m B|.8 I 1 : ^ I ! I I |;r;:,;;:| Let H represent the head of water, , on the orifice A B, in a thin plate; r, the radius, A O = B of the orifice A B; y, the radius, C E = E D, of the cross-section C E D, talcen at any point, E ; X, the distance B of the point B, from the centre O, of the orifice ; d X, an increment of length of the vein ; V„Hf., the velocity of the liquid, in the piano of the orifice, A B; V, the velocity of the water, at any point, E, on tho axis of the vein ; g, the acceleration of gravity, per second ; * y, the heaviness of water, or weight of one unit of volume. We know, from experiments with liquid jets, from, say J inch in diameter upwards, produced under various heads, up to, say 10 feet, as a matter of fact, that if a jot or vein of water is interrupted at any point whatsoever, the last particles of liquid immediately in front of the interrupting body rise as high, vertically, and reach as far, horizontally, in vacuo or even in tho open air, as if the continuity of the vein had not been broken. We may therefore take for granted, that the whole energy e, which the hydros- tatic pressure exerted on the top covering or sides of a reservoir, is capable of developing, through a given oriflue A B, in the unit of time, is invariably imparted to the spurting water within the reservoir, before tho liquid particles pass the piano of that orifice, and the assumption that this is also the case for vertically descending veins, projected through orifices in the horizontal bottom of a reservoir is not unrea- sonable, lionce, if gravity be abstracted outside of the reservoir of supply, the measure of an element, (/e of this energy, must bo the same for all sections of one and tho same vein. But, in general, the amount of energy e, stored in any moving mass is repre- sented by the product of the square of the velocity v, the volume of the body, and its heaviness y, divided by twice the acceleration of gravity, viz., 2g ; we must, there- fore have, in any theoretically perfect liquid circular jet, uninfiuenoed by gravity after leaving the orifice, tho relation. v'^ 2 f Constant quantity for every elementary slice or ''^ — 2fl^'' — 1 sheet of liquid contained in the vein. Whence it follows, that in general : ^dxyz -Ttr'^dxy 2g 2g by considering nr^dx to be tho increment of the volume of liquid discharged or ejected from the reservoir of supph', during the unit of the time t, which corresponds to dt. Now I found, by direct measurement (See Table III) : 1. That the area of tho section of greatest contraction of a liquid circular vein projected vertically downward, through an orifice 04 inch in diameter, under a con- ^ant head of about 3 inches is : 7tr2„,„.=0 66l07tr2; Tconi. standing for the radius of tho circular perimeter at the section of maximum con- traction. 2. That the square of the velocity, (V„Hf) in the plane of the orifice is : V2,„if,=(0'6662)2 2gli = (0'4438) 2gU. Whence, admitting that in a perfectly liquid stream, or in a continuous stream of infinitely small sensibly equiaistant bodies, tho velocity must vary inversely as the 88 lie orifice aOBD, contre O, no of tbo E; on tbo le unit of iquid jetp, to, say 10 my point errupting ^en in tho le hydros- capable of imparted tbe piano ascending [)ot unrea- 3 measure le and the is repre- ly, ana its UHt, there- ivity after barged or }rresponds cular vein ider a con- imum con- .-r-''MS^)='-'<^'°>- area of the vein ; we obtain at the section of maximam contraction, for the square of tbe velocity, u„„,. '^^^ =2^H (•4438) (.^^3^ We mnst therefore necebsarily have, for tbe energy of every element of volume of the liquid vein, under consideration : de={'Om H TtrVoy- -4438 H n r^dxy. Now this result is clearly impossible or absurd of itself, and cannot obtain unless we admit: That in tbe plane of tbe orifice A B, the intensity r„,if. of the moving force is less than that i com a* the section of maximum, contraction, in tbe ratio of 0*4438 to 1 0157 and increases gradually from tbe former to the latter place, whether or not tbe vein bo interruptod at any point, whence we are led to the conclusion that : 4369 i™, , or ieo„t =2'2885 t„Hf. must obtain either on account of the mutual 'orif. interference ot the jammed up liquid particles, or in consequence of some other corresponding molecular bction or owing to a combination of some such actions. Again, Table 17 shows that for a vertically descending vein projected through a circular orifice 0-482 inch in diameter, under a head of 3 inches : 7t r^cont. =0-6912 7tr2 and hence : ^ tonl. whence ; Y\„,=2gE. (•6803)2=-4628 (2^B) =•4688 (2,H) (.,-«^,) l-=2,H(:||?)=0.95ai(2,H) c/e=0 952l H7tr2^.i7=0-4628 H (nr'^) dafy, «cont. [n.ACOQ 'orif. ^ UO-. SlOU Now the increment of volume moved forward successively at every instant remains clearly invariable so long as the pressure in the reservoir is kept at a uniform intensity ; the vein having to lengthen out sufficiently at every step to providu room for each new accession to its fold. Therefore, since the sum total of the increments of acceleration generated by the moving force whilo overcoming both the inertia and unimpaired cohesion of the liquid particles, must also bear to the sum of the increments of acceleration , Accunmlated while this force has to contend inertia of matter.the un< merely against the I unceasingly varying mean ratio of y/i^-{-i„x to x^i^s+icX in order that both these conditions may be fulfilled simultaneously there remaimu no alternative but for the areas of the cross>sections of the vein to vary inversely as this ratio, viz., we must have always : — Mi' if: H it \m \m 111,?- i . :*i ■ > ', ; 1 -I As nothing definite is known concerning the laws which govern the variations of the ratio of t'o to t'e, in order to simplify this formula and all others based thereon, let us divide both the numerator and denominator of the fraction in the second member of i this equation by i\ and also by ti and further substitute i for ■° — when we obtain 1/2= r2 ^^ ^+'^ (a) whence we deduce, for the fundamental equation of the curve whoso revolution about the axis E X generates a conoid similar to the theoretical naturally contracted fluid vein A O B I) E C, abstracted from gravity : ti — f — I — . (b) Now granting — as many experiments made with jets of medium sizes, pro- duced under heads or pressures, neither very small nor very groat, tend to prove — that the energy generated per unit of volume of the liquid issuing from an aperture in a reservoir under --^. oxajax^xy- 1-.- conditions of flow, is in general pro- portional to these heads, and denoting by (i^j the ratio ^.^"!L_ between the \auh/ n head due to the actual or experimental velocity of efflux V,n„i -^ and the head H=0 X V.Auuy the total height of liquid pressing un the orifice A B, we have for the velocity at this orifice : \A01i/ whence we deduce for the velocity vnnr *t any section C E D : Vtinr = VI coeflT. \aob R{x\i&) Vu-\-xx Bat in general, when t represents the time, 'p the acceleration, X the space described, V the velocity acquired, the following f&'ndr jaontal relations hold good for all variable motions, viz. : d\} dv. V dx dt= —'P V Consequently, if in order to allow of distinguishing the theoretical, vertically descend- ing and ascending veins from each other, we substitute successively, in the last series of fundamental relations : for t,—t, t, t, t d a for p,—p,p,p, t d a for v,—v, v,v, , t d a for y,—y, y, y, t d a , ; we will obtain : 1. For horizontal jets abstracted from the action of gravity ontpide of the reservoir (which for swift jets is very nearly the case for a length of trajectory equal to a a couple of diameters or so) : yf= V(,0 39 -\- i X (.0 s -h ^ V5 s + e X (a) (cocfTV orif / i' s + I a; — (.0 C) 1 ono) i, UD (a) ) H%=\- dx \lc s + i a; ) (D (10 (2t) (3t) (4t) As all available experiments bearing on the subject, notably those recapitulated in Table X, seem to point to the fact that the mean value of the ratio _?. of the respective alternating intensities of the moving force varies, with the absolute ve- locity of the water or the pressure in the reservoir and the area or radius of the cross- section of the vein, i was introduced in the above equations to denote generally (a)> this mean ratio inside or outside of the reservoir between any two sect^ions A OB and E D and i' to indicate the same mean ratio proper to the portion of vein lying within the reservoir between the plane of the orifice A O B and the plane of rest R S. (Sec Fig. 8.) j/t is a minimum for a; = oo , when it becomes equal to r 4//v\_ y, is a maximum for a; = — i' s when it becomes equal to oo ; Ut is a mini- C) mum for a; = — i s, when it becomes equal to o ; u, is a maximum for a; = oo, when the velocity becomes equal to : C) Pi is a minimum for a; = oo when it becomes equal to o. |) is a maximum for x = — C) when it becomes equal to oo. a) f, = 00 , both for a; = 00 and for a; = — H s, ' o 2° In vertically descending circular veins projected through simple horizontal orifices, where the acceleration p,„ is always equal to the acceleration p„ of the theo- retical horizontal vein, plus the acceleration g, produced by the never-ceaaing force of gravity, in addition to that due to the hydraulic pressure stored in the reservoir, we have — 'Ml ■m ^^■'• * I H', 40 /'.i = 7't+i/=' Cp,i dx^'Cipt + ^r ) rfx = I \dvt Vi-\-gjdx = Iv? + gx = \ 2 H- <7X = i rft;aVa=it>^ whence — »d = ^ yj + 2^x = I i' s -{■ i X C) (i) + ?j7a; (2,) R d 4 (3d) i' s + f X G) (0 /• da; t. /dx ^'(D^((;)^ + 1 + '^ (4d) i' s -{- I X (0 (a) y^ IB a minimum for a; = oo , where the radius of the vein becomes infinitely small, theoretically speaking, i/d is a maximum for : i' s -\- i X + 2gx (5c,) viz: for — X= 2 H (§)(g-)^+t( m (■ l(6d) (D when the ordinate y^, becomes infinitely great, the velocity v,,, being a minimum and equal to o. Urt is a maximum for x =co, being then also infinitely great, theoretically speaking. 3° In the vertically ascending vein, where the retarding effect constantly pro- duced by the force of gravitation is on the contrary inflecting, the liquid filaments outward and diminishing their previous inward inflection towards the axis— r^-Pt—9 = t' S+J X (I) G) t ( i' s + X { i' 5 + 1 x\2 VG) G) ; (cocffv f onf/ I fp, dx^j'(,pt — g)dx =j'(dvt V, — g)dx=: Ivl — gx^ dv^ v. = JuJ (la) infinitely ^ I (6.) minimum loretically antly pio- filamonls = it # whence — \/ i' s -\- i X ^ 0) (^ w 0) rVfe) fl y.= V ©«(a.'+^)-^ 0)0 l' S + I X (a) O dx ''(^''(o'^')'^ <^-> i' 3 -f- i a; G) (0 '(^"i'o'^n- l' 8 -\- i X O (.0 '■JC)=0, (5.) viz : when — x=. h (a) Again, y,, is a maximum ard at the game time v,, a minimum, when— a) Ha)' \ a) ' (6.) d i' s + i X (a) (a) = ) ax -I X' (I) whence — I I" Votif/ I ^"V (a) (a) C) V m [roi; vel head orii fV Ca) .(.0 C) (P.) (a) (a)/ (a) All the experiments made bearing on the question of viscosity and mutnal inter- ference combined, seem to point to the conclusion that the loss of velocity head, caused by this complex resistance increases, in some measure, with tbj head, and diminishes as the area of the orifice or cross-section of the vein increases, but in obedience to what precise laws the variations of the coefficients c(heLi) &^^ i take place, is not easy to establish from the experimental data on record. Outside of the reservoir, the fluid molecules are not directly subjected to pros- sure, comparatively to what takes place inside ; but the resistance of the air has also to be taken into account. Horizontal jets produced under heads varying from 1 foot upwards, with circular orifices, varying, say, from 1 to 7 inches in diameter, are said to reach, according to all authorities on the subject, which have come into my hands, to the end of the same distance measured from the orifice, as if the greatest '41 velocity of the jet at or noar this orifioo was the same as that acquired by a heavy body after failing freely through a space equal to the mean height of the water sur- face in the reservoir above the opening in its side. It does not yet appear to be absolateiy establlBhod, that the horizontal projections of jets formed in circular orifices, which are pierced in thin plates, invariably coincide with those of a solid body having a velocity equal to \/2gH7 According to Weisbach, the coefficients of velocity increase with the heads and Micholotli's experiments on horizontal jets go to show, on the contrary, that they diminish as the heads increase; thus, while for a head of 7} feet the coefficient of velocity was found by the latter to be 993, for'a head of 23J feet, it was only 983, with the same orifice. This matter is still involved in much uncertainty and must remain so until some philanthropically disposed Goveinment, wealthy corporation, rich nobleman or merchant prince may choose to take sufficient interest in the advancement of hydraulic science, to sot apart the funds required for making conscientious and syfltematio collections of all reliable experimental data having a bearing on this subject, which are to bo found in existing works and archives, and to organize a pro- per hydraulic service, amply provided with all the necessary apparatuses and appli- ances, for the purpose of filling, with the results of ft-esh experiments, the numerous gaps which must inevitably be found to declare themselves after the work of com- pilation is completed and for verifying such results of old experiments as might appear to be of a doubtful character. The following table (XIII.) shows the values of (Jl ) for eflux in air, which were arrived at by different experimenters, with various orifices and heads, and also -cocff V the corresponding values of f ..eads. ) the coefficient of velocity head of efflux in tho •k orif. ' plane of a circular orifice in a thin plate. 4» I Is g = CO o o S o o V pq < Ci : a ; s ^ - ^ mZi a- 2 S 2 o Or J3 _ II ^|*^lllO»t-OOOaO- O •-" N en •«»i 10 CO t- 00 o> PI fq oj c» e ; • t : : • : * • • i : 1 : :_c ♦ •5 13 9i a * ^ »> "t* m o 00 *• >»o - uoq: l& ^.■A •-^ o II -I'^.S I ^ »0 CO CO CO C3 CO CO ss oooooooooooooooooooo a i? t»» _ s :: :: 8 CO^C«0»-!«5-H;Of»«3W«J>MO>©0>00 '9 aot-oc^<0?4^-ao s 25 OJt- o ^ o S3 d l-H C^ i-« t-l ivsja ap lO ap Ip O O O M w r-t ^ tf) »-( 1-4 f-4 p-4 r-t ^H c^ CO o o o o >-i rt .^ i-H iH f^c4C4e^ nnm* ^ ■<* «»" C0t-OQ( ^ ^ ^ ' — *3 S .a 11^ o 2 d g pi « ^ ® 111 I ■^ ® s O V ■5 fell a 9 S 00 CL^ 5= ®^ cS ^' e oQ •c i t^ o Sa|« o 2 s g •-- '^ 2 S ® ® 2 ® Si 2 o ^ o« o ^'*'§« 0} ^ ^ >1» «>«H S ® •si- '5.1. M pa PQ -•I EH 1 •c o 03 'St'H g o ► O si*"* © 1-3 •*; "O M a o ® 6 oooooooooooeoooo Mof^cpcp^f^M^minoo^ I CO M c^l < . _. ^ _;«-•« — CI Ob <0<0COCOC0COCDCD M U ■s i» » s o g. •- - - o_ » ^ l-l r-l I-* 1^ '-I l-H fH "'•'in .:■.' 46 Mr. John Neville says, at pa^ro 55 of the 3rd edition of his worlc : " It may be remarked, in passing, how universal the coefficients '613 to '628 are for all forms of orifices in thin plates; or with the oatside arrises chamfered. Indeed, the coefficient '62 may always be used with certainty for practical purposes, for every orifice of this kind (round or square), whether at the surface, in the form of a notch, or at the sides or bottom of a vessel, if the section of the approaching water be large in proportion to the area of the discharging orifice or notch. By co- efficient, of course, is here meant that decimal which, multiplied by the theoretical value, gives the practical result; and this is substantially the same for notches and orifices sunk below the surface." It is evident, judging by the coefficients given in Tables XIII. and XIY., that the case is quite different as regards theoretical computations. All the arguments advanced thus far in support of the theoretical formation of the vend cmtractd, as al)0ve, are based on tne teachings of phenomena per- taining to veins generated through circular orifices in thin, perfectly flat plates. Not- withstanding this, it is readily perceived, upon reflection, that no reason exists why the principles deduced from the enquiries instituted should not al»o hold good for veins projected through all kinds of circular orifices, viz., whether effiux takes place through a plane at right angles to the direction of motion or through an interior, cylindrical, divergent or convergent tube, without touching the sides. That there is something abnormal in connection with this feature of the theory proposed, which requires to be looked into and cleared up, appears from the following considerations : FJ|.0 I' t^ li^-lO It is well known that when the axis I X of a stream A 6 C B makes an acute angle E O I or a with the wall E A O B F, as in Fig. 9, the contraction is smaller, and when the axis I X makes ^n obtuse angle L! O I or ai.,with the wall E A O B F, as in Fig. 11, it is greater than in the case of a vein projected through an orifice A O B pierced in a flat plate EAOBF, where the angle E 1 = a, is a right angle, as shown in Fig. 10. 4Y Borda, Bidone aod Weiubacb have found that when the angle E l=:a^ in fig. 12, reaches 180*^, the coefficient of contraction is reduced to a mean value of 53 — and in two of his experiments Bidone obtained coefficients as low as 0-50 net rly. Dr. Weisbach made a series of ozperiraents with a great number of mouth-pieces, 2 centimeters or 0**787 inch wide, and under pressures varying from 1 to 10 feet ; the results of his experiments with respect to efflux, were as follows : Angle E 1. 180' 167}° 136° 0-677 112J° 90° 67}° 46° 22J" 11 J° 61° 0° Coefficient of efflux. 