HYDRODYNAMICS. 



491 



Pi IE 



rccxvrn. 



PO 



Since the ^locity with which the surface AD de- 

 scends is a* the square roots of the altitudes, then, as 

 the velocities are proportional to the times, the times 

 in which these altitudes are described will also be as 

 the square roots of the altitudes. Hence, since 13, 11, 

 10, fee. are the times in which the different heights are 

 to be described, the height* should be as 12', 11', 10, 

 or as 144, 121, 100,81. 



The exact time of describing each part 6f the alti- 

 tude A is easily deduced from the formula. If the 

 time is to be one hour, then we must proportion the 

 area B of the base, and the height A of the vessel, 

 to the area A f the orifice ; so that 1 hour = 



B<3^ /_!>.. hour = "^-; from which 



"VK ISAv/g 



nriy of the three quantities B, h, and A may be found, 

 when two of them are given. 



SCHOLIUM. 



In practice, the 1 1 first divisions should only be em- 

 ployed, on account of the effect of the funnel-shaped 

 cavity upon the regularity of the discharge. See p. 

 437, 506. 



Paop. IX. 



If a prismatic or cylindrical vessel ABCD is kept 

 *h 



:ly full, it will discharge twice as much water 

 as it contain* in the time that it takes to empty itaelf 

 completely. 



It follow* from Prop. VII. that the time in which 



p J L 



it empties itself is , h being equal to the height 



A Y g 



m o ; and from Prop. V. that the quantity ef fluid dis- 

 charged in the same time, when the vessel is kept con- 



stantly full, u ***'* x 2 Ay's* = 2B x A; but 



A V g 



this quantity is doable of the prism ABCD, which is 

 equal to B x A- 



SCHOLIUM. 



In practice, the effect of the f 

 must be considered. See 



of the funnel-shaped cavity 

 Chap. III. Sect. V. p. 506. 



Paop. X. 



If water is discharged from two prismatic or cylin- 

 drical vessels, the times in which their surfaces descend 

 through similar heights, will be in the compound ratio 

 of the areas of the bases, and the difference between 

 the square roots of the height of each surface at the be- 

 ginning and end of its motion directly, and inversely 

 as the area* of the orifice*. 



Let 

 sels 



* the corresponding quantities in the two ves- 

 be distinguished by accents, then, by Prop. VI. 



ordividingby ^. 



Pnop XT 

 Rop - AI - 



Discharge 

 of 



To determine the quantity of water which is dis- 

 charged in a given time from a large rectangular ori- 

 fice in the side of a vessel. 



Hitherto we have supposed, that the orifice from 

 which the water was discharged, when it issued from 

 the side of a vessel, was so small, compared with the 

 diameter of the vessel, that every part of the orifice 

 might be considered as at the same depth below the 

 surface. As this supposition, however, is inadmissible 

 in the case of large orifices, we must now suppose a 

 large orifice divided into an infinite number of small 

 rectangles, (if the orifice is rectangular,) and regarding 

 each of these as an orifice, all the points of which are 

 equidistant from the fluid surface, we must determine 

 the quantity of water discharged by means of the pre- 

 ceding propositions. The sum of ail these elementary 

 quantities will then be the total quantity of fluid dis- 

 charged during the given time. 



In order to shew the mode of doing this, by ageome- PLATE 

 trical construction, we shall take the case of a rectangu- CCCXVHI. 

 lar orifice as given by liossut. Let LNOM be the given v f- * 

 orifice in the vessel ABCD kept constantly full of water. 

 Draw X/, i z, infinitely near each other, and parallel to 

 LM, so as to form the elementary rectangle \Z **. 

 Then if RI is the height, which may be considered as the 

 distance of all (he point* of the small orifice from the sur- 

 face AD, the quantity of water discharged in the time 

 t will be SXZ x It X l</g X </M- In order to find the 

 mm of all these elementary quantities, construct upon 

 the axis RV the parabola RT, whose parameter is p, and 

 produce KM, I / . . : . VO to Y, S, i, and T. The small 

 parabolic area IS * i may be considered a* a rectangle 

 = ISxI- But IS=v'KlXv'p: hence IS x Ii=l/X 

 v/RI x </p- Calling e this parabolic area, and o the 

 elementary quantity of water which flows through the 

 elementary orifice \Z si, we shall have e : 9=! f X 

 1 ' x </p : 2 \Z x 'vV X 1 1 X \/HI, which gives us 



ixv 



SCHOUVU. 



In practice, the effects of the contraction of the fluid 

 vein miut be considered, as in Chap. HI. Sect V. 

 p. 308. 



. 



/P 



If we can determine, therefore, the sum* of all the e't 

 or the parabolic surface KVTY, we shall easily deter- 

 mine the sum of the tf or the total quantity Q, which 

 i* discharged in the time / from the aperture LNOM. 



Complete the rectangle RVTH, and draw SG, tr pa- 

 rallel to VR. Now the area RHT is composed of the 

 element* SG, *g, which are proportional to the square* 

 of the distances RG, Rr: (see CONIC SECTIONS, Sect. 

 IV. Prop. XII. Cor.) Hence these element* increase 

 as the sections of a pyramid, whose summit is R, and 

 whose height i* RH. Consequently the form of all the 

 GS's which make up the area RHT, are equal to 

 i RH x HT ; therefore the area RVT = ^ VR x VT. 

 The space KVTY, or the sum of the e't, i* therefore 

 = j(RV X VT RK X KY);but 



Hence we have 





Q= ?(RV X VT - RK X K Y) x 



If we substitute in thi* expression v/VR X \'p instead 

 ofVT; v/RKxv/P instead of KY ; and if we call 

 VRsH, and RK=A, and XZ=6, we have 



In thi* expression, g it always =16.087. 



