HYDRODYNAMICS. 



497 



Ifedavge 



CHAP. III. 



*?**'" AccorNT or EXPERIMENTS ov THE DISCHARGE or 

 J*"*^- WATER FROM VESSELS THROUGH SIMPLE OiuricEs 



AND ADDITIONAL Tf BBS. 



SECT. I. On the Vena Contracta. 



, UM I \ the first Chapter of Hydraulics we have explained 



M coo- the cause of the contraction of the fluid vein, or vena 

 *** coxtracta, and have stated in general the effect which 

 contraction produce* upon the quantity of water 

 discharged from vessels. ' The following are the mea- 

 sures of the contraction according to different authors ; 

 A being the area of the orifice, and a the area of the 

 section of the contracted vein : 



Sir Isaac Newton, . . . 



I'oleni, ..... 



Highest found by Bossut, . 



Mean of six experiments by Bount, 



Lowest found by Bossut, 



Bernoulli 



lotti, 



Dii B'j;it, ..... 

 ttiri, ..... 



Evtelwein, .... 



The measures given by Boasut were taken by a pair of 

 spherical compasses, with which he measured directly 

 the diameter of the contracted vein, which he found to 

 preserve the same diameter fur tome lines. The alti- 

 tude of the water in the reservoir which Bossut used 

 was 1 1 feet S inches. He measured the vena contract* 

 also, when the water issued by vertical orifices placed 

 4 feet below the surface of the fluid, and he obtained 

 the very same result*. 



The ratio of A to a is however by no means constant. 

 It undergoes perceptible variations, by varying the 

 form and position of the orifice; the thickness of the 

 plate in which the orifice is made ; the form of the ves- 

 sel ; and the velocity of the issuing fluid. 



SECT. If. On the quantity of Water discharged by Ori- 

 Jlcet of different Formt, from Vtuelt kepi constantly 



Discharge 

 of Water 



the included air to escape as the water was poured in. 



which was the mark to shew when it contained exact- 

 ly the number of cubic feet. The second tube allowed 



TABLE I. Shetein* the quantity of Water discharged in 

 one Minute by Orijtces differing inform and ]>ositiun. 



Iw order to determine the quantity of water dis- 

 charged by orifice* of different forms,' M. Boswt em- 

 ployed very clear water, which was excellent for drink' 

 ing. The experiment* were made at Meiiere*, in the 

 beginning of September, in very fine weather, which 

 only information from which we can conjecture 

 the temperature of the water. The orifice* were per- 

 forated in plate* of copper half a line thick ; the time 

 was ascertained sometime* by a seconds watch, and at 

 ether time* by simple second* pendulum ; and the 

 most exact measure* were ued in ascertaining the 

 quantity of water discharged. The fundamental mea- 

 sure which he employed wa* a hollow cube of copper, 

 having one of its sides exactly ix niche* within. It 

 contained the eighth part of a cubic foot when it wa* 

 entirely full, and on its four interior face* were traced 

 four vrrtical scale* for atcrmining the quantity of wa- 

 ter which it contained when it w.i full. M. Boasut 

 also used other two measure*, one of which wa* a small 

 barrel containing exactly a cubic toot, or eight times 

 tit* contents of thr ctilncal vessel ; and the second wa* 

 barret containing < '-feet. Each of these bar- 



rels had two mull tubes rising from their upper end, 

 through one of which the water was poured, and upon 



VOL. II. PART II. 



From these results we may conclude, General 



1 . That the quantities of water discharged in equal result*. 

 time* by the same orifice from the same head of water, 

 are very nearly as the areas of the orifices ; and, 



_' That the quantities of water discharged in equal 

 time* by the same orifice* under different heads of wa- 

 ter, are nearly a* the square roots of the corresponding 

 height* of the water in the reservoir above the centre* 

 of the orifice*. 



If we call Q, q the Quantities of water discharged 

 in the same time from the two orifices A, A' under the 

 same height of water in the reservoir ; 9 and Q' the 

 quantities of water discharged during the same time by 

 the same aperture A, under the different heads of wa- 

 ter h, h', we have by the first of the above results, 

 Q : q = A : A' ; and by the second, q : Q' = t/h : i/h' ; 



A' X Q Q' -/h 



from which we obtain a = -- j , and q= T,, ; 



A 'v 



then since ^ - ^-, we hare by Euclid, B. V. 

 A 



Q : (^ = A v'* : A' y/A', that is, in general. 



3. The quantities of water discharged during the 

 same time by different aperture* under different heights 

 of water in the rtacrvoir, are to one another in the 

 compound ratio of the areas of the apertures, and of the 

 square root* of the heights in the reservoirs. 



This general rule may be considered as sufficiently 

 correct for ordinary putpoae* ; but, in order to obtain a 

 great degree of accuracy, Bossut recommends an at* 

 tention to the three following rules. 



1. I" ridion is the cause, that, of several similar ori. 

 fices, the smallest discharge* lets water in proportion 

 than thoie which are greater, under the same altitude* 

 of water in the neserx-oir. 



S. Of several orifices of equal surface, that which ha* 

 the smallest perimeter ought, on account of the friction, 

 to give more water than the rest, under the name alti- 

 tude of water in the reser 



3. That, in consequence of a slight augmentation 

 which the contraction of the fluid -.tin undergoes, in 

 proportion a* the height of fluid in the reservoir in. 

 creases, the expence might to be a little diminished. 

 3 R 