0-641 546 0-606 0-632 0-684 0-763 0-882 0-924 0-949 0-966 As a small loss of velocity always takes place daring efflux, he estimates that the coefficients of contraction are from 1 »o 2 per cent, greater than the coefficients of efflux. Under a head of tHwct 2*475 inches I found that the coefficient of efflux through an orifice of 0-416 inch in diameter with a sharp edge in a wall whose sides were inclined at an angle of 157^° to tlie axis of the vein, was as high as 0'598 instead of only 546. Furthermore, under a head of 2-65 inches, the coefficient of efflux obtained by me for a jet formed in an orifice 0-405 inch diameter, pierced in a wall inclined to the axis of the stream at an angle of 135° was as high as .657 when the aperture, instead of having a sharp edge, was surrounded by a flat rim about ^^ of an inch wide in the plane of this orifice. (See Table Xli, page 31). For the present, however, it is not necessary to attach much importance to these comparatively small variations in the coefficients of efflux and contraction; the broad fact remains, that both the coefficients of efflux in the orifice and coeffi- cients of contraction are variable with the degree of inclination of the sides of the truncated cone A B F E, whose small base A B, constitutes the orifice, to the direction followed by the current or axis of the stream. The fluctuations of these coefficients are due, as several oxporimenters have remarked, to the fact of the molecules which flow towards the orifices having to sufi^r various deviations from the initial directions followed by them while finding their way through the orifice, to form the corresponding^^ in each case. In this respect, vi2. : as regards deviation from the directions^oUowed in order that the maximum amount of vis viva may be produced which may be designated as the normal direction— the molecules flowing through a circular orifice in a thin flat plate — are clearly not an exception to the general rule. That is to say, some of the molecules which are between the plane of rest K K S, and the plane of the aperture A O B (Fig 8) and particularly those lying nearest to this latter plane, must neces- sarily be deviated to a small extent from tbe normal direction just described, and it is oT^ident also that, in its passage from the reservoir outward, through an orifice in a thin plate, the liquid stream is not strictly confined, inside of the reservoir, within a truncated conoid resembling that which is generated by the revolution of the curve determined by equation (ij on its longtitudional axis X E, Fig 8. It will be observed that 'ven in this, the simplest kind of orifice, tho free efflux of the liquid is somewb at interferer' with, and friction against the metallic envelope being abstracted, the vel jcity in the plane of the orifice must be slightly smaller and contraction outside ol the reservoir correspondingly greater than if tho flow had taken place through a oonoidal mouthpiece, so proportioned that within it motion would diminish gradually^proceding from tho plane of the orifice to the plane of rest-Hsolely by virtue of the continuous increase of the field of action, in accordance with some fixed law, toward the interioi* Qf the reservoir perpendicularly Ik". •'•■■•5 ■ * ..-^ ■ t *- * I '4 48 to the plane of the orifice, as compared to the motion imparted by the original impulse to the first elementary layer or sheet of liquid which leaves this plane on the aperture being cpened. It is therefore evident that even if water was devoid of viscosity and if absolutely no resistaii'^e was encountered in the passage through the atmosphere, nor friction of any kind generated, a vein projected through a circular orifice in a thin plate with a sharp edge, under a constant head or pressure, could yet not be called a theoretically perfect fluid jot, to wit: a jet composed of a succession of elementary fluid sheets, detached from the body of liquid contained in a state of rest within the reservoir, with gradually increasing velocities and free from all lateral disturbance by extra- neous contiguous molecules. The head K X (Fig. 8), the cross-section C D, and its distance E E, from the origin of motion or plane of rest BS within reservoir, being given, the corresponding perfect fluid circular vein may be defined to be the stream possessing the greatest possible amount of energy to be obtctined under the conditions imposed, at the given cross-section as well as at the section of maximum contraction. Now, a stream or vein flowing through an orifice in a thin plate, under a com- paritively small head of say, 5 or 6 diameters or thereabouts, cannot differ sensibly from the theoretically perfect conoidal stream just defined — more especially the portion outside of the reservoir hence the coefficients of velocity of efiiux and con- traction corresponding to such an orifice — viz., the ratio between the actual velocity of the liquid in the orifice and that due to the head must coincide very nearly with the theoretical coefficients of velocity of efflux and contraction corresponding to a maximum production of living force and may, therefore, be taken as the measure of these latter, very little error being made. Again, we have already seen that the largest coefficient of velocity of effux, in air obtoined with an orifice in a thin plate, is about 0*70 ; this figure (or say Yf with Newton) may, therefore, be considered to be the true value (nearly) of the coefficient of velocity of effux of the corresponding perfect theoretical vein, viz.: it may be con- sidered that one-half of the head of water in any reservoir is essentially consumed or utilized in ejecting liquid through a simple orifice, and the other half in generating additional velocitj' or vis-viva. Finally, by adhering to the principle verified by experiment, within certain limits at least, that the energy developed is proportional to the head or pressure in the reservoir, the probable theoretical coefficieut of maximum contraction of a natur- ally contracted vein composed of perfectly fluid matter, in which case no loss what- ever of force could t.'xke place, is thus found to be equal to ^)~=0-8408, not at a dis- tance equal to the radius of the orifice, or so, from the reservoir; but at an infinite distance from the same. APPLICATIONS OF THE NEW THEORY. COMPARISON OF TUEOBETIO.IL COMPUTATIONS WITH EXPKRIMKNTAL RESULTS. After constructing the fundamental formula required to determine theoretically the motions, forms, &c., of the moat elementary kinds of circular contracted liquid veins that are formed through an orifice in a thin plate, I will now attempt to employ some of these equations in thenumericalcompu^ationof quantities and dimensions, previously established by means of actual measurements of veins of water produced ia nature, and of the corresponding discbarges in a fixed length of time. ^ In this manner, I may perhaps succeed in removing some of the ground for hesitation, respecting the acceptance of the hydraulic theory presented above, which the want of concordance of theoretical with experimental results has not, without good cause, proved to bo in many similar instances. Distrust as regards the soundness of the hydraulic theory here presented, would be the more natural, as I found the use of complex and comparatively obscure phraseology unavoidable when endeavoring to describe the effects produced on an infinite number of molecules of matter, liable to change their relative positions at every instant— by an agent, wh je action is not directly perceptible to the 49 tonch nor measurable, such as proves to be the force which holds together the cod< stituoDt elementary particles of every mass of liquid, the reality of whose influence is apparently incapable of being rendered manifest to our senses in any other man- ner than throDgh the variations of form and pressure brought about by it in various kinds of liquid veins and moving fluid bodies. On account of the limited number of reliable experimental data of the proper kind that are available at present, it is not to be expected that I should be able to furnish numerous examples of successful applications of the fundamental equations above laid down, to the determination of the forms and other properties of all the different kinds of fluid veins to bo met with in nature,aB well as of the discharges from tubes, pipes, &c. Indeed, I was forced, in nearly all the cases exemplified, to content myself with computing mere rough approximations to the quantities and dimensions sought ; but although rough, the results will be found to be indicative of the soundness of the principles of the new theory. 1, would obscure ed on an itions at to the HORIZONTAL JETS. The first experiment which I have chosen in this connection, for comparison with theory, is one of a truly original and scientific character. We owe it to the initiative of Mr. T. Trudeau, the present Deputy Minister of the Department of Bail- ways and Canals, of Canada, so justly distinguished for his learning and scientific attainments, who is for ever taking the greatest interest in the advancement of those branches of the natural sciences, which are more especially connected with the duties of the important office which he so ably fills. In order to obtain an infallibly correct representation of the form assumed by the contracted vein at its exit from the reservoir, Mr. Trudeau conceived the happy idea of having a photographic view taken of 's liquid vein projected horizontally through a circular orifice A B, ., plate II, 0'530 inch in diameter, under a constant head or pressure of about 14 inches. This orifice was pierced, on the lathe, in a polished brass plate C D ^^ inch thick, being flaired out from 0*530 inch in diameter at A fi, on the outer face, to about 4 inches in diameter atC D, on the face within the reservoir, so as to form a conoidal cavity resembling, as near as could be judged by a close inspection of the outflowing fillets, to the inner portion of a contracted liquid vein projected, under an equal head of water, through a circular oriflce in a thin plate, having about the same diameter. By this arrangement it was possible to photograph a far greater length of tho more important portion of the vein, than if the orifice had been pierced in a thin plate reduced to a feather edge, from the outer towai-ds the inner face, viz., that in contact with the water. It will also be noticed that, formed in these conditions, the vein outside of the reservoir must have presented a profile differing less from that of the true theoretical fluid vein referred to at page 48, than under any other circumstances, and the contraction must undoubtedly have proved smaller than in the case of a corres- ponding vein projected under the same head through an orifice in a thin flat plate. On the other hand, this mode of proceeding gave rise to some uncertainty as to the precise location of the origin of the nearly theoretically perfect fluid vein thus obtained, and therefore, also, with respect to the exact diameter of the cavity in the plate corresponding to this origin or, more properly, the plane where the velocities due to the forces/io and/i, are equivalent. This difficulty was got over, however, by fixing the value of the coefficient of contraction, viz.: c^ = i^, approximately at 0-83 — at a distance of about one diameter from the orifice— (this number being the mean value of tho coefficient of maximum contraction of a vein projected through an orifice 0*482 inch diameter, under a head of 3 inches, found by direct measurement. See Table lY), on the ground that the contraction of a vein produced under a head BO small in comparison to its diameter, must also have proved nearly the same as the corresponding contraction in a theoretically perfect fluid vein, viz. .* one unaffected by either friction, or resistance of the atmosphere, and otherwise undisturbed in its natural forward movement. 'i >.lh m '•r',s I . •'<[ l> :'; ll '• 60 From the negative obtained, which was much smaller than the natural size of the vein, enlarged views were made in a solar camera, the actual diameter of the vein being in this manner augmented from 0-50 inch to 8*36 inches. These pictures were skilfully executed by Mr. S. McLaughlin, the experienced photographer of the above named Department, so that an outline of figure, sufficiently clear and sharp, was ob- tained, to allow of accurately measuring, by scale, the coordinates of the curve forming the longitudinal profile of the vein under consideration, for a distance ot about § of an inch or 1^ diameterafrom the plane of the orifice. A fa c simile of this profile, together with an approxim'ate^ction, MiiHiffrinca|^MMnmMM|f of the brass plate, is given in Plate II ; and Table XV, which here follows, shows the lengths of the ordinates com^ puted by means of equation (1,) side by side with those measured on the photogra- phic record. TABLE XV. S.S o S ■S a ^1 o-fi "a a o UU3 Inches. —0 9893 —0-7600 —0-6000 0-0000 +0-3380 +0 5000 +1-0000 +1-6000 +2-0000 +3 0000 +4-0000 +6-0000 +6-0000 +7 0000 +7-5000 +8 0000 +90000 +10-0000 +11-0000 +12-0000 +13-0000 y (Ordinate perpendicular ' to axis of vein, mea' sared on photographic record. Designation on plate II. OA ■338a 6b "Ic" Tid lo" ~W 'W 'W ir IT T5K" IT dm Ton Ho T2p T3Q Inches. Inches. 4-2800 4-2100 4 0600 3-9700 3 9000 3-8200 3 7650 3 7450 3-7250 3-7100 3 7060 3-7050 3-7100 37150 3 7170 3-7220 3-7260 i. o «'aartioleB drawn iato the theoretically perfect conoidal otream, between the orifice in a thin plate and the plane of rest B S, being a factor of distur- bance of which it is imposBible to form an estimate, that it can well be arrived at other- wise than by making repeated trials with moath-pieoes variously proportioned. There can be no doubt, however, but that the distance O E=0'9893 a) determined in Table X y is slightly shorter than it should be. If we took for granted that the law, according to which t apparently varies, ia general, the conditions of such variation might possibly be directly combined with the other relations already established, and new equations more generally applicable to the class of veins under consideration could then be constracted. Sach a course would, however, tend to bury effectually oat of sight, under what Mr, Travtwine has chosen to call mathematical rubbish, perhaps not altogether with- out some reason, fundamental principles which are, of their own nature, far from being easily discerned and understood, even when exposed and described in the fullest and clearest manner possible. I have, therefore prefetred, not to attempt such algebraical combinations at present, contenting myself with introducing in the applications of these formulae which here follow such values of t^v\ as would be required by the particular circumstances of the cases considered, keeping constantly in view that in general : the larger the head or pressure in comparison to the orifice, (1) the greater the value of i'^vn in accordance with the law just enunciated, (2) the greater the protrusion of the vein from the orifice A O B, whence (3) the less the distance s=olf from the plane of the orifice to the plane P Q, where perfect equilibrium between the liquid particles ceases to be dis- coeif vel head orif turbed, whence also (4) the smaller the coefficient of the velocity head of efflux Ii through an orifice in a thin plate in comparison to unity, which is that of the velocity due to the fall of a heavy body through a space equal to the total head of water in the reservoir above the orifice. M VEailOALLT DESOINDINQ VIINS. The new theory was applied as follows to the determination of the value of i.y^ at several points of the vertically descending circular vein projected under a head H:^2 99 inches through an orifice in a thin plate 0*4 inch diameter, which I measured with points mounted on a diaphragm, as already described, the dimensions used being those given in Table III. The numerical value given to t'^vx s, which represents the distance between the plane of the orifice and the plane of rest within the reservoir, is that which was determined experimentally, as explained, by introducing a cylindrical pin or rod 0*185 inch diameter into the reservoir, from above, opposite the orifice, approaching its base by means of screw motion, towards the plane of that orifice and establishing the lowest or limiting position of the base of the rod for which the volume of water discharged in the unit of time remained a maximum with a constant head —the cylinder being raised a small distance at a time and the corresponding discharge measured in every position. As this limit was reached approximately when the base of the cylindrical rod stood 0*24 to 0-25 inch above and Back of the plane of the 0-4-inch circular opening in the thin plate, I put, accordingly, s=0-25 inch. Substituting, therefore, in th3 following expression for t .^a /coeff.V (a.) in terms of y, x, H, r o \ orif. I, which is deduced directly from equation (3^) viz : \l) = (coeff.V /coeffA /coeffA /coeff.V x'y^—xr^k^XjS respec plate the the di orifice verticj discba lor a the liq wilhsti fillets i which by the '''here (9) I iZll •Pai 0i m Htreatn, diBtur- A other- There Q Table aries, ia ted with tpUcable ler what ler with- ture, far ed in the mpt Buoh g in the would be onstantly (arisen to law jast se AOB, CO to the to be dis- (coeffv be velocity )f water in ilae of i/y-j der ahead ', which I dimensioDS etween the which was oal pin or the orifice, t orifice and r which the 1 a conBtant rrcBponding proximately )ack of the dingly, i Q of y, X, H, r (cncir.v hi;;Ii| as found in (.rif,7 Table XIII, and for the coordinates y and x, sncoossivoly, the dimensions obtained by direct measurement, as given in Table III, we find — TABLE XVI. Abciasa measured from plane of orifice in thin plate down- ward. l-OOO im 2-635 6-635 10 635 16-636 yd Ordinate. 0-1616 01480 0-1416 0-1210 01120 01036 '(;::) 0-29737 0-37099 0-42937 0-35735 0-43650 0-43807 Remarks. These two values of I'/v.-v do not seem to be in harmony with the others. It may be remarked, however, that a very slight error in the measurement of the diameter affects the value of i/v.\ considerably. Those results scorn to indicate that t^y v increases simultaneously with the velocity, and nearly as the square root of this velocity. Moreover, that for a mean diameter of about J-inch and a velocity of say 120 inches or 10 feet per second i,^y. = 0*44 nearly, in a vein projected through an orfiico in a thin plate. A portion of the differences obtaining between the values of j^a at various depths is, however, due to the tact of the plane of the theoretical oridce not being coincident with that of the orifice in tho thin plate. It is not usual to find that restrictions are made by authors on hydraulics respecting the uniformity of the discharging power of an orifice pierced in a thin plate ; taking into account the position of its plane in relation to the horizon and the direction of the stream. No doubt, practically speaking, under the same head, the discharge through an orifice in a thin plate remains constant, whether this orifice lies in a horizontal, vertical, or any plane inclined to the horizon or vertical. From a theoretical standpoint, however, I am inclined to believe that the discharge through sach on orifice, the head being constant, must be slightly greater for a vertically discending vein, especially under small heads, than it would be if the liquid stream followed a horizontal direction at its exit from the reservoir, not- withstanding the incrca8ed converganco and consequent mutual interference of the fillets in the immediate vicinity of the plane of the orifice outside of the reservoir, which are due to the additional acceleration suddenly imparted to the fluid particles by the action of gravity, VERTICALLY A80ENDINQ JETS. Dr. Weisbach gives, in his admirable treaties of Mechanicb*, the following table here the heights reached Dy vertically ascending jetB projected through orifices in (9)1 *Page 880, Vol. 1, English translation, Weisbach's Mechanics, by Ooxe. Van Nostrand, New York. 7 1 54 thin platos of 1 and 1*41 centimeters, vie. : *394-inch and '691inch in diameter, under heads varying from 10 to 70 foet, are indicated. TABLE XVII. i 'h I Height /(, due to velocity, in feet. Feet 10. Feet 20. Feet 30. Feet 40. 