Let AENL (fig. 174.) be the vessel, and on the side AE describe the semicircle AKE; let also the orifice be at any point B, and draw BK perpendicular to AE, meeting the semicircle in K; then, the distance to which the water will spout on a horizontal plane passing through E, is equal to twice BK; that is, EF-2BK.

For the horizontal velocity of a fluid at B, is that which is due to the altitude of the fluid above B; and since the curve BF described by the fluid, is, by art. 385, a parabola whose parameter is 4AB, abscissa BE, and ordinate EF; and by the property of the curve, (prop. 3. Hutton's Conics, book 3.) 4AB × BE=EF2; we have EF=2√(AB × BE): but by the nature of the circle, AB X BE BK'; therefore, EF=2/BK' -2BK.

565. Cor. 1. If ED AB, then DHBK, and the fluid will spout to the same distance through an orifice at D as at B.

566. Cor. 2. If AE be bisected in C, and CI be drawn perpendicular to AE; then, because CI is the greatest ordinate to the semicircle AKE, the distance EG=2CI is the greatest: that is, if the orifice be made at C, the middle of AE, the water will spout to the greatest distance possible.

567. Cor. 3. Because the distance to which the water spouts depends entirely upon the altitude of the water above the orifice; if the vessel stand upon an eminence, the distance to which the water will spout from B, upon the horizontal plane efg, passing through the bottom of the eminence, is as ABX Be; and the distance to which the fluid is projected on the plane efg is a maximum when the orifice bisects the altitude Ae.

568. Cor 4. If the side of the vessel AE be inclined to the horizon in any angle, and the velocity and direction of the spouting fluid be known, the distance, &c, may then be found by the rules already given in the theory of Projectiles.

MISCELLANEOUS EXAMPLES.

Ex. 1. If a vessel 10 feet high be kept constantly full of water, and a small hole be made at the distance of 4 feet from the top; to what distance will the water spout on the hori 'zontal plane on which the vessel stands ?

Answ. 2(24)=9.79796 feet.

Ex. 2. If the depth of a sluice be 10 feet, and there be a 'hole in the bottom of 2 feet in area; what quantity of water will be discharged through this hole in one minute, and with what velocity does the water run out; the sluice being kept constantly full?

By art. 440, the quantity discharged is 8.0208 ath; where a=2 feet, t=60′′ and h=feet; therefore, in this case, the quantity discharged in one minute is 8.0208 × 2 × 60 × 10 3044.4123 cubic feet.

And the uniform velocity of the effluent fluid is v= √(4×16×10)=25.364 feet per second.

Ex. 3. If there be rectangular slit, 2 inches wide and 6 inches deep, at the top of a sluice, kept constantly full; what quantity of water will flow through this orifice in one minute?

Here ha foot, a=2×6= of a foot, and t=60"; therefore, by art. 448, 8.0208 × 1 × 60 × √(1)=18.905 cubic feet is the answer.

Note. In these two last examples, the velocity of the effluent water is supposed to be equal to that which is acquired by a heavy body in falling through the whole altitude of the water in the vessel above the orifice. But (art. 447.) the result thus obtained must be diminished in the ratio of 2: 1; or, according to experiments made by BossUT in his Hydrodynamique, those results should be multiplied by,62, and the product will give the true quantity discharged nearly. For more on this subject see book the fifth.

Other works on this subject, which may be consulted with advantage, are, Newton's Principia; the Hydrodynamique of D. Bernouilli: the Traite des Fluides by D'Alembert; the Motion of water and other Fluids by Marriotte; Vince's Hydrostatics; Gregory's Mechanics, and the various Encyclopedias: particularly the elegant article on the Resistance of Fluids in the Encyclopedia Britannica.

PNEUMATICS.

BOOK IV.

SECTION I.

On the properties of common Air.

569. Defs. THE word pneumatics, according to its original acceptation, signified air; but it now denotes the doctrine of AEROSTATICS, the object of which is, to determine the motion, equilibrium, and density of elastic fluids in general; it, however, treats more particularly of the properties of common, or atmospheric air.

The atmosphere is that body of air, which surrounds the earth to a considerable altitude; viz. about 45 miles, as determined by Astronomers from the duration of the twilight.