Problem 8.

To find the Solidity of an Hyperbolic Conoid.

Def. An Hyperbolic Conoid is a Solid made by the Revolution of a Semi-hyperbola about its Axis. *

Rule.

To the Square of the Radius of the Bafe, add the Square of the Diameter in the Middle between the Top and Bottom; this Sum multiplied by the Height, and the Product multiplied again by .5236, will give the Solid Content.

Example.

Suppofe ABCDEF be a Hyperbolic Conoid; the Semi-diameter AF of whose Base A E is 52 Inches; the Diameter in the Middle BD 68 Inches, and the Height CF 50 Inches, what is its Solid Content?

Operation. A F 52

2704 +□ BD 45247328

× CF 50366400 × 5236 = 191847.04 Inches, the

Solid Content required.

An Hyperbolic Conoid is a Solid whofe Sides are straiter than a Parabelic Conoid, yet more curved than a Cone.

Problem 9.

To find the Solidity of the Fruftum of an Hyperbolic Conoid.

Rule.

To the Sum of the Squares of the Semi-diameters of the Bottom and Top of the Fruftum, add the Square of the whole Diameter in the Middle; this Sum being multiplied by the Height, and that Product again by .5236, will give the Solid Content.

Crample.

Suppofe ABCD be the Fruftum of an Hyperbolic Conoid; the Semi-diameter A T of the Bottom measures 16 Inches; the Semi-diameter BH of the Top 12 Inches; the Middle Diameter m d 28.17 Inches, and the Height HT 20 Inches, what is its Solid Content?

Operation. AT 256 +0 BH 144 + omd 793.5489 = 1193.5489 × HT 20 = 23870.9780 X.5236 12498.84408080 Inches, the Solidity required.

*

Note. As many Houfhold Utenfils are in the Shape of fome of the foregoing Figures, as, for Example, Tuns and Tubs in Form of Fruftums of Cones or Conoids; Furnaces and Coppers in Form of Parabolic or Hyperbolic Conoids; Cafks in Form of the Middle Zones of Spheroids, Parabolic Spindles, Double Fruftums of Parabolic Conoids, and Double Fruftums of a Cone; the Quantity of Liquor contained in each may be easily afcertained by dividing (as before in Planometry and Stereometry) the Solid Content in Inches by

282 for Ale Gallons.

231 for Wine Gallons.

2150.42 for Corn Bufhels.

With refpect to Cafks, it may be difficult, on Account of the different Bending of the Staves, to afcertain_exactly the Form to which they belong; for though the Dimenfions of feveral Cafks may be exactly the fame, yet their Contents will be very different, as is clear from a Sight of the following Figure.

* The rifing Crowns of Stills are Segments of Spheres; the remaining Part generally the Fruftum of a Parabolic Conoid; Bowls and Basons are generally the Segments of Spheres, and measured accordingly.

Suppofe ABCDEF to reprefent a Cafk; then, it is evident, that if the outer curved Lines A B C and D E F are the Boundaries or Staves of the Cask, it must of Course hold more than if the inner and ftraiter Lines were the Bounds and Staves of it, yet the Dimenfions of the Bung Diameter B E, and Head Diameters AF and CD, and the Length LH, are the fame in all the Casks.

E

H

If the Staves of the Cafk are very much curved or arching, as the outer Line in the foregoing Figure, it is fuppofed to be in the Form of the Middle Zone of a Spheroid, and its Content may be found by Problem 3d.

If the Staves are not quite fo much curved or arching, as reprefented in the fecond Line in the Figure, it is taken for the Middle Zone of a Parabolic Spindle, and is meafured by Problem 7th.

When the Staves are but little curved or arching, as the third Line in the Figure, it is fuppofed to be in the Form of the lower Fruftums of two equal Parabolic Conoids joined together upon one common Bafe in the Middle, and its Content may be found by Problem 9th.

If the Staves are quite ftrait from Bung to Head, as the inner Lines in the Figure, it is then confidered as the lower Fruftums of two equal Cones joined together upon one common Bafe, and its Content may be found by Problem 8th. in Stereometry.

Note. Cafks made in the firft Form hold the moft; and thofe of the laft Form hold the least of any other Kinds.

But fince we can only at laft guefs, as it were, at the Variety or Form which the Cafk belongs to, the easiest and beft Way of finding its Content is to be preferred in Practice, which is to find fuch a mean Diameter between the Bung and Head Diameters as will reduce the Cask to a Cylinder equal to it, which may be done by the following

Rule.

Multiply the Difference between the Bung and Head Diameters by .7, or by .65, or by .6, or by 55, according as the Staves are more or lefs arching; add the Product to the Head Diameter, and that Sum will be a mean Diameter, i. e. it will be the Diameter of a Cylinder, whose Length and Content are equal, as near as can be, to that of the Cask.

Example.

Suppofe a Cafk whofe Bung Diameter is 31.5 Inches; the Head Diameter 24.5 Inches, and its Length 42 Inches, what is its Content in Ale Gallons?

Operation.