32-68 33-77 Feet 60. Feet 60. Feet 70. Height of jet projected through circular orfice in a th n plate 0-384 inch^l centimetre in diameter Height cf jot projected through circular orifice in a thin plate fi655 inch=: 1-41 centimetres in diameter 9-61 9-716 18-31 18-69 26-98 26-76 38-12 1 39-72 42-66 44-63 46-30 48-68 The reduced elevation of 46-30 feet above the plane of the orifice, to which a jet of 1 centimetre is said to reach, when the head of water in the reservoir is 70 feet, is, of itself, very remarkable and cannot well bo accounted for Eolely by the resistance offered by the air, and the so-called resistance encountered during the passage through the orifice, while admitting, in accordance with the theory based on Toricelli's theorem, that the vein should rise to the level of the water surface in the reservoir. Let us suppose the coefficient of resistance g* produced by the passage of the vein through the atmosphere to be equal to that of a plane surface moving through air, the area of which is equal to that of the cross-section of the vein at every point of its path, viz., to 1'25, according to Du Buat and Thibault.* As air, at the ordinary atmosphere pressure, weighs about ^J^ of water or, say twice as much, viz., ^^0, to make ample allowance for any air that may be carried along with the vein, the diminution of the effective pressure of the water due to the passage of the jet through the atmosphere is thus roughly, for 70 feet head of water, 70 X 1*25 X ^'o 0=0*218'? feet. Hence, the jet should rise to 69-78 feet, or thereabouts, instead of only 46'30 feet, if the atmosphere was the only resistance to be overcome. Another proof of the fallacy of attributing to the resistance of the air, the greater part of the difference between the head doe to the velocity actually generated in a fluid projected through a simple orifice, and the total fall from the surface in the supplying reservoir to the centre of this orifice, is obtained by comparing Michelotti's experiments on horizontal jets, with those of Dr. Weisbach, on vertical jets. According to Michelotti, jots issuing from &n orifice in a thin vertical plate, 0-889 foot::^9 668 inches in diameter, under heads varying from 7-51 to 23-59 feet, and Eassing thoroforo, roughly, from 33 to 23 foot through the air, are said to be projected orizontally in each case to a distance equal, within 1 per cent., or less to the cor- responding ordinate of the parabola which would be described by the jet if its horizontal velocity near the plane of the orifice was equal to that due to the head. Weisbach's experiments on vertical jots formed in an orifice 1'41 centimetres or, say f-inch diameter, under heads of 30 to 40 feet and passing 26-75 to 33*77 feet through the air, go to show that the heights reached by the jets will fall short, in each case of the height of the water surface in the reservoir above the orifice, from 11 to 16 per cent. I am aware, of course, that a vein formed through an orifice of 9-688 inches is very much larger than one through on opening whose diameter is only § inch or so ; but £ cannot see how even this large difference of area could render the proportional resistance of air ^nv ten to fifteen times greater in one case than in the other. As for " the resistance during the passage through the orifice " to which frequent allusion is made in works of hydraulics, I confess that I fail to conceive how it can be possible for any round hole pierced through a plate so thin that it may be •See BngliBh translation Weisbach's Hechaaics, page 1031. 55 considered to be devoid of thicknees, to offer resistance to bodies passintr through it when ejected from a vessel, no matter what may be the rate of the motion imparted to them. But thou it may be, of course, that after having assumed that theorotically the liquid particles must of necessity acquire, at the short distance of, say one radius of the orifice, in front of the said orifice, a velocity equal to that due to the full from the surface of the water to the centre of this orifice, the authors, when saying "during the passage through the orifice," mean to refer to the time occupied by the water in passing from within the reservoir to the section of maximum contraction and velocity, or to some other point.* *lf some such broader meaning is attributed to the expression "during the pas- sage through the orifice," 1 must acknowledge that it is well suited for smoothing over the difficulty of reconciling the shortcomings of a defective theory with the irrefutable arguments supplied by properly substantiated experimental truths. Although I have not found it practicable, up to the present time, in directly employing equation (6,) for the computation of the height /i, to which a jot will rise vertically in the air under a given head, I am satisfied that the groat difrorencos between the height to which the jots experimented on by Dv. VVeisbucb roso and the corresponding elevations of the water surface in the reservoir of supply, mast be attributed chiefly to the decrease xtt the «afiffiasBt9#=:Cte velocity head of eiUux, • i^^)j^^^^±itliS^J^^X^''^ \iy ^^"° ^' P"'' ^'^™ '"^"" '" «'•*'"' velocities and from large lo small orifices. The following attempts at applications of equation (6,,) for the purposes of dis- covering what values have to be assumed for i^,x for arriving at the heights to Dr. Weisbach's jets projected through an orifice 0'394 inch diameter, roso heads of 10 and TO feet respectively, am olao the diamelmie at thu uppat ' absurd or which under O B t w omitioo of tb ase j a t i^ go to show that this formula does not lead to results. In the case of a jet formed in an orifice of 0-394 inch diameter, under a head of 10 feet, we may, judging by what we have seen, put i^y^ s=r='^fA=0'iy'7 inch= ,,;^',,)= 0-612=0 372, without much risk of material error. These orif/ numbers being substituted for the symbols in equation (6,,), it is found that in order that a; may be 9-61 feet, i^y^ must be equal to 0*40 nearly. When the diameter of the circular orifice is 0*394 inch and the head 70 feet, wo can put ify^ s the distance from the plane of the orifice to the plane of rest, equal to 0-6r,or say 0-01 feet; al8OlhJ^i|=0'682=0'3364. Upon the respective symbols being replaced by the corresponding nambers in equation (6^) we find that in order that x may be 46 30 feet i.y. must be equal to about 0-50 . The mean values of i^y^ thus established roughly. viz., 0-40 and 050 are not absurd or unreasonably low or high, when compared with the mean value of this quantity (0"4096) in the horizontal vein projected through an orifice 63 inch in diameter under a head of 14 inches which was photographed, and with that (0-44) in the vertically descending vein projected through an orifice 0- 4 inch in diameter under a head of 2*99 inches, which I measured directly with the pointed screws mounted on a diaphragm &c., &o., as explained. It is not improbable that vertical jets produced under great pressures through orifices in thin plates rise in the air to elevations much below those which jets issuing under the same pressure from properly proportioned cono'idal-mouth pieces would attain, on account of the interference with Tree efflax from the reservoir, arising in each case from the fact of the body of liquid intervening within the vessel between the surface of the conoidal form that would be assumed by a theoretically perfect vein and the inner surface of the orifice plate being drawn up in the jet spurting through the orifice in the thin plate. "ill il I: i^ .1. if 'I ,.^!i; *> '% M As stated before, for practical purpoaos, the ooeflSciont of dischargo proper to an orifice in a thin plate may be considered to be invariublo, whatever direction the stream flowing through this orifice may follow ; in point of fact, however, the discharge through such an orifice must be leHs, especially under small heads, when the water flows vertically upwards than when^foUows a horizontal direction at its exit from the reservoir, notwithstanding the gradual spreading of the fillets, which takes place necessarily in such ca^te from the plane of the orifice to the upper end of the vein, being the result of the action of gravity in a direction contrary to that of the motion of the liquid. Lorgna savs, in article L of his " Phisico Mathematical Theory," etc.: — "It is observed that the quantity of water supplied by a vertical jet, in a fixed time, through a given orifice, and under a permanent bead, is much smaller than that which would issue from a reservoir in the same time, through the same orifice, pierced in a thin plate in the side of this reservoir, under the same head." (See the comparison of these discharges in the tables given by M. BosBut, in his Hydrodynamics, Part II., Chap. IV. DI8CIIABQE THBOUail CYLINDRICAL AJUTAGES OR TUBES. Foleni has made known the singular effects of cylindrical tubes two centuries ago; and the determination of the cause has been a serious stady with jcientt'sls <. ever since. If we prevent or destroy, artifically, the inflection of the fillets of a naturally contracted horizontal vein projected through a vertical orifice in a thin plate O R (Pig. 13), by causing this vein to flow through a cylindrical tube O E S T, added to the reservoir at the orifice, so as to completely fill the tube, the velocity acquired by the liquid, and consequently the discharge, in a given time, under a constant head, can be arrived at — if the effects of gravity outside the reseivoir are abstracted or neglected — in the manner hereafter described, by supposing the natural fluid fillets to be spread over the full cross-section of the tube in a uniform and continuous manner, by virtue of their attraction towards its sides, in every part of the cylindrical space from O to S^which is not strictly the cose in reality, however, as we will see pre- sently. In these conditions the ever-varying ratio between the two velocities which are due to the forces /j'o and fi^ in the natually contracted vein, is continually trans- formed into the constant ratio of unity or 1, through the intervention of the capillary attraction of the metal, wood, glass, etc., of which the tubular envelope is made, the acceleration due to the force//o being increased, and that duo to the force //y simul- taneously diminished in a corresponding manner. Thus, if the acceleration due to the force /?o is continually increased, along the trajectory of the naturally contracted vein abstracted from gravity, in the ratio of 1 to j, the total amounts of momentum due to two sensibly constant mean forces fi^ and fi^ being necessarily the same under all circumstances, at the end of equal times, independently of any transformation whatever, which the constituent factors of mass and velocity may be subjected to within the tube, by virtue of the attraction of its walls — while the momentum is being generated — it follows, that the relation : / vcIV I ratio I- Inat- I- \vein / v/«(v.) So -t- a; Aa';)'" + *'(a':)^ which holds good for any point P of the naturally contracted vein, situated at a dis- tance X from the orifice, measured in a direction parallel to the longitudinal axisy^ will become transformed or converted into the relation : suTtt^ST It at a dis- na] axisyv. (Vcl » r.illii 1 .yliil I l/«(v.)S.. + a; + (/»■(»■)«„ -h t(v.)^- i/^v.)S„ + «(v.yx) whonco we doduco tho equation ; l^(«(v.)S„ + <(v.y"j: (U) and Iho value ofj in terras of «„, i and x, viz. : J= -s., '.X a, + \ +i\/l^+x (12) Now, if wo loavft tho acceleration duo to tiio force fi,, entirely out of conBidera- tion for the present, it will bo seoD that tho total velocity which is duo to the force //„ in the natural contracted vein projected through an orifice in a thin plate, at the inatant when the water reaches the point P, bears to the total velocity duo to the iorcejfi„, as increased by the lateral capillary attraction at the iniur surface of the cylindrical envelope, the ratio of v/iVv)So4- 8yv.\X to i/#-HT^7^^41 /«; + ' (14) EXAMl'LE 1. By usinp a cylindrical tube, such as that roprosonted in (Pig. l6), 18 olJ fronch lines = 1'5985 inchoH in diamotor, but only 54 linos = 4''79.'i5 inchoa long, Vonturi obtained under a constant bead of 3'2'5 frencb inches = 34 61'7t) onglish inches, a discharge from the reservoir, bearing to that passing under tho same head, through a circular orifice in a thin plate having tho same diameter as tho tube, the ratio of 41 to 31*. The delivery of 4 cubic foot took tho eamo time, viz , 31 seconds, when the tube was 57, instead of only 54 lines. (See page 136, exp. 6, same work.) In the case of the vein projected under a head of some 14 inches through an orifice in a thin plate 53-inch diameter, which was photographed «„, was found to be approximately equal to 057 r, r being the radius of the orifico. If we assume, therefore, s^ to vary nearly inversely as the square root of the velocity, we can hero put «„= -57 r( ^77==)= say '45 r := say 4 00 lines. Again, we may ^ I 34 '64 ' allow, in the absence of any more precise data, that for a diameter of 1*6985 inches, and a head of 34 64 inches, i^^y. has nearly the same value as for an orifice of 0*4 inch diameter, and a head equal to 34 64 x (x^kS = ^^7 ^''^ inches, when, ac- cording to experiments Nos. 15, 16, 17, 18 and 19, of Table V, we may put approxi- mately t .,v =ct = 0*42 on an average, along the portion of natural vein 64 lines or 4.7955 inches long, which corre^iponds, as regards position with reference to the orifice and reservoir, to the cylindrical tube. Substituting these numbers for the respective symbols in the last equation (14), we find the computed velocity ratio c u,yii„. to be 1-26, as against ||=l"3.i y, siniplu obtained by direct experiment, indicating a deficiency of about 5 per cent, in tho computed velocity. While a small part of this difference may be the result of the disengagement of the fluid particles produced by the attraction of the sides of the tube, and of tho transverse action of gravity during the passage of eacb sheet of water from the reservoir end OR (Fig. 13), to the other extremity S T, of the tube, the greater portion of it is, in all probability, due to the fact that the filaments of the naturally contracted vein are not dispersed in a uniform and continuous manner over the entire cross-section of the cylinder, as was assumed, at least for a length of one diameter or so beyond the face OE of the reservoir. The actual condition:^ of the flow through the simple tube are rpparently intermediate between the theoretical con- ditions upon which the above computation is based and those of a stream flowing through a divergent tube of the form or ST (Fig$. 14)added to a mouth-piece orO R having the shape of the naturally contracted vein. wTS^ * See Gzperimental Eaquiriea, coacerniog the principles of the lateral oommunioation of motioa in fluids, applied to the explanation of Tarioui hydraulic phenomena, by citizen J. B. Venturi, trans- lated from the French by W. Nicholson ; second edition, included in Tracts on HydraulicSj edited by Thoi. Tredgold, page 134, London, printed for Josiah Taylor, 1826, 1 for tho "•oJ (U) ilJ fronch , Vonturi inohoB, a I, througli lO ratio of )nd9, when k.) d rough an 8 found to vo asBumo, locity, wo , wo may of 1-6985 D orifice of when, ac- it approxi- U lines or nco to tho ation (U), |l=l-3i ont. in tho agement of and of tho irom tho he greater 3 naturally over tho ;th of one of the flow retical con- xm flowing iece orO R XXAMPLI 2. Buff* found that with a short cylindrical tube ^{^ ■"^h in diameter and /^ inch long the coofilniont of discharge was 0-861 under a head of 2^ inches. Ab the coefficient of discharge into air through a simple orifice of tho same diameter as tho tube and under the same head, may bo takon at 65 nearly, the ratio of the discharg- ing capability of tho tube to that of the simple orifice in a thin plate is 86l-4--650=l'3246. Wo may in this case put approximately s„=:'9r='135 inch, and j='41, whence, substituting these values tor the symbols, in the above formula and 0*50 inch for /, we find this ratio to be equal to 1*23 nearly. The difference between the observed and the computed coefficient of velocity is therefore '0946, indicating a deficiency in the latter coefficient of some 8 per cent., due to the causes just described. The increased discrepancy of 8 per cent, as compared to that of 5 per cent, in example 1, is, I presume, due here to the greater transverse effect of gravity in the oylindrical vein — during its passage from iho reservoir to the outer end of tho tube, with the compuratively small velocity gonoratod by a head of 2^ inches. 1 have taken tho liberty to introduce here in extenso a chapter from Hydraulic Tables, Coefficients and Forraulic, by John Neville, Esq., Civil Engineer, M.R.I.A., &c., &o., on the conditions of flow, &o., in short cylindrical tubes, with and without entrance contracted by a diaphragm, wherein a metlfod is suggested for calculating the discharge from such tubes. This course was followed with a view to convenience for reference, &c., in perusing some remarks which I have ventured to offer respect- ing some of the statements, etc., contained in the said chapter. At pages 160 to 164 of Mr. Neville's valuable work, 3rd edition, dated London, 1875, wo find the following: — The contracted vein o r is about 0-8 times the diameter O R; but it is found, notwithstand- ing, that water in passing through a short tube of not less than H diometer in length, fills tho whole of the discharging orifice S T. This is partly effected by the outflowing column of water carrying forward and exhausting the air between it and tho tube, and by the external air tlion pressing on the column, so as to enlarge its diameter and fill the whole tube. ^Yhen once the water approches closely to the tube, or is caused to approach, it is attracted and adheres with eome force to it. The water between the Annalen der Phidk and Chemie Von Pogftendorf, 1839, Baid 46, page 243, or Neville's Hydraulic Tables, coefficients and forniuliu, page 148. Third Elitioa. London, 1875. tion of motion 'enturi, trans- ic3i edited by 60 I El^.lS i^Ja-Sgj?jf: 3 =s 11 xjo-'<-STi olil riviicb UiiciiK tube and the vend contractd is, however, rather in a state of eddy than of forward motion, as appears from the experiments of *Venturi with the tube shown Fig. 15, giving the same discharge as the simple cylindrical tube (Fig. 18.) appr area put, thet the V coeflB tract is oq I! 