HYDRODYNAMICS. 



489 



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Cor. 5. If the fluid, when it issues from the orifice 

 should continue to move uniformly with the velocity with 

 which it issues, it would describe a space equal to 2 m o, 

 in the same time that a heavy body would fall through 

 the height o . 



In studying the preceding proposition, the reader 

 must consider it as giving the velocity, not at the ori- 

 fice iuelf, but at the vena coxlracta, where the velocity 

 is greatest. Sir Isaac Newton having found that the 

 velocity at the vena contracta was that which was 

 due to the whole height of the fluid, and that the ve- 

 locity at the vena coxlracla was to the velocity of' the 

 orifice a* </2 : I, or as 1.414: 1, necessarily concluded 

 that the velocity at the orifice is only that which is due 

 to half the altitude of the fluid. 



PROP. III. 



If a cylindrical or prismatic vessel, of which the ho* 

 rizontal section is every where the same, is filled with 

 fluid, and empties itself by an orifice, the velocity with 

 which the surface descends, and al<o the velocity with 

 which the water issues, is uniformly retarded. 



Since the velocity of the surface of the fluid is to the 

 velocity at the orifice, M the area of the orifice is to the 

 area of the horizontal section of the veMel, or the ares) 

 of the surface, the velocity of the surface mutt vary a* 

 the velocity at the orifice. But the velocity at the ori- 

 fice varies M the square root of the height of the fluid 

 in the vessel by Prop. II. Cor. 8. ; consequently the veto, 

 city of the surface must also vary according to the height 

 of the fluid, that is, with the space through which it de- 

 scends. But as the velocity of heavy bodies projected 

 upward* varies in this manner, the velocity of the fluid 

 surface most be uniformly retarded in the same manner 

 as heavy bodies. 



POP. iv. 



If a fluid issues from a cylindrical or prismatic vessel, 

 whote horizontal section is every where the same, and 



in which the fluid i - always kept at the same height, 

 the orifice will discharge t*ice the quantity contained 

 in the vessel, in the same time that the vessel would 

 have emptied itself. 



As the surface of the fluid is uniformly retarded, and 

 as its velocity becomes nothing at the bottom, the space 

 which the descending surface would describe, with 

 the first velocity, continued uniform during the time 

 that the vessel take* to empty itself, is twice the spmce 

 that the surface really describes in the time in which 

 the vessel empties itself. In this time, therefore, the 

 '|uar>' 1 discharged in the former case is twice 



that which ii discharged in the Utter case, as the quan- 

 tity discharged when the vessel is kept constantly full, 

 may be measured by what would be the descent of the 

 surface, if it coold descend with the velocity with which 

 its descent commences. 



The preceding demonstration is given by Vf r Vince. 

 M. Bossut deduces the proposition M corollary from 

 formula- which express the quantity of water dischar- 

 ged under the circumstances stated in the propoti* 

 uco. 



TOU II. PAT II. 



PROP. V. 



Di.<eli*rge 

 of Fluids 



from 

 Orifices. 



To determine the quantity of water discharged by a _ 

 small vertical or horizontal orifice, the time of discharge, 

 and the height of the fluid in the vessel, when any 

 two of these quantities are known. 



Let A represent the area of the small orifice m n; W PI.ATK 

 the quantity of water discharged ; T the time of dis- cccxvm. 

 charge, H the height m o of fluid in the vessel, and g = Kig> 1. 

 16.087 feet, the space described by gravity in a second. 

 Then since, by dynamics, the times are as the square roots 



of the spaces, we have y^.. v'FT = 1 second : ^5, the 



time in which a heavy body would fall through the 

 height H. But since the velocity is uniform, the space 

 described will be double in the time that a heavy 

 body would describe the height H, and therefore a 

 column of fluid = A x 2 H will be discharged in the 



tirae^/_- Now, as the quantities of fluid discharged 





in different times are proportional to the times, we 



bnreS AH : W = J ^L : ,. Hence, 



2AH SAH/x/T H 



" = ~7ff~ x 7ff~ fi ' andsmce ^ 



wehaveW=2A< v 'gH 



r =v / H, 



A= 



H= 



Cor. By means of these formula-, we may determine 

 the quantity of water \V" which is discharged in the 

 same time T, from any other venel in which A' is the 

 area of the orifice, and H the altitude of the fluid for 

 HOC* / and g are constant, we shall hare 



W'=AVTT:AVH'. 

 POP. VI. 



To determine the time in which the surface of wa- 

 tt r in a vessel will descend through a given height, 

 where the fluid is discharged through a small orifice in 

 the bottom. 



Let A BCD be the vessel, and let it be required todc- p tAT1 

 termine the time in which the surface of the fluid de- CCCXVI1 

 scends from AB to RS. Draw MN, ^ , parallel and Pi* . 

 infinitely near to each other, then since P T is infinitely 

 small, we may consider the height as constant du- 

 ring the time that the lamina of fluid MN p . flows 

 through the orifice; and consequently its velocity N uni- 

 form. The time I, therefore, in *hich the height I'w'u 



\t \ w B 

 described, will, by Prop. IV. let = - "? _ for 



in the present case W = M X x P , and H = P m. In a 

 nimlar manner we may obtain the times /', /" for all the 

 other elementary lamina: into which the turn ABNM 

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