1130

THEORY AND PRACTICE OF ENGINEERING.

Book II.

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superficies of the circle, anti 4d the circuit of its square; whence we have — : ddC, 4d>

which shows that the ratio of the waste to the natural expenditure of a square orifice threelines square may further be expressed by -fg. Tims when we wish to know the proportionfor 3ny square orifice, we must multiply the denominator of ^ by the side of the square

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reduced to lines : for example, for a square of an inch, we shall have ———, or —.

On tke Measurement of the Water flowing through rectilinear and vertical Orifces. —

Fig. 1714.

It has been demonstrated that a prismatic vessel continually filled with water has each ofits faces pressed in a horizontal direction by the films of water which it sustains; conse-quently, if this surface be pierced with several holes, II, K, Sec in the vertical E F, the waterin issuing will be driven in horizontal directions, with velocities which may be expressedby the roots of the heights E II and E K, or by the corresponding ordinates III and K Lof a parabola E I G, since the property of this curve gives a''EH, a^E K : III, KL,Supposing the perimeter of this parabola 60 feet, the ordinates III and K E will express,not only the ratio of the velocities of the water, but also the real velocities per second ofthe threads which issue through the small holes H and K : then knowing the heightsE H and E K in feet, inches, and lines, we shall have, by the assistance of the first table,the values in feet, inches, and lines of the ordinates III and K L.

It follows, that if all the threads of water which issue through one of the orifices H or Khave the same velocity, the natural expenditure per second will be equal to a column whichwould have for its base the plan of the orifice, and for height the ordinate which answersto it.

If we suppose the axis E F of the parabola ELO divided into an infinity of equal parts,they will compose an infinite arithmetical progression, or the smallest term of which willbe zero, and the greatest the height E F of the water, which will express at the same timethe number of terms of this progression. Drawing through each point of the division anordinate H I or K L, beginning from the summit E, all these ordinates being in the ratioof the roots of their abscissa?, or those of the terms of the progression of the parts of theaxis, the sum of all these roots will be found in the same manner as those of the ordinateswhich compose the superficies of the parabola, by multiplying the axis E F by two-thirds thegreater ordinate F G. We shall have also the sum of all the roots of the correspondingabsciss®, or that of all the terms of the progression, by multiplying the axis E F by § VE F,2 E F

or a/E F by-.

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To demonstrate the same rule, independently of the parabola,let us consider the right-angled isosceles triangle E F G, whoseheight E F being taken for that of the water, all the elements M Nwill compose the terms of the preceding progression, or all thedifferent heights of the water, taken from its level to the bottomof the vessel. To have the sum of the roots of all these elements,call E F, h; E M, x; thus Mm will be dx , which being multi-plied by Vx will give dx^x = x\dx for the sum of the rootscomprised in the differential plane Mm N n, the integral of whichgives or \x x Vx, or \h x <\/^,when x becomes equal to h.

If a fluid, according to Mr. Vince, issues from a cylindrical orprismatic vessel whose horizontal section is everywhere the same,and in which the fluid is always maintained at the same height, the orifice will dischargetwice the quantity contained in the vessel, in the same time that the vessel would haveemptied itself. As the surface of the fluid is uniformly retarded, and as its velocity becomesnothing at the bottom, the space which the descending surface would describe with the firstvelocity, continued uniform during the time that the vessel takes to empty itselfi is twicethe space that the surface really describes in the time in which the vessel empties itself: inthis time, therefore, the quantity of fluid discharged in the former case is twice that which

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Fig. 1715.