'tJj I54.IC where 6 R=6 E, 6 S=6 S, S :t=s' T. If the entrance be contracted by a diaphragm, as at O E, Fig. 14, the water will also generally fill the tube, if it be only sufficiently long. Short cylindrical tubes do not fill when the discharge takes place in an IS ne( wate: watei veloc sudd( due U This fY|.« exhausted receiver, but even diverging tubes will be tilled under atmospheric pressure when the angle of divergence 0, does not exceed 7 or 8 degrees, and the length be not very great nor very short. When a tube is fitted tc the bottom or side of a vessel it is found that the discharge is that due to the head measured from the iurface of the uater to the lower or discharging extremitij of the tithe. It must, however, be sufficiently long, and not too long, in order to got filled throughout. Guilgliolmini first referred this effect to atmospheric prepsure, but the tirit simple explanation is that given by Dr. Mathew Young, in the Traneactions of the Eoyal Irish Academy, Vol. VII., page 66. Ventiiri, also, in his fourth proposition, gives a demonstration. The values of the coefficients for short cylindrical tubes, which are given (page 156), have been derived from experiments. Coefficients which agree pretty closely with them, and which are derived from the cooffioients of discharge through an orifice in a thin plate, may, however, bo calculated as follows ; Lot C bo the area of the * Venturi found (1) thut throuRh an orifice R pierced in a tliin plate in side Hm of a reservoir, whose diameter waj 18 Frencli lines (old system of measures)— 1.6985 English inches — 4 French cubic feet = 4.8334 English cubic feet of water aro diEiiVi»v«iain 41 seconds under a head of 32.5 French inches == 34.6476 EOKlish inches. (2.) He fitted to this orifice the conical mouth piece R, ][^,|^Q r, of the proportions shown in Pig. 15, and having nearly the ^ form of the natural contracted vein, when under the same head the same quantity of water was discharged in 41! seconds. (3. ) By introducing the mouth-piece o, r R (Fig 16) alone into the cylin- drical tube Fig. 16, as shown in Fig. 18, thQ same volume of water was discharged in 32 6 seconds. (4.) To the mo'ith-piece 0, r R, he added the tube S S T IV0 (Fig. 16), and the duration of the fl^w, all other things being equal was only^l seconds. (6.) He replaced the compound tube OoiiS^TrkOby the simple cvlin- drical tuba Fig. 16, having the same diameter and length, and the eflluz of 4.8384 feet look place again in 31 seconds. (6.) fjastly, when he had amended the porfabuL- o 8 T r (probably by rounding the angles at and S) the time required for disoharsing the const-tnt quAntity of 4.8384 oubic feet was reduced to 3O seconds, under the same head of ?4 6476 inches. (60.) from (61) loss 8 cient diami cooffi whici Whei and t (. ..nu as the coefficient of velocity becomos equal to tho coefficient of discharge when thoi o is no contraction, in such case this coefficient which we call cof, is oxpref-o^i by the formula (-) -^■={i-^^+(J__-,)f When the diaphragm is placed in a tube of uniform bore, then C=A and (62i) cof.=^^ -A"^^, ac. a and the loss of head in passing ihe diaphragm becomes; (62i) /A ^ 2 2j. * Vide Sir Robert Kane 'a traDBlatlan of Rahlman's boolt on norizoatal WTater WbeelB, p. 49. 8 62 It is evident from the equations that — and c, depend mutually on each other, and that they cannot be assumed arbitrarily. When the approaching section C is very large compared with the area A 1 M^ «of-=< t+/_L_i (63) If r, =0-64, the last equation gives cof.=='812 ; if r,=-60l cof.= -833; if c, ='6 17 co/.=-847; and if r,^=62l cof.=z-S5G. These results are in excess of those derived from experiments with cylindrical short tubes, perfectly square at the ends and of unilorra bore. As womo loss, however, takes place in the eddy between or, Fig. 14, and the tube, and from the friction at the sides, not taken into account in Ih « above calculation, they will account for the difference of not more than from 4 to 1 per cent, between the calculation and experiment. If c^ be assumed for calculation equal '590, then cof. equals '821; and as this result agrees very closely with the experimental one, c, should be taken of this value in using the loregoing formulae, from (60) to (63 for practical purposes. The thickness of the diaphgram itself and the relation of that thickness to the diameter, as well as the form of the orifice a, are necessary elements in the consideration of this question." REMAEKS. Considering that the natural contraction of the liquid vein projected through a simple oridco, is destroyed gradually in a cylindrical tube, from a point between the orifice 11 in the reservoir, and the section of maximum contraction or (Fig. J4r) up to the point to which the tube must extend to furnish a full stream, the water in this contracted section or, cannot, it seems to me, bo looked upon as striking suddenly against the body of water between it and the end section T S, hence the consequent reduction in the total head cannot bo exactly the amount of pressure corresponding to the ditt'erence between the total theoretical velocity due to the full head and the actual velocity of the stream at its exit from the tube. Streams passing from short cylindrical tubes into the open atmosphere inva- riably carry a certain quantity of air along with them, and in order that air may be able to mix with the water, it is necessary that the absolute pressure of the vein at the mouth of the tube should be different from that of the atmosphere. From this circumstance it must not bo inferred, however, that the presence of atmospheric air, orsome other gaseous fluid in the tubes, is essential, in order that the filling of the same may take place, with the resulting increased discharges in comparison to those afforded by simple orifices of equal diameters and under the same respective hydros- tatic pressures ; the air or any other gas that may bo in the tubes, no doubt, assists in causing these to fill with water, but that is all. The statement that " cylindrical tubes do not fill when the discharge takes place " in an exhausted air receiver," is apparently incorrect, for Mr. Hachette guts he is certain of having produced the phenuraonaof additional tubes under such « receiver, in vacuum.* The same experimenter also managed to obtain a clear, contracted vein within a cylindrical tube 0,1332 ft. diameter, and 0,3117 ft. long,which was perforated near its middle and quite around its perimeter with a dozen small boles; but this o])eration, it is stated, had to bo performed with great caution, as a slight agitation was then sufficient to produce contact, causing a flow with full tube to take place. I have seen no detailed description of the experiments made by Mr. ilachettc. It would be interesting to knog; wUst the pressure was in a cylindrical tube running full, say at a distance of half a MiSwror so f>om the orifice in the reservoir, when the pressure in the receiver of the air pump was down to near 0. According to the theory 'See Spon's Dictionary of Engineering, page 1,901. 63 of Daniol fiernouilli, that the presBuro which u fluid exerlu againnl tho Hides of a tube in which it moves, is equal to the head, minus tho height due to the velocity of tho motion, the absolute pressure in Mr. llachette's tube, near the spot pointed out, must, under such circumstances, have been less than 0, provided that the head of water used in making the experiment exceeded, say 1^ times, the nmall tension which could not be eliminated from the receiver, — that is to fay, tho exhausting power of tho stream must have boon greater than the minimum power of aspiration capable of producing or forming what is termed to bo a vacuum, viz., a space devoid of ponderable matter of any kind, air included. Now, the internal condition of such a stream of water must be different, at least as regards ahsoluto tension, from that of the space freed of all rastter, which wo call a vacuum^ 'the question therefore presents ittell : In what manner does an increase in tho power of exhaustion, of a liquid vein touching tho sides of a cylindrical tube, sffoctthe conditions of molecular equilibrium of the sub- stance, if any, that fills a space enclosed by a vessel placed in communication with the tube, after all ponderable matter, air included, is exhausted therefrom. However this may bo, I am inclined to believe that the increased discharge afforded by cylindrical and divergent tubes, is entirely due to tho spreading action brought about by the adhesive or attractive properties of their sides or envelopes, by virtue of which the relations between tho inertia and attraction o* cohesion of tho particles of pona^.^hle matter moved, are continually modified in tho tube during the gradual enlargement of tho sectional area embraced by the stream, tho tendency being to create an absolute vacuum — and that the pressure of tho atmofr side of a reservoir, it istound that the discharge IS that due to the head measured from the surface of the iraler to the loiver or discluirging extremity of the tube,' ho must i lean, no doubt, a cylindrical tube fitted to a convergent conoidal mouthpiec« havin; the form of the contracted vein, for ho refers to Venturi's fourth p-"^^-'itiou ) s a proof of tho correctness of this law. ■^0 simple cylindrical tube, fig. 16, ;n ndod tube, a little less time was conditions — all of which tubes length along tho axis — does space between the envelope of . k s.. . the 64 IH The velocities at the lower ends of such tubes, fitted to conoidal convergent month pieces, happen to agree tolerably well with those acquired by solid bodies after these have descended freely through spaces equal in each case, to the head measared from the surface of the water to the discharging extremity of the tube. 3Vm coincidences So, however, in my opinion, no more possess the Aa^tSLunen^aX- cIkArAcia:£ which it is sought to attach ilietvly than that other, if anything, more generally accepted so-called hydraulic law ; " the velocity of a fluid at its passage •' through an orifice, made in the side or bottom, or top of a reservoir, is the same as ♦' thai which a heavy body would acquire in falling freely through a space equal to " that comprised between the level of the fluid surface in the reservoir and the centre " of that orifice " — the acceptance of which experimental indication as a natural law, the celebrated Lorgna has conclusively ~»1i«wji not to be warranted by the facts and truths elicited by properly directed investigation.* In their attempts at theoretical demonstrations of the law just enunciated, mod- ern authors, in genera), shelve all difficulties apparency without any scruples, by constructing a reservoir to suit themselves, viz., one having its sides joined with the orifice of efflux by easy convergent channels of approach, in order that, they state, the parallelism of the moving sheets or layers of liquid taken perpendicularly to the axis of the sJjpam can be considered to bo perfectly realized ; it is clear, however, that this is in reality equivalent to dodging past the contracted vein, which, however, unwilling they may be to admit it, remains the stumbling block in their way. DISCHARGE THROUGH DIVERGENT AJUTAGES OR TUBES. 1. Tubes A B C D A, applied directly to the Wall op the Resevoir, without THE Intervention op a Conoidal Mouth-piece, uavino the pobm op the Natural Contracted Vein. 1* If, in addition to the ab- straction of gravity outside of the reservoir E G H I (Fig. 19), it is assumed, as was done in the case of cylindrical tubes, that the fluid filaments of a nnturally contracted vein issu- ing through an oi-ifice A B, are dispersed in a uniform and continuous manner over the en- tire cross-section of the diver- gent tube A BCD, fitted to the cif ce A B, in the reservoir, as shotrn in Fig. 19 at every point of their path along the axis O K, through this tube — not- withstanding that this hypo- thesis is even perhapo a little further removed from the true conditions of the otilux through divergent tubt.: unprovided with oonoidal mouthpieces, than il is from the conditions of the eftlux through cylindrical tubes — the coelficioiit of eitlux or dis- oharge proper to a divorgt>nl tube such as A B C D, viz., the ratio of this discharge to that afforded in the same time and under the same head, by an orifice A O B, in a thin plate, can bo determinwl as follows: Here, as in plain c\ /.udnoal tuboH fitted directly to the wall of a reservoir, the ever-varying ratio beiwoen the velocities which are respectively duo to the forces //„ and/4 in the naturally contracted vein, is continually being transformed through • See tranBlalion of first two chanters of big " Physioomatheraatieal Theory of tho motion of liquids isBuing from oritices Id reservoirs " appended hereto. V' V' vergoDt i bodies he head le tabe. ;, more passage same as equal to le centre natural L by the ed, mod- iples, by with the state, the ly to the iver, that however, , WITUOUT OF THE ) the ah- oatHido of (Fig. 19), I done in cal tubes, jnta of a vein issu- C8 A O B, liform ami ^er the on- tho diver- fitted to reservoir, I at every ng the axis ibo — not- hia hypo- tho otilux is from tho ax or dib- i discharge A O B, in lorvoir, tho the forces od through he motion of 65 the intervention of the capillary attraction of the sides of each tube; the force//,, being increased, not only in the tubes which are absolutely divergent, but also for tubes whose sides have a loss convergence than those of a mouthpiece having the form of the naturally contracted vein — and the force /^'e being simultaneously modified in a contrary sen&e. If, therefore, the force /!„ is transformed into j fi^, j being any positive number whatsoever, greater than unity — considering that the total amount of momentum which can be developed in an element of mass by any two forces in the unit of time, or during any fixed period of time, mukt remain constant, so long as there is nothing added to nor subtracted from the sum of the forces— tho expression : which represents, in a general way, the proportional velocity j;,, or velocity ratio of tho motions duo to the two forces fi^ and fi„ at any point of the naturally con- tracted horizontal veins abstracted from gravity outside of the reservoir, in terms of the abcissa x — becomes converted in the divergent tube into: But here this fraction is not uniformly equal to unity, as was tho case for cylindrical tubes. In all tubes in general, all other things being equal, the proportional (not the actual) velocities, or the velocity ratios i\, of the moving fluid, evidently vary, along the axis, inversely r.s tho areas Tlif of their circular cross-sections, viz. : as — wthat -7- = -V— where v^ is tho velocity ratio corresponding to the ordinate y and v\, that V ___ corresponding to the ordinate xj'. But when the length O E = x- of tho tube A R C D is reduced to o, viz. : when this tube is removed altogether from the reservoir, and the fluid passes through the orifice AO B, we have for the proportional velocity or velocity ratio: «^.= »^*/v) So -f + ^h-\ So + (v)S. -f- /^v) ^ - J^v;^.. + i^.^J^ ^ J' '(,0'° + 'CO^'"'' 1 r r" (r + mx)- whence : and: 'o X (17) 66 ■' '1 is In ■1 SubBtituting tborefoiy^in tho expresBion which roproBentfi, au N»o + explained in tho case of the cylindrical tube, tho ratio between tho absolute number of liquid molocules passing the plane of tho orifice A O B, in u thin plate, during a given time, and that flowing through tho corresponding base A O B, of any tube of the length x, during tho same time — the value of ^ iu^t found in terms of X' for the symbol, wo obtain for the velocity v/^,|i"\in the small base A O B, of any conical divergent tube A B C D, whose length O E = /, applied directly to the side of a reservoir, viz. : without contracted mouthpiece : Let us now apply this formula to the determination of tho velocities at the bases next to the reservoir, of some oftho conical divergent tubes oxporimented with, for the purpose of comparing the computed ratio between tho velocity in an orifice pierced in a thin plate and that in the small base of the tube, in each case with tho corresponding velocity ratio deduced from experimental data. EXAMPLE. By fitting immediately to the side of a reservoir, viz.: without intormodiato contracted mouthpiece, a divergent tubo, whose length OE=l = d'2l2i inches, was nine times its diameter A B=:2r= 1*0236 inch at the small end, the flare of its sides AD, BC, being 5°-<)' and tho diameter of the largo base D G = 2 (r = 7nl) =l'844l inches, Eytolvoin found that with a constant head of 2-3642 feet = 28-3'7 inches, the coefficient of discharge for the base A B, was 1-18, the theoretic dis- charge being 1. As already done in other cases, we may here assume, without risk of erring materially, that .«„ varies inversely as the square root of the velocity — consequently, as for 14 inchcn head I found s„ to be equal to from 0-54 to 057 r or so, we have for a head of 28'37 inches, s„ = 0'57 r \U = ^ 2917 X 1^ = say, 0-25 inch. 2-308 28-37 Again, judging by tho results entered in Tables V and XIII, wo may put approxi- mately ify-. = 0-43 and also c \A O I!/ = 0-630= = -3969. Substituting these values for the symbols in equation (18), wo obtain 1'21 for tho coefficient of discharge at the base A O B, through tho tube A B C D, in place of 1-18 found by Eytolvoin. ^'■B- I applied directly to the horizontal bottom of my circular reservoir, viz., without the intervention of a conoidal convergent mouth-piece, a conicully divergent brass tube 12 inches or nearly 29 diaraofors long, whose small bn«e was 0-422 inch in diameter and the largo base 1*333 inches, the total angle of divorgonco "f the sides being thus 4° 22', and found the coefficient of discharge under water to be, on an average, 1-12 in tho small base of the tube, with an effective head or diff'erence of level between the water surfiicos of tho supplying and receiving reservoirs of 1-30 inches, as compared to u theoi-otio discharge of 1 at the same place — and 1-723 as compai-ed to the actual discharge under water through an orifice in a thin plate. failii paBs 2° G as si ratio poin absti that virti the AB x = wke romi pror 61 I am not certain, however, that this tube was effectivo over its whole length failing which, the lower portion must have proved more of an obstruction to the paeeago of the water than otherwise. of a (18) Tunss C D E G applied to the shall base D O, OP THE Conoid AL Mouth-piece A B C D, iiavinq nearly the form op the naturally Contracted Vein. Here the expression •%/l/v\So "T^ + Vif which, as shown, denotes correctly, in general terms, the ratio of the velocity or motion proper to any point on the axis of the naturally contracted vein, abstracted from gravitv outside the reservoir, to that in the orifice A O B, becomes transforraod by virtue of the lateral capillary attraction of the the tubular envelope, only after this natural vein A B C D, has doflcribed a portion of the trajectory a: = OE, Fig. 20, viz.: into =ii««*iA.^ Fit. 9 y^AA 'wSl '2f ii^i ,fcl(* mm' iMill' 'llll M U t ' run SO wbere x' represent* K H tbo len^ih of the divorgont tnbo. Siom^i when the lenjijth of tho divwRpent tobe fs reduced to •, " removed altocrether an(^ the tlaii.. past^e: only througk lib* aaalkfiHB ASC X^^ pi-oporlional -nakctty >r ^eloci)^ ratio is sioLply, as itaaHi dHwRB, m/mi to yl'o «.+JC ^i^.-^ *^+'(:f I. ' \ f 68 and, moreover, as the velocity ratios corresponding to any two sections D and F G, of the compound tube A B G F, must vary inversely as the squares of their diameters or raoii, we have the following relation : /a)'"+'^+^'+A|'c)^° +ko ^+ka) ^'-V'a) *"+^© ^+'QJ^ 1 'en ^°'^\i) '^"•"'(a) •^^' DE» (19) "~FH' fr' + mx')» where r' represents D E, and m the tangent of half the angle included bolwoen the sides D V and C G, whence we deduce : ; = If now wo substitute this value of ^' for this symbol in the expression : (20) V V */v\ 5oT*fT\ ^"T'/'vN J^ *(a)*""^'a) ^+*a)^' which, as previously explained, represents the ratio which the absolute number of fluid particles, considered as solia molecules, that pass in the unit of time through the orifice in a thin plate A O B, as well as through the section of maximum con- traction DEO, bears to the number of particles that flow, under the same conditions and daring the same time, through the corresponding bases A O B and D E ot the compound tube A B C G P D A, wo obtain for the velocity in the small base DEC of this tube : tuhc 1::= '2gH< — — ^^ _==z= ^ X velocity 1 / vel<»city \ natttrni I w I uriticu 1 eonlractecl I A I DEC I vein at / I mouth- I Dli / \ piece / (21) II standing for total head of water on the orifice A O B, and g for acceleration of gravity. EXASIPLE 1. I applied to the bottom of mj' circular reservoir of about 4 inches diameter, a cono'idal mouth-piece A B C D (Fig. 2C), having nearly the form of the contracted vein issuing from an orifice in a thin plate 0*4 inch in diameter. At the small base C B, of this mouth-piece, where the diameter was only 0-313 inch, I added a conical divergent tube JD F G, a;*=9'96 inches long, along the axis B H, and measuring 0*319 inch diameter at the small end G D, and 0*892 inch at the largo end F G, }l the angle of divergence botwoen the sides C F, D G, being therefore 3" 18'; on account, however, of the slight difference of 0003 inch between the diameter D, at the small base of the mouth-piece and the corresponding base of the divergent tube, the angle of divergence between the base C D of the mouthpiece, and the base P G of the tube was actually 3° 20'. In three experiments, under pressure heads of 13*6 and 15*1 inches, I found the mean coefficient of discharge under water, through this tube, to bo 2 028 at the base C D, while, with the same heads, the corresponding coefficient of discharge of the mouth-piece A B C D, alone was only '975, on an average under water, for a head equal to, say (2*028)'Xl4 inche8=58 inches, whence it is clear that the discharging capability of the compound tube A B D G F C A, was 208 times greater than that of the mouth-piece alone. In this instance, A O=r=02 inch, D Fi=r'=1665 inch, O E=a;=l'00 inch, B H=a;'=996 inches, P H=r'+ mx'=(i-U6 inch, m=tangent 1° 40 =:-029097. So may approximately be taken at 0'56r=0"112, judging by its value in other cases, and, by inspecting Tables I, II, and V, it will be seen that we can put i,v\= (cocflr. y u" Hc ), the ratio of the theoretical velocity due to the head H, to the mouth- j velocity of efflux through the orifice B E C, of the contracted mouthpiece, under a / cocfl. \ ' / velocity I head of from 65 to 60 inches, also conSid =1, nearly. \ vein at / \ 1>E, / If we substitute these numbers for the symbols in the last equation and divide by 2g H,we find by computation 1 '973 for the coefficient of discharge or velocity through the base DEC, against ^ 028, by experiment. The discrepancy between the computed and the observed coefficients of dis- charge is probably duo to an unavoidable want of accuracy in some of the factors which were introduced into the computation, and a part of the excess of the latter over the former co-efficient iri also to be attributed to the great disengagement and consequent diminished mutual interference of the fluid particles moving within such tubes, in comparison to what takes place in the naturally contracted vein issuing from an orifice in a thin plate. Furthermore, the profile of the mouthpiece differed slightly from that of a perfect naturally contracted vein formed under a uniform pres- sure, or in the open atmosphere, the said embouchure being a little more convergent than the vein. Example 2. Theoretrical determination of ratio of velocity in small base of divergent tube with cycloidal mouthpiece, experimented with, in 1853, at Lowell, Mass., by Mr. J. B. Francis.^t> ^ArorWif^/ fc/oei^y ^'telo AeacL >■ Mr. Francis, the celebrated American hydraulician, fitted to the vertical side W Z of a reservoir, a cono'ldal mouth-piece a I'O foot in length from N to R, formed by the revolution of a semi-cycloid A U, generated by a point U, in a circle O, 0'635 foot in diameter, rolling along the base A Rt, as shown in Fig. 21, with a cylindrical prolongation U C D V, O-l foot long from U to C, having a diameter of 1017 foot between these two points. To this compound mouth-piece he joined, in a horizontal f)Osition, a divergent cast-iron tube C D L K, made in four parts b C d 6, each 1 foot in ength, screwed together and ground smooth inside on a mandril, with emery, but not polished, having the form of a frustum of a cone, 0*1454 foot wide at B F, and 0-4085 at K L, with sides E K, F L, containing an angle of 6° 1, joined to the cylindrical portion U D V", of the mouth piece by means of an arc of a circle of about 22*69 feet radius, tangent to both the conical frustum E K L F produced, and the cylinder U C D V. Although the discharge took place under water, the tube proved to be effective only for the first 3 feet, viz : up to I J, or probably for a lengtn intermediate between 3 and 4 feet. 9 10 I W v :i ■li l''^i ?;i n The following cbaracteriHtio results obtained uro extracted from Table XXVII, of experimental data given at page 221, Lowell Hydraulic experiments, 3rd editioD, 1871: TABLE XVIIJ. Nos. of experi- ments. Orifice in thin plate and parts of the compound tube used. See fig: 21. Diameter at the place of dischaige. See fig : 21. Differences of level between the water surfaces of the supplying and receiving reservoirs, or effective head il producing the discharge. tlaximum ratios of velocities at smallest section to velocities due to the beads. Feet. Feet 94 Orifice. 01017 0-0916 0-6642 96 II 01017 -4835 0-6797 99 II 01017 1-0242 0-6915 97 II 01017 1-4987 0-5928 2 a 0D = 0-1018 0-0340 0-8183 6 11 II II 0-2300 0-8626 11 H i< II 0-6590 0-9367 18 II ■1 11 1-6168 0-9i39 37 ab EF = 0-1464 0-8644 1-6919 49 a b c G H = 0.2339 1-0999 2-1643 62 abed IJ =0-3209 1-1772 2-4306 78 abode KL =0 4086 1-2823 2-4213 iff ' ''V After making various deductions from the results of bis 101 experiments on tb . discharge under water through the divergent tube and mouth-piece just described, Mr. Francis discusses, at pages 126, 127, 128 of his work, the application of Ber- nouilli's theory in connection with the large coefficients of efflux or velocity arrived at by him, as follows : " According to Bernouilli's theory, the velocity of the water at its final discharge from the tube should be that due to the head ;* in experiment 62 this velocity is 8-7018 *0all A. the area of the section, and V the velocity of the water at a 6 (Pig. 21), B the area of the section, and v the velocity at e d; A = the head or difference of height of the surface of the water in compartments X and T. The motion having become permanent, we have -. A V = B ». The volume of water included between the sections a b and e i in the small time t will move to d b' e' d' ; the volume included between the sections a' b' and e d is common to both positions, every particle in one having its counterpart in the other, both in position and velocity. In finding the change in the living force in the two positions, we need only consider the volumes a a' bb' and ec^ d d\ These volumes are equal, and assuOiing the water to be pure and at its maximum density, the weight of each ia 62-382 A V ( pounds. The living torce of the volume a a' 6 6' is 62-3f2A V< ya ce' dd' is & 62-382 A V t if The increase of living force in passing from one position to the other being 62 -382 A Vf („2_va) 3 O) 'V Ull IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I ;f iiM iiM 2.0 1.8 1.25 1.4 16 4 6" ► ^ .||fi = 0-0630 ft, bed: ^.|M^ = 00631 ft. Also, b: 0-57r=0035'796 ft, be: 056 r = 003528 ft., for tube a for tube a s„=for tube a for tube a for tube a a; =for tube a bed: 56 r: b: 1-0 ft., a b c : 2 ft., abed: 3-0. :003534 ft.. X for tube a b : 0-94, a b a b c: 0-945, c d : 0-95 for efflux under water ; these last factors being taken in excess of those found by Mr. Francis for corresponding heads, as per Table XVII J, on account of the greater efficiency of the mouth-piece for the increased velocities generated by the divergent tube. By substituting the above values successively for the symbols in equation (21), we obtain, after dividing by V2 g H, the following ratios of velocity at smallest section to velocity due to head ; the tubes, as already stated, being considered as true frustums of cones, viz. : For tube a b: 1-3606, " " a be: 1-8523, " " abed: 2-0793. The same ratios computed in accordance with hypothesis (6), are found to be — For tube a b: 1-3590, « « a bo: 1-8514, " " abed: 2-0693. These three ratios are deficient in comparison to those derived from Mr. Francis' experiments, nearly 18 per cent., for each of the three tubes. This uniform discre- pancy L attribute to a supplementary conversion of acceleration into mass effected in the excessively convergent cyclo'idal mouthpiece (as compared to the theo- retical contraction of the natural liquid vein = v^^ or -8408), simultaneously with the reduction of the absolute pressure in the said mouth-piece through the agency of the divergent, tube. In the case of the convergent mouth-piece, I conceive the pro- cess of transformation of the elements of energy to be the reverse of that obtaining in the divergent tube ; in the latter the liquid is attracted by the sides, while in the 7ft the tubes iroctly by valent as Btively by aboat tbo 1 not have ion for the lUODV, jccur very he results t, without od): the point IS under :'ancis for icy of the tion (21), smallest sidered as 1)6 — '. Francia' m discre- s effected the theo- usly with agency of e the pro- obtaining ile in the former its adherence to these sides is diminished; the pressure is, however, reduced in both instances. Notwithstanding the unavoidable want of accuracy in some of the factors which had to be used in connection with the practical illustrations of the working of the new theory given above, it is evident that this theory leads to results much superior, in point of concordance with observed facts, to those obtained with the aid of the theories now in vogue ; some of these latter results seem to me to be in direct con- tradiction with the actual state of matters as established by careful observations. DlSCHABQK THROUGH CoNIOAL CoNVEROENT TOBES. Although this class of tubes is as simple in conformation as the diverging tubes, the conditions under which the flow of liquid takes place through them, are variabla not only with the degree of convergence of their sides, but also with their length. 1st. In tubes such as A B £ I A, whoso sides A I, B K converge less at every point of the axis O J, than the corresponding naturally contracted vein of equal length A B M L A, projected under the same head through an orifice ^ in a thin plate, whose area is equal to that of the large base A O B of the tube, and of a length O J = / less than that for which j is a maximum, the fluid is unceasingly compelled to follow the sides A I, B K of the tube, as in the case of a plain conical divergent tube added directly to the reservoir without mouth-piece. Formula (18) is therefore directly applicable to all such tubes ; the distance J, from the orifice or large base A O B, at which the convergent tube A BC D ceases to act in a similar manner as vergent tube, and where j relation : is a maximum, being determined in general the di- by the r dx I 2i^^.s^ + x+iyy.x + 2 V^ (D fr/'v\"o'^ "* •*'*/v\ ' , "T l/-y\^ I ilfl 1 + l + l (a) + \l)' , + l=(v)So + 2i.y.x /c) T 1/v\SqX T XI^^\8q T X r^\X' X 1 dx x(r 4- mxy + 2i,^.xr''(r + mx) ar 4mi.y^xrXr + mx)' rv ^kO' (22) + i,yyxr*(r + nxx) j — | i.^Jr + mxy -h 2i.^.T\r + mx) X (r + mx) — 4w^^^v^a;r^ \ 1 2i.^^s„ + x + i .x -I- — j So hyp. log. x\ \r + mx\ =o I kcow of no experiment made with tubes so conditioned. 2nd. When the sides A D, B C, of a tuble A B C D, converge at every point more than the corresponding outside portion of the naturally contracted vein A L N PUB, projected through an orifice in a thin plate equal to the largo base A O B of the tube, or when they converge less than this naturally contracted or theoretical vein only for a part O J of its length, as in the tube A B C B, and for the remainder J E of the distance O E, from the large base A O B to the small base DEC, more than said portion of contracted vein A B P N A, it is clear that here also the same as in divergent tubes, motion assumes a permanent state in the tube taken as a whole, only after the initial fluid sheet occupying the plane A O B has passed the section DEC, contrary to what takes place in the naturally con- %9 ■M W % I 'i tracted vein, in which the conditions of motion in the posterior portion A B M L A can evidently not be affected by any change that may tako place in those of the fluid particles passing at D £ C. In all such tubes, any difference existing between the velocity of the fluid issuing from the tubeat the small baseDBO, and that of the naturally contracted vein ABMPNLA, at the corresponding section N D B C P, is the result of an artificially increased, or partly increased and partly diminished velocity, due to the force fie, viz., of that corresponding to^lhv\8o + * in the said natural vein. This transformed velocity may, in general, be represented by \rrf° "^ •'"^' where j is a number greater than unity for increased velocities, and less than 1 for diminished motion, or rate of progress of the vein, as regards the force /'■<.. The expression : _jL_L_--____- , which, as yj\i r -^\:f already staled and explained, represents, in general, the velocity ratio Vp of the motions due to the forces /tc and fi^, at any point of the naturally contracted horizontal vein outside of the reservoir, is converted in the convergent tube A B D, into: ^\lfo+jX and the same as for conical divergent tubes, we may put : r' where r stands for the radius A O =: B, and m for the tangent of half the angle of convergence ; whence we deduce : J = ' {(^(Jw°+^af+ J^o'"^'')}-'^)'" "^ (r-my) If, now, we substitute this value of; in the expression : (23) X V 'Ca)'°"^*af which indicates the relation between the absolute number of particles that pass in an orifice in a thin plate having a diameter equal to A B, and those passing at he large base A B, of the convergent tube, we obtain : coefF, \ / coefT, (coeff, \ I coelT, \ vel \ / vel I large 1^1 orilfAB 1. base A B l~~l ■" thin I ' convergt I '" 1 plate. / tube. / \ / (20 ^^\l)«o + i^.J-^ (l+ (r—mly } xl 'Qf^'O^ where I is substituted for a;= O E, the length of tube. BML A f tho fluid mingfrom SIPNLA, ireasod, or 5., of that velocity aator than >f progress iiich, as Vp of the contracted a A B D, r I the angle (23) s that pass passing at (20 1 11 . Without a thorough knowledge of tho laws governing the variations of i^v\aQd So, it is impracticable to determine accurately, by computation, the velocity at tho small base C D of the tube. Moreover, on account of the sharp turn of the liquid fillets about the angle of tho junction of the tube and reservoir, it is probable that these do not adhere to the sides of tho tube before striking against the same, wherefore a part of tho officionoy assumed for the tube in oonstructing formula No. 24 is lost, and the discharge is also affoctod by friction. The approximate determination of tho coefficient of efllux for one of tho conically convergent tubes, experimented with by Messrs. D'Aubuisson and Castol, referred to hereunder, was undertaken chiefly for the purpose of showing that tho above formula) lead in the right direction. With a tube 1'161 inch in diameter at tho large end A B (Fig. 22), 0'61 inch at the small end D, having a length E O = I'bib inch = nearly 2-6 diameters of the small base and sides A C, B D, inclined at an anglo of 40°, 20', the coefficient of efflux for the small end was found, by experiment, to be 0*87 under a head of 9*84 feet. (melt \ i>;'f 1 = 0-62 ; also, r = 8836 inch, I = 1-575 inch and m = tang 20°, 10' = 0*36726. We obtain, by using formula (24) : (coeff \ / cocIT \ / cocff \ S I = 0115* and ( &| ) = ( S X ^^ = conv / I cony I I cony f U'ul tube / \ tube / \ tube / 0-9686. On the Flow op Liquids thboucih Oblonq Orifices in Thin Plates. Numerous experiments were made by Messrs. Poncelet and Lesbros, at Motz, in 1826 and 1827, upon efflux through large rectangular orifices, pierced in a vertical brass plate 0'1575 inch thick, so as to obtain a perfect contraction of the stream. The widths of these apertures were generally 7'8737 inches, and in some cases 23'62 1 1 inches, while their heights varied from 0-3937 inch to 7-837 inches. Although these experiments are, with good reason, considered to be the most accurate available for practical purposes, on account of the uncertainty, as regards the effective head and nature of the contraction of the vein, arising from the fact of a depression taking place during efflux, in the water surface of the supplying rosor- yoir, immediately behind the vertical side or partition which contains tho orifice, they are obviously not suitable for use in connection with theoretical investigations. The only experiments I know of which appear to me to have boon made in the proper conditions and with the requisite amount of care, to be serviceable for theo- retical purposes, are those by Messrs. Castel and D'Aubuisson do Voisins, with rectangular orifices 0-328 feet = 3936 inches long and 0033 feet = 0-399 inch high, pierced in a vertical partition ; the ratio of the length to the breadth being, therefore, equal to 9 9398. The mean results obtained by these engineers are given in tho following table: f |t/vvSo+^ ■^Hiy 10 78 TABLE XVIII. 4 !1 Cd h M Coefficient o*" discharge or velocity, tao theoretical Depth of upper Depth of lower D=:lcl\/2g{lfi-hh velocity due to side of side of the mean pressure of orifice below water orifice below water = Discharge per i{^) surface. surface. second. 1 on the orifice being equal to unity or 1. >i feet. feet. cubic feet. 1 0-0491 0-0821 0-01607 728 2 0-0819 0-1149 0-1946 0-720 8 0-1147 0-1477 0-2242 0-719 4 0-1476 0-1805 0-2497 0-716 6 0-1804 0-2134 0-2723 0-710 In common with the last-named and other experimenters with oblong rectangular orifices and the like, I found, under a small head of about 3 inches, that the coeffici- ents of efflux or velocity proper to annular and lunular-shaped orifices, are invariably greater than those corresponding to orifices which embrace the full urea enclosed within the circumference of a circle. 1. When ratio between the breadth and the mean length of the annular space or opening formed by introducing a cylindrical rod, 0*185 inch diameter, in the reservoir opposite an orifice in a thin plate 0-4 inch diameter, was 8*55, the coefficient of dis- charge was about 0*7256, with the base of the cylinder in the plane of the orifice ; this coefficient became, however, reduced to 0*68, when the cylinder protruded through the plate 0-2 inch beyond the plane of the orifice, as shown in table VI. 2. When this ratio was increased to 20-10, by introducing into an orifice 0-482 inch diameter, a disk 0*355 inch diameter and 0-048 inch thick, the coefficient of discharge rose to 0.7948 for the upper base of the disk in the plane of the orifice, and to 0*8098 for the lower base in the plane of this orifice, as per Table VIII. When the ratio between the mean length and breadth of the ring-shaped aper< ture was still further augmented to 80*35, by introducing the disk just descrilbed into an orifice 0*384 inch diameter, the coefficient of discharge rose as high as 0*8907 for the lower base of the disk in the plane of the orifice, and 0*91 for the upper base in this plane, as per Table IX. 4. When the discharge took place through the lunular-shaped opening left between the circumference of a cylindrical rod 0*185 inch diameter and that r>f an orifice 0*4 inch diameter, as shown in the figure at the head of Table VII, the coefficient of discharge was 0*7016 while the base of the cylinder coincided with the plane of the orifice, and about 0-663 when the rod projected 0-2 inch below this plane. In all these experiments of mine, however, the contraction was probably modi- fied, and, to a small extent, destroyed along the longitudinal face of the rod or disk introduced into the reservoir and let down below the plane of the orifice, for which reason the discharge proved, perhaps, slightly larger in each case than it would have been, if the stream had been allowed to contract freely all around the perimeter of the orifice. If the larger coefficients, obtained in the four cases just referred to, are cor- rected for this want of completeness of the contraction of the stream — approximately sharge or ioreucal e to lure of ') ng equal rl. Bctangular le coefflci- invariably a enclosed r space or reservoir mt of dis- le orifice ; protruded e V I. ifice' 0-482 afficient of )rifice, and aped aper- i deBcrioed 1 as 0-8907 upper base ening left hat (»f an VII, the with the this plane. )ably modi- rod or disk , for which would have Brimetor of o, are cor- »roximately w in accordance with the empiric rules given by some authors, tho> become reduced, respectively, from 0-7256, 0'8031, 091 and 7016, to about 0-700, 0-77, 0-85and 068; they remain much higher, however, in any case, than the coefficients which are proper to a circular orifice of equal area for efflux under the same head. There is no apparent reason why the first slice or sheet of liquid leaving the orifice at the instant it is opened, should move off faster, under the same pressure, from an oblong than from a perfectly round or circular orifice in a thin plate, and I see no other cause for the increased discharge obtained than the following : When one elementary slice or sheet of liquid of the oblong-shaped vein tends to detach itself from the next succooding one, and that, owing to the intermittent action of the resistance or force of cohe8ion,tho motion of the liquid particles, or filiets.bocomes accelerated, and consequently the total area of the moving stre'-.m correspondingly diminished, the increased rate of contraction in the direction of the longest of the radii, which extend from the perimeter of the oblong orifice or vein to the centre of figure, as compared to that taking place along the shorter radii, produces, together with a change of form, also an enlargement of the sectional area embraced by the spurting liquid vein through the admixture of air with the water or otherwise— when the conditions of flow become similar to those of divergent tubes. Liquid PaEssuRE, Motion, Eneroy, &c. Pressure is most frequently generated in liquids, whether in a state of rest or in motion, by gravity acting on a large number of particles superimposed to one another ; but it also often results from the action of a piston moved by some exterior force. No matter how generated, it may be considered in the light of an artificial increase, in the natural force of repulsion co-existing with that of attraction between all molecules. When the force of attraction is artificially increased, instead of that of repulsion, the result is the opposite of pressure, viz , dilatation or distention or exhaustion. Liquid motion and energy are, in all cases, governed by the differences between the forces of attraction and repulsion obtaining at the origin and along the path of the stream. If a pressure^, has to be applied during the small space of time d t, in order that a liquid particle may describe, within the sphere of molecular oscillations, the small distance d x, necessary to overcome the force of cohesion, together with the inertia of the shid particle — accoixling to the laws of uniformly accelerated motion^— another pressure np, will have to act during a length of time=:d t -r^to cause the same par- Vnp tide to describe the distance dx, that is to say, the number of times which one and the same apace d x, is passed over in the unit of time, say one second, by successive molecules, varies as p/, of the intensity of the pressure to which the particle is sub- jected. In the case of a liquid vein issuing from an orifice in a reservoir by virtue of the action of gravity alone, the absolute velocity varies therefore, as the |/ of the depth of the centre of pressure on the orifice below the surface, being theoretically equal to 0,7071= p/ J of that attained by a body after having descended freely through a space equal to the said depth, wherefore, abstracting all causes of incidental perturbation, the energy of such a vein is directly proportional to the pressure or Dead on the orifice. This constitutes the badis of the generation of the absolute velocity and energy proper to a liquid vein taken as a whole, thus : if a circular vein having a mean diameter of say 1 inch between two points, A, B 1 foot apart, of its trajectory, and formed under a water column pressure of 1 foot, takes say ^ of a second to travel freely from A to B, another vein of tho same dimensions betwQeeu the said points, but 80 ■I'l^jU ■■'■nj .4 •fl H l! :v« J genoratod by a hydrostatic pro89uroof4 feot, yet in every other respect formed under the same conditions aa the first Jet, will fill up the space of 1 foot, referred to between A and B in | = ^^ second :— wherefore the quantity of water supplied by vein No. 1 will boar to that afforded by vein No. 2 the ratio of I to 2, and energy will be developed in the ratio of 1 to 4. The absolute rate of motion or velocity just referred to, which is proper to the whole of the elementary liquid slices of which every jet may be conceived to conHist, is quite distinct, however, from the rate of progress of one and the same elementary sheet of liquid in assuming different positions successively along the path of the stream. It is by this relative motion or rate of advance, that the outline of the conotJal space swept out by the contracted vein and the distribution of pressure in tubes are essentially controlled. The relative velocities of an elementary vciumo of liquid ejected from the reservoir corresponding to the area of the orifice, are governed by the- elementary impulses or increments of acceleration which are imparted, in rapid succession, to the increment of vein considered, from a state of rest all along its trajectory ; these impulses having to ovt/come alter- nately cohesion and inertia combined and ««.-re^ccai»ert» «!•»«■ — asXIJrea^^ ;^ explained in another part of the paper. In the naturally contracted vein the pressure is null, oro — from the theoretical orifice, which is situated at the plane, where the total acceleration or velocity, generated by the small impulocs -Afffli^S. against cohesion and inertia combined, is equal to the velocity due to the impulses expended in overcoming Tela cei- inertia alone — to the end of the vein outside the reservoir ; from the said orifice to the plane of rest, the pressure gradually increases, becoming equal to that due to ' the full head at the said plane. When a divergent tube is added to a conoidal mouth-piece, having the form of the naturally contracted vein, the molecular force of attraction is increased so as to produce a dilatation or distention in the liquid filling the mouthpiece, which pro- bably diminishes in intensity, from the smallest section to the theoretical orifice, and thence to the plane of rest, where the full hydrostatic pressure again obtains. In the divergent tube itself, the exhaustion decreases gradually from the small to the large base, where it is reduced to a minimum. Thus, if the total velocity generated, by the addition of the divergent tube, at the smallest section, is to that obtained at the same place with the mouth-piece alone, in the ratio of 2 to 1, the force of attraction will be increased by a quantity equal to 2^ — 1=3 times the pres- sure due to the head of water on the centre of pressure of the section of the tube. If the same divergent tube was added directly to the reservoir, viz., without the intervention of a conoidal mouth-piece, the force of attraction would also bo increased, but to a less extent. In a conically convergent tube, or over-convergent mouth-piece, of any descrip- tion, added to the side or bottom of a reservoir, with or without natural conoidal mouth-piece, the force of repulsion or pressure diminishes during the flow of the liquid from the large towai'ds the small base. In order that the whole volume of liquid may pass at the large base, which can be ejected through an orifice having an equal diameter, by virtue of the pressure in the reservoir, the force of attraction must be increased in the same manner as in the divergent tube, and vice versa, if the force of attraction is increased at the small base of a convergent tube, by the addition of a divergent tube, the discharging power of the former and of the two tubes combined is increased as compared to the power of a natural conoidal mouth- piece, having its orifice at the small end equal to the small base of the convergent tube. CONCLUDING EBMARKS. It was in the year 1645 that the Italian mathematician, Toricelli, enunciated the theorem which bears his name and may be stated as follows : — " Generally and making abstraction of every obstacle or all cause of perturbation, " the velocity of a fluid at its passage through an orifice made in the side of a 81 formed irrod to lied by energy 'oper to aivcd to and the ilong the e outline prossuro (montary le orifice, eleration red, from ottO altar- ee^a^r . beoretical • velocity, id inertia rercoming orifice to at duo to bo form of ,sed so as which pro- jal orifice, ure again from the il velocity , is to that 2 to 1, the the pros- B tube. ithout the d also bo ny descrip- al conoidal flow of the volume of having an f attraction iersa, if the bo, by the of the two dal mouth- convergent nciated the (rturbation, side of a *' reservoir, is the same as a heavy body would acquire in falling freely from the " height comprised between the level of the fluid surface in the reservoir and the " centre of that orifice." About the year 1738, Daniel Bornouilli propounded his theory, vi/. : — "At any " point of a system of hydraulic conduits or pipes, the absolute total hood or pressure •' is composed of the pressure of the atmosphere, the actual hydrostatic pressure or " head, the head due to the velocity of the watof and the head consumed by friction " and other resistances encountered between the water surface of the source of supply " and the point considered. Ever since, it would appear to have been the constant aim of all hydrodynami- cians to determine the nature and intensity of the resistances to be overcome under all possible conditions, by making numbers of experiments varied in a thousand ways, from which empirical coefficients of friction, contraction, velocity and etllux could bo deduced and formulas based thereon. If, despite all the labours and pains taken by eminent men of all ages to place the science of hydraulics on a solid basis, there is still room for much improvement, judging by the discrepancies which exist between experimental data of apparently similar nature, furnished by different authors and the variations in the formulas given in works which are all held in equally high estimation, as also by the failure of water works systems to prove equal to the requirements of the services which they wore calculated to perform, it must be attributed, I think, to the fact of no one having apparently thought it necessary to take into account, independently of all resistance caused by friction, sharp curves, sudden enlargements, etc., the influence of the force of cohesion or aggregation which unites the fluid molecules into one homogeneous mass, and prevents their isolation. If I have alluded to the shortcomings of the theories advanced and of some of the experiments made by the learned authors whose names are mentioned and others, it is certainly not with any intention of making disparaging remarks respecting the arduous labours performed by them, but solely as a mean^ of assisting in the advance ment of a science the principles of which are still imperfectly understood, and, in hopes of attracting men of science, endowed with greater powers of penetration, and more generously favoured, as regards spare funds and time, than I am, to consider the suggestions thrown out herein with a view of placing the theory of hydraulics on a sounder basis. APPENDIX. PHISICO-MATHEMATIOAL THEORY OF THE MOTION OF LIQUIDS ISSUING FROM ORIFICES IN RESERVOIRS, BY MR. LE CHEVALIER LORGNA. INTRODUCTION. It is not to be denied that certain parts of natural philosophy owe everything, so to speak, to the mathematical science^and that other parts are much indebted to them, for, these sciences have fortunately rendered tractable things, into which neither reason nor experience, alone or combined, would ever have been able to penetrate so far. But in a great number of other instances these sciences have really not been of any assistance towards making a forward step ; unless we are prepared to accept, in the case of natural things, that which will never be, viz. : the truths of computa- tion for truths of fact, but which has taken place to a singular extent in those instances where the character and conditions of the object are totally changed when 82 4 1 •'v by abatraotion, it is Btrippod of overy thing tbat constitutes it— oh nai'iro domands that it sbould be, in the struoture of the worlot In point of fact there is not, for oxarapio, on the intimate affections and raotionH of compreHsiblo and iDoompreBHiblo flatdfl, a theory founded chiefly on mathematical principles which, as might happen in mathematical philosophy, could lay claim also with an equal right and above all exception, to a place in the natural science of nature. And if such meanft of invcHtigation were to fail us, what other course would there be at our dispoHal for penetrating deeply into the study of this science, considering that the constituent principles of the objects are unknown to us and that the various characteristic properties arc closely interwoven with very obscure and imperceptible forces. If I do not mistake, the mode of proceeding which seems most appropriate is that of very attentive observation and reaooning, makiitg a judicious use of one and the other by the methods of decomposition and composition —to wit, by the methods of analysis and geometry and by profiting also, in case of need, of the symbols of the one, and the figures of the other, but invariably as instruments only, and when the things or their parts can,without being disfigured,assume the character of simple homo- geneous quantities, be subordinated to mutual relations, and even bo represented to the bensos, under the abstract figares of geometry. Would not that be the true use of mathematics in connection with natural philosophy ? It is not meant that all suppositions are excluded from this manner of philosophising ; it is sufficient that such assumptionb bo loasonable and reasonably admissible in physics — as the postulates are in geometry, and not ideal and arbitrary or made for the solo purpose of adapting the object to the laws of computation. No doubt, this method is not that which is most followed, because it is not the most accepted, nor the easiest — and that it is much more convenient and pleasing to human pride to pretend having found than to find out actually what nature performs. It is for this reason that Mr. D'Aembort has not hesitated to declare that now-a-days every thing is accomplished by means of suppositions and computations. However, that may be, if it is not the simplest, this method is undeniably the surest and it leads to the truth, or at least to results which are not very far removed from (he truth and which time does not obliterate so easily as it obliterates inexorably our comments It is upon these principles that I have undertaken and effected this in- vestigation, us by trial, and as well as it lay in my power — of the motion of liquids within and outside of the reservoirs where they are maintained at u constant level during the flow. The principal properties which distinguish the liquids from any other known fluid, to wit: natural incompressibility, perfect mobility and the very strong affinity of aggregation commonly called reciprocal adherence of molecules — exert an influence on their affections, that without having a particular regard for these properties as indicated by the phenomena, we can ne\ . nope to attain sound knowledge as regards the very complex irregularities of their motions. The only time, it appears when we may dispense considering these properties, which are the cause of particular actions taking place among the molecules, one upon the other, is when none of them are disturbing the general movement; in this circumstance it is permitted to view the liquid in the light of an imperfect fluid and to subordinate it in a manner to the laws of dynamics. In such case, for example, I have thought a liquid vein in motion could be imagined to be established whose molecules are continously urged on with a uniform velocity in one and the same direction ; and by this means I have endeavoured, in another paper which will be found in this volume, to bring under the dynamical laws, the appreciation of the permanent impulsion of liquids against plane surfaces. But in every other condition of things, if the properties enunciated exert an essential influence on the phenomena, it will be necessary, in order that the theory may not be wrong, that it should always be based on facts and that it should invariably be directed in the path pointed out to us by these experiments alone wherein liquids have acted naturally and such as nature has constituted them. 83 ids that notions matical ' claim ionce of Id there Bidering various •ceptiblo ^e is that and the sthods of lis of the vhon the )le homo- trosonted I natural m»ner of lasonably jitrary or ia not the leasing to performs, ow-a-days [lowever, est and it from the rably our ■d this in- of liquids ,ant level er known jg affinity influence )ertie3 as ta regards ears when )articular ) of them d to view ner to the n could be a uniform ^voured, in dynamical le surfaces. ,n essential may not be variably be lin liquids I do not know if I have succoedod in my undertaking, as was my intention, but in any case the failure will be duo to my want of ability and not to the method which I have laid down for my guidance. CHAPTER I. NATURAL PHBNONENA. I. Phemmenon 1. — If a perennial vein of water A B, (Pig. 1) flows into a reservoir placed underneath and having any form whatever C D E P, in which the orifice of the bottom G H, where the incoming water is to escape, ia smaller than the area of the cross-section of the vein A B, it luUl he noticed that a certain quantity ofw'ittir ia first spilled and spreads over th ) closed bottom G D, H B, and then, niter fi certain t'ne, the liquid assumes a height Buci< as D T, above the bot- tom, tho surface being continually agitu'(^d by the influx of the vein, {iud once the efllux is equalized with tho influx, the water-level I K, remains static niiry, as long as the same con- ditions continue to subsist ; neverthe- less, the flow here is lu corrupted in the direction of tho vein at L M, and continues its course until after the liquid issues from tho orifice G II. If. Phenomenon 2. — And if several openings, smaller or greater than G H, are pierced in thin plates of metal, which can be applied to the bottom D E, it will be remarked, that oy using openings getting smaller and smaller, tho surface L X, ia formed and maintained at a level more and more elevated above the fixed bottom D B; on the contrary, hj applying successively to the bottom, orifices getting greater and greater than G H, the permanent height D I, of the water above the bottom diminishes more and more, and even disappears entirely when tho vein A B, flows freely past the bottom D B. III. Phenomenon 3. — But, if the inflowing vein is received in a recipient N O, placed quite close above the surface I E, pierced by small boles, so that tho water may descend in very small fillets, it will be seen that the surface I K, remains sensibly horizontal during the flow, as if the body of liquid I D E K, was stagnant. IV. Corollary I. — It ia therefore evident that the liquid spilled and spread on tbe bottom D E, is an ever-flowinf liquid. Oorollaiy II. — And that the surface I K, is the limit of tbe overflowing. 84 Corollary III. — ^And, as on the ooe hand, the sensible horizontality of the surface I £., daring the flow, and on the other, the successive flowing necessary to supply the quantity discharged through the orifice G H, give rise to a sensible state of rest on one side and of motion on the other, the undisputable result of these phenomena is, that the condition of the flo./age I D E E, is a certain singular state which par> ticipates both of rest and of motion, and which is, consequently quite distinct from the absolute state of either rest or motion. V. n Scholium, — We shall see hereafter how these few certain phenomena, which are the real axioms of natural philosophy, enlighten reason and guide it In finding out possibly the properties of liquids issuing from orifices in reservoirs, when the water contained therein is maintained at a constant height above the level of the orifices. It is a decisive stop in this very obscure matter, to have discovered, as we shall see, that the state of the liquids in the interior of vessels is in a state of overflowing and that this state is mixed and distinct from that of rest and motion taken ia their absolute sense, but participating nevertheless of both. B'U, before going any further, let us examine other phenomena which will show us more manifestly what is the use of these flowages, by moving their limits farther and farther away from the orifices of the vessels, while expelling the liquids succes- sively through smaller and smaller orifices. VI. 1^ M m Fig. A m '"I'li ^ _.<:..,L.i-L-v-v-v---v- i:i;:i D K -V-\- -T- T- I I ^^2S!^ZSOSlA rt Phemmenon 4. — Let a glass recipent A B D (Pig. II) be pre- pared, in the bottom of which an opening a b, is made. Let a perennial vein greater than the opening a b, continually throw into this vesssel, during a given time, a given quantity ol water, and let the the water held back overflow into the vessel up lo the elevation B G, there to asRume the horizontal sur- face F G, the limit F G of the flow- age being marked carefully on the glass. This being done, let the vessel A B D be removed from under the vein E, and after having let out some of the liquid through the opening a b, let this orifice be closed, and in place of the water wasted lot about an equal quantity of common oil be introduced. After this let us bring the vessel again under the vein E, so that the water may fall on the oil and traverse it to arrive at the surface of the water lying below ; then let the orifice a b, be opened anew. After the lapse of a certain space of time during w^ich one can see the oil ascend and descend alternately, it will be noticed: 1. That the surface of the oil stands still a little above the limit G F, indicated by the water, and the efflux of the water through the orifice becomes again permanent, as before the introduction of the oil. 2. That the surface of the water below the oil assumes and retains con- stantly a sensibly horizontal position, such as H I. 3. That the water introduced traverses the oil as if passing through a filter and enters the body of water underneath without producing any undulation therein, merely compensating for the discharge through the orifice a b, under the head afforded by thd two heterogeneous liquids, as the 85 surface supply of rest mena is, lich par« from the hich are ling out e water orifices, shall see, ring and absolute rill show \ further s succes- a glass 1) be pre- )f which I. Let a than the irow into m time, a nd let the low into tion B G, ontal sur- the flow- y on the the vessel om under aving let rough the orifice be the water his lot us le oil and )rifice a b, )ne can see face of the efflux of troduction tains con- introduced inderneath discharge liquids, as was the case under the permanent height B G, of the originally overflowed homo- geneous liquid. ' \rii. Phen. 5. — If this experiment is repeated with an increased quantity of oil, it will be seen that the surface of the water lying underneath can bo lowered permanently nearly down to tho plane of the orifice a b, especially if this orifice is' very small. Yet the oil continues to rest on top of the water, internally troubled on account of its affording a passage to the water which goes to compensate for the discharge, but remaining, with respect to itself, as a mass of stagnant liquid ; and, moreover, it will be remarked that the greater the quantity of oil poured upon the water the more will the surface of the oil rise above the level J K, originally established. Phen. 6.— If another apparatus is used, and in- stead of being introduced immediately into the oil — as in Fig. II, if the water of the perennial vein B be con- ducted separately through a pipe down to the surface I H, of the body of water lying below the oil, as shown in Fig. Ill, it will be observed that as soon as the flow becomes per- manent: 1. The oil on top of the water remains motionless, as if it was a solid body, even though the surface of the water should descend close to B C — which is admirable to behold. 2. The surface f g, of the oil remains, as be- fore, above the level F G, — assumed by the overflown water, and yet a little higher ,on accountof the vol- ume of oil displaced by tho pipe kept immersed therein during the efllux. 3. Fin- ally, if, in the various at- tempts, an account, as cor- rect as the circumstan- ces will permit, is kept, whether of the quantity of oil introduced or of the quantity of water expelled from the vessel by the oil pourod in it, it is found, if I made no errors in taking these very delicate measurements, that after the permanent flow is established, the weight of the oil introduced remains constantly a little greater than that of the waier expelled ; this, it has appeared to me, should be attributed to the adherence of tho oil to the sides of the vessel, owing to which adherence tho pressure of the oil on the water underneath is somewhat diminished. IX. Corollary I. — In the meantime, it remains decided by those phonomena that the velocity of the water issuing from the opening a b, cannot, at all, be due, as was 1 1 1 ! i ! 1 ,V.»«gi8fJ..,.,^r»na f< I: V- 1 i J: . i^:t^ i : I J:! 86 thought by Newton, to the actaal descent of the liquid from its permanent surface F G, to the plane of this same orifice a b — whilst it mi^ht also be due to the various other falls from different other elevations, such as I H (Figs. II and III), which is absurd ; considering that there is a downward motion onlj in the water at the bottom of the vessel, and none whatever if the superimposed oil was substituted for water — which oil is quiescent and fixed in its position during the flow. Corollary II. — And because the oil acts on the water lying below it, per modum unius (as a whole), like a loaded piston pressing upon the surface I H, of the water, it is evident that the pressure exerted around the opening fpressione circumfusa alford) is not merely that of the perpendicular column, having this same orifice for a base, as was thought by M. M. '. arignan, Hermann and many others, but, indeed, that of the whole body of liquid. For, since it is possible to bring down the surface I H, of the water, more and more towards the bottom, simply by increasing the height of the superimposed oil, and keeping up a uniform efflux by the introduction of water through a pipe or tube, as above indicated — and considering that the oil never acts otherwise than a piston, exerting an equal pressure on all points of the infinite sec- tion I H—^it follows that the action of any column,what6oever, of definite dimensions, is uot possible, nor can a determinate descent or fall take place, as was demonstrated in the preceding corollary. Corollary III. — It is proven by the phenomena that the water maintained within reservoirs, at a uniform height above discharging orifices, is an overflown liquid and that in this overflown state, the pressure exerted by the mass of liquid around the opening acts like a piston to eject the water through this opening, and that consequently, the force which the water has at its exit from the vessel has not, any more, been imparted to it, by virtue of its actual descent or fall from the surface or limit of the overflow to the opening, than it has been produced merely by the pressure of the vertical fluid column having the opening u>r its base. We can, therefore, plainly understand why the limit of the overflow rises the more above the level of the orifice, as this orifice diminishes in area — and that it falls lower and lower as the orifice is being enlarged, vanishing entirely, together with the overflow itself, when the supplying vein passes freely through the opening. Corollary IV. — Furthermore, we can now clearly understand in what manner acts the sensible rest existing within overf own liquids, about the sensibly horizontal position of whose surface or limit of overflow there is no more any doubt, considering that it is principally the pressure which urges on the liquid towards the orifice. And this kind of downward movement of the liquid which, nevertheless, takes place in this overflown state, appears clearly to be but the successive reflux of the molecules towards the orifice on account of the successive compensating substitution of water for the water which flows out, this being a reflux which must make itself felt throughout from the bottom to the upper limit of the overflown liquid owing to the very delicate yieldingness of its parts, without actually expelling the water through the orifice. As to the manner in which subsist and are verified sensible rest in a body of overflown liquid and an interior downward motion having no part in the produc- tion of the flow in the oiifices, it will supply the argument of another special exposi- tion which will be made further on. X. Phenomenon 7. If once the flow from the glass receiver A B C D (Fig II), which contains nothing but overflown water up to the level F G— has become permanent, small pieces of Spanish wax, or of some other similar body sligh^y heavier than water, are dropped into the vessel along its sides, we observe that the small pieces of solid matter descend slowly towards the bottom in a nearly vertical direction — until having reached a point very close to this bottom — their path becomes visibly inclined and curved towards the opening, and when making their exit they all pass close to the edge of this orifice, forming a sensible determinate acute angle with the bottom. This phenomenon has been first observed by Mr. Daniel Bernouilli, after- wards by the " Abbe " Bossut, as may be seen in their excellent treatises of hydro- dynamics, and I have punctually repeated and verified this observation last year. the sarface varionB jhich is bottom (vater — modum i water, I alforo) • a h&&o, , that of I, of the t of the >f water iver acts Dlte BOC- Lonsions, nBtratod lintainod veiflown of liquid ingi and has not, e surface jT by the riBes the at it falls r with the nner acts lorizontal asidering e orifice, kes place molecules of water itself felt the very rough the n a body e produo- ial exposi- 11), which lermanent, ivier than all pieces jrection— es visibly Y all pass with the uilli, after- of hydrO' year. XI. Phenomenon 8. — Having gathered and measured the quantity of water which passed under different permaneat beads in the reservoirs, through the orifices, whether pierced in thin plates or provided with additional tubes, it has thus been found that in all the experiments made by the most careful and trusty expoi-imontors — the velocities acquired by one and the same fluid issuing through the same tube or orifice pierced in a thin plate — bear to each other the sub-duplicate ratio of the permanent heights of the fluid above the centre of the orifioe. The more recent observations, viz., those which, through Royal munificience are just after being instituted on a grand scale at Turin (Michelotti, Sper. IdrauUa, e mem. deW Ac. R.pergli anni 1784-85), concur with all the observations made in bygone times in proving the truth of this law, so that there is perhaps not a single natural phenomenon so con- stantly established as this one. Corollary — Therefore, from whatever elevations a heavy body at rest may descend freely, it can acquire, at the end of the motion, tho actual velocities of the water issuing from the same orifioe under different permanent heights of liquid in the reservoir, and, as according to the thooryof uniformly accelerated motions, these velocities are to each other in the sub-duplicale ratio of the said heights, whatever they may be, it is unquestionable that the permanent heads, under which the water has run out with the said velocities — must be to each other as the heights through which a falling heavy body would have acquired the same velocities at the end of the fall. CHAPTER II. ENQUIRY INTO THE STATE OF OVERFLOWN LIQUIDS IN RESERVOIRS. XII. Prop. I. — The surface of a liquid abandoned to the free action of gravity, and constituted in perfect equilibrium in the vessel of any form whatsoever, which con- tains it, is horizontal or perpendicular in all its points to the direction of gravity. See the proof of this proposition in the worsen hydrostatics. XIII. Prop. II. — Reciprocally, a liquid contained in a vessel, of any form whatsoever, and abandoned to the action of gravity, whose surface is at every point horizontal or perpendicular to the direction of gravity, is in perfect equilibrium. XIV. Corollary I. — Therefore, if a liquid contained in a vessel is but sensibly consti- tuted in equilibrium, its surface will be only sensibly horizontal or perpendicular in all its points to the direction of gravity. Corollary 1 1. And, reciprocally if the surface of a liquid contained in a vessel is BODsibly horizontal all over, or perpendicular to the direction of gravity, tho whole system will be sensibly in equilibrium. XV. Prop. III. The surface of the overflown water contained in reservoirs whence the liquid issues through orifices pierced in thin ptates, fitted into the side or bottom, and wherein it is maintained during the flow at a uniform height above the centre of the orifices— remains always sensibly horizontal. See Phenomenon 9 § III of the foregoing chapter, 88 Corollary. I Therefore such a syatem of overflown' water maintains itaelf daring the flow sensibly in a state of equilibrium in the interior of the reservoirs (§XIV.) Corollary II, But as in the interior of the reservoir,a motion must exist, in order that the efHux may be compensated for, there is not the shadow of a doubt (§ lY) but that the condition of this water is a mixed state which partakes both of con- tinuous sensible rest and continuous motion. XVI. Prop. IV. This being so, to define the law and the natural symptoms proper to this state of overflow of the water in the interior of reservoirs. Considei'ing, in the first place, that in the permanent state we must suppose the efflux of the water through the orifice to be exactly equal to the supply at the upper part of the reservoir, it is unquestionable but that the outflow and the influx must take place simultaneously, otherwise, either, on the one hand, the outflow would not be uniform, or on the other hand, the upper limit ol the overflow would not be con- stant. It is therefore indispensable that in the overflown liquid mass the passage of a quantity of water equal, neither more nor less, to that which issues through the opening or to that which comes in at the limit of the overflow, must take place and bo verified at every instant; and as the whole body of the liquid is homogeneous, the water which comes in does, therefore, not pass by filtration through the overflown water, as it did through the oil (§ § VI.VII), but flows over immediately and spreads itself through the receiving water in the vicinity of the limit of the overflow, and it cannot reach the orifice to leave the vessel without the water which precedes it, and which is successively closer to this orifice having progressively made way for it. Hence the verification of this passage is effected by the successive translation and nearing to the orifice of the gradually anterior molecules. But on account of the perfect mobility of the water and the very delicate yieldingness of its parts, this efl'ective interior motion cannot take place without the whole mass up to the exterior surface or limit of the overflow being afifected by it. Hence there cannot exist in this mass absolute permanent rest cor permanent equilibrium between its parts — and consequently we cannot have an absolute permanent horizontality at the surface* Nevertheless, it is a fact (Phen. 3 ) that this surface maintains itself sensibly horizontal during the overflow, sensible equilibrium exists, therefore, between the parts of the water which is in the overflown state and consequentiy sensible rest in the whole system. But if there is, in this water, so constituted, neither an uninter- rupted continuity of equilibrium nor of rest, because, contrary to fact, the surface should remain continuously and absolutely horizontal, nor yet an uninterrupted con- tinuity of unstability, because, likewise, contrary to fact, the sensibly permanent horizontality of the surface could not subsist either, as in the imperfect fluids, it is necessary that in this singular condition of the water a perpetual succession of states of equilibrium and unstability should occur. Hence, motion and rest, viz., unstability in the parts and return to equilibrum, must, necessarily, be successive. But, again, the horizontality of the surface and the egress through the orifice appear to be sensibly continuous. We must, therefore, conclude that the successive passages from rest to motion and vice versa, are, as much as can be so, a sudden operation of nature, instantaneous, very rapidi Therefore, the law and the systems proper to the overflown state of the water in the interior of reservoirs consist in the existence, within the overflown body of water, of a periodi- cally variable condition, or of a particular kind of successive p'eriodioal passages from momentaneous rest to momentaneous motion, and from the latter again to rest- so that neither the rest of the system,iw«iii -wTixck results the sensibly continuous and permanent horizontality of the surface, nor the descensional motion which gives rii^e to the sensibly continuous and permanent reflux of the molecules towards the orifice — appear as if interrupted to tne eye sight. Whence, it is evident of what nature is this mixed state, as we have stated, (§ IV), which participates of rest and motion, and is as distinct from either the absolutie state of rest or the absolute state of motion as these two states are distinct from one another, and unique of its kind. Q. E. D. are. 89 daring iXlV.) a order of con- oper to pose Iho le upper X must )uld not be con- ssage of ugh the iace and sous, the verflown I spreads w, and it 38 it, and ^ for it. Lion and Qt of the irts, this J exterior b exist in irta— and Bur facet sensibly ween the e rest in I uninter- Q surface pted con- ermanent uids, it is i of states uilibrum, e and the therefore, as much •efore, the nterior of a period i- passages 1 to rest— sontinuous hich gives wards the ave stated, either the re distinct XVIT. • Scholium. — There is, therefore, no definite or undetermined size of reservoir, nor any kind of vessel to which the law which wo have jnst defined, is particularly limited. Whatever may be the form of the vessel wherein the liquid has an estab- lished, permanent surface, and whatever may be the opening through which it flows out unifoimly, the liquid is always in a true state of overflow, and when in this state, neither the size nor form of the vessel, nor of the opening, enter into considera- tion. This is the characteristic property by which it may bo recognized and dis- tinguished from other states. XVIII. Prop. V. — The actual velocity of any molecule whatsoever, which traverses the mass of overflown water, durinef efflux, is always infinitely small. For, as there is to be a successive passage from rest to doscensional motion, and from the latter to rest, and so on, always alternatively, during the flow, all the small spaces described successively by a molecule will always intervene between two sta- tionary periods, or periods of rest; and, consequently, there cannot be any descending molecule, in the act of falling which did not start from rest in the immediately pre- ceding instant. But there is no determinate force which can impart, in an instant, a definite velocity to any body starting from rest. Wherefore, the actual velocity of any molecule whatever, descending through the mass of overflown water, will bo, of necessity, infinitely small. Q. E. 1>. XIX. Corollary I. — If we suppose, therefore, a liquid which flows out with an infinitely small velocity, as soon as the efflux is permanently established, that sensible equili- brium exists between the parts of the system. Corollary II. — In this state, therefore, which is that of the overflow, it is also quite evident that the law of sections, reciprocally proportional to the velocities, can- not strictly hold good in the overflown mass, as it does when the liquids move freely. For to make sure of such a law obtaining within the mass, it would be necessary either to use vessels of a defioite form and size, which the nature of this state does not require, or to subordinate the momentary velocities of the molecules which tra- verse the mass to a law quite different from that which has really been shown to exist — which velocities are alternately extinguished at the renewal of equilibrium, and revived at the cessation of the same —and the alternative action being very per- sistent and imperceptible. Whence, it follows that the theories of the most illustrious hydrodynamicians on the motions of liquids issuing from orifices in reservoirs, are, perhaps wrongly founded on thi< law, which is necessarily excluded from the state of the overflow. XX. Scholium. — It is very difficult to reconcile a continuous acceleration of motion in the overflown water contained in reservoirs with the phenomena, and especially with those which show us openly that the velocity of the flow is due to the pressure of the water around the orifice, and never to the actual free fall from the upper limit of the overflow to the place of egress. The momentary stations, owing to which the sensible equilibrium of the parts is renewed at every instant, while they interrupt, at every instant, the downward course, preventing the velocity acquired by the mole- cules from being retained by them, and removing, at its origin, all acceleration — are, at the same time, those which give rise to an interior sensibly uniform but always elementary velocity being revived at every instant of rest, which constitutes an admirable economy of nature certainly well worthy of being developed and clearly pointed out, if I have succeeded in doing it properly. XXL Scholium. — Henco, so long as the water contained in the vesseld is in an overflown state, the system of the mixed state which we have defined, is preserved (§ XVI). and the velocity of the molecules can never be definite nor receive a determination. In order that this forever elementary velocity, and which, as we have said, always reappears after rest, may receive a determination, the water mast pass from the overflown to the free state, which is truly the state wherein the water is not prevented from flowing with the velocity and in the direction of the motion which animates it, whether on account of the natural motion or owing to the forces by which it is soli- oiled to move on. XXII. Scholium. — Because it has been demonstrated (§ XVIll) that the celerity d c of any molecule whatever, passing through the mass of overflown water, is always indefinitely small, and that, besides, dynamics have shown to us that the initial velocity of a free point excited by any power whatever g, is proportional to the product g d t, of the power g, by the indefinitely small space of time d t, during which it remains applied to the same point, if any molecule whatever of overflown liquid solicited by the pressure around the orifice (§ IX, Coroll. Ill) becomes a free point, and that we call g the force or pressure which excites it, the velocity of this molecule in the instant d t, will be expressed by the product g d t. Tht^refore, this velocity which was d c, indeterminately in the state of overflow, becomes g d t, in the free state, and is determined by the equation d c=g d t. Hence, at whatever point of the overflown system this passage of the molecules from the state of overflow to the free state may occur, we will always have the equation : — (A) dc=g d t=o, XXIII. Corollary I. — It is therefore demonstrated that equation (A) cannot hold good within the mass of liquids maintained at a uniform height in reservoirs in the actual and effective state of overflow such as they are in, and that it is applicable only to the free state ; that is to say, when in overflown liquid masses, the passage from the former to the latter state takes place. Corollary II. — And, therefore, remainins^ firm in the resolution to make no mental distinctions nor pliable hypotheses adapted to the laws of computation, but to con- clude only what the phenomena or the rigorous reasoning lead us to conclude, we see, from what all that has been presented heretofore, that the motions which are com- monly attributed to overflown liquids by hydrodynamicians are inexorably excluded from their midst. XXIV. Scholium. — No one perhaps, has come so near as Mr. D'Alembert to recognizing, in the liquids enclosed in vessels, the state of overflow which participates of the two states of motion and rest and which is yet essentially distinct from either. It is sufficient to examine the principles upon which he has bus^d his theory of the motions of fluids to be convinced of this. And truly our equation (A) (§ XXII) which draws legitimately its origin from having taken cognizance of this state, might be used as a fundamental principle for solving all the problems of this illustrious geometrician, if a simple hydrodynamical speculation was my aim. But then a state of motion only would be assumed all through and not the actual state of overflow, which is the object aimed at, wherein thi«> equation can in no way hold good. (§ XXIII). We see by this, in what condition of things his theory agrees with the facte, viz., by supposing that the fluids ..re not in a state of overflow, but that they flow without the alternatives of descent or movement and equilibrium, which alternating actions destroy all acceleration and all continuity in the motions. di verflown § XVI). lination. I, always irom the irevented mates it, it is soli- •ity d c of B always x\ velocity luct g d t, t remains >licited by d that we lie in the 9 velocity n the free tint of the to the free hold good the actual ble only to »e from the no mental )ut to con- id e, we see, ch are com- y excladed recognizing, i of the two ither. It is eory of the ) (■§ XXII) state, might B illustrious len a state of rflow, which (§ xxiii). th the facte, i they flow alternating Scholium. — But for fear that by proceeding further with this enquiry which could easily be done, I might confound the objects, and render obscure the very clear ideas which we have just formed respecting the interior condition of liquids in the state of overflow, I will now explore, guided by the steps which have already been taken, the exterior movement of thetie liquids after they have passed from the over- flown to the fr«e state; and this will form the argument of the next chapter. In the two remaining chapters (3rd and 4th) of his " Phisico-Mathematical Theory," Lorgna treats of the motions of liquids after they have emerged, as he says, from the state of overflow existing within reservoirs, through orifices pierced in their sides f^r bottoms, and of the contraction of the stream in horizontal, vertically des- cending and vertically ascending jets. After explaining in what manner the liquid molecules issuing from orifices in reservoirs, wherein the liquids are maintained at a constant height above the centres of those orifices, are solicited by natural gravity and by thecoaotion of the pressures around the orifices combined, the author manages, by an ingenious train of reason- ing, to fix the height due to the actual velocity in an orifice pierced in a thin plate at : 2H X 2 (V^^~?y= 0472127 H and arrives at : 2Aa\^ a( -472 a) — y Y x + -472 a\ = o or for the equation of the hyperbolic conoid of the contracted fluid vein —-where A repre- sents the permanent height of the fluid above the orifice, a the radius of this orifice, y the radius of a cross-section of the vein taken at any distance x, from the plane of the opening. Putting a = a; = 1, in the last equation, it becomes : •472 A — i/Y 1 + "472 aW 0, whence : ('472 A )'* f radius DE (Pig. 8) of circular cross-section of vein at ^ (14- •472A)'< — 1 distance of, say, § diameter of the orifice, from its plane, which is the formula of the hyperbolic conoid of Newton . The curve traced out by the extremities of the ordinates (y), calculated by means of this formula is, however, utterly at variance with the profile presented by the naturally contracted liquid vein, the contraction of which is much greater than that of the corresponding computed vein-form, an clearly shown by Venturi in the following table extracted from his " Experimental Enquiries." at a) Authors of Experiments. Poleni (de Oastellis, § 3S) Hichelotti; Sperim. Idraul., Tom. I., Bzper 46; Tom. II, Exper. 4 Bossut (Hydrodjn, Art. 437, Exper. S)..... Venturi, with 35 inches charge and a horizontal circular orifice, 18 old French lineasl'SQSS English inches in diameter Value of DE (Fig. 8) found by actual measurement 79 085 0-818 0-798 Value of D E (Fig. 8) calculated by the preceding formula. 0-97 0-99 0-99 0-984 92 " It is evident," eays Ventari, " that the contraction of the vein, as found by experiment^ is incomparably greater than can be produced by the acceleration of gravity, even in descending streams. But what can we say of horizontal and ascending jets, in which assuredly the action of gravity does not take place, bat in which, nevertheless, the contraction U obier red nearly in the same manner as in descending currents ? The contraction ot' the stream is therefore very different from the Newtonian hyperboloid." Yenturi fbrther adds : " Desirious of proving that the vein does not possess the whole velocity arising from the height of the fluid above the centre of the orifice, Lorgna relates the experiments of Kraft,''' which are not applicable to the question, because they were made with cylindrical pipes, and we have seen that such pipes always destroy part of the velocity of the fluid ; consequently we cannot establish any rule from them which shall apply to orifices through thin plates.f He wishes not to determine the velocity of ascending jots by the height to which they rise, because he is apprehensive that the preceding part of the stream or jet is urged, and supported oy the succeeding part nearly to the height of the charge. Never- theless, if we intercept the jet all at once, the last portions of water fly to the same height as those which preceded them, without having any continued column of the fluid below to follow and support them ; these last portions must, consequently, have received, at their passage through the orifice, all the velocity which was necessary to raise them nearly to the surface of the fluid in the reservoir." •Acta Petroo. vol. VIII. fTorcelli took notice of this difference at pafi^e 168 of liia works, " quoties eumque autem aqut per tubem lalentem deeurreniper anguttint irantire (lebuerU,/alta omnia reperiei." f- ■ I ■.... -a' I found by ileratioD of zontal and lace, but in nner as in y different possess tho the orifice, le question, such pipes »t establish He wishes they rise, t is urged, je. Never- to the same lumn of the nsequently, which was ir." item aqut per A mmmmmmmmm