To find the Solidity of a Wedge-fig. 35.

RULE. To the length of the edge of the wedge de add twice the length of the back ab; multiply this sum by the height of the wedge df, and then by the breadth of the back ca, and of the product will be the solid content.

To find the Solidity of a Prismoid-fig. 36.

RULE. Add the areas of the two ends a bc, def, and four times the middle section g h, parallel to them, together; multiply this sum by of the height, and it will give the solidity.

To find the Solidity of a Sphere-fig. 37. RULE.-Multiply the cube of the diameter by .5236, and the product is the solidity.

To find the Solidity of a Spherical Segment-fig. 38.

RULE. To three times the square of the radius of its base a b, add the square of its height cb; then multiply this sum by the height, and the product by .5236.

To find the Solidity of a Spherical Zone or Frustrum-fig. 39.

RULE.--To the sum of the squares of the radius of each end ab, cd, add of the square of the height bd of the zone; and this sum, multiplied by the height, and the product by 1.5708, will give the solidity.

To find the Solidity of a Spheroid—fig. 40. RULE.-Multiply the square of the revolving axis cd by the fixed axis ab; the product, multiplied by .5236, will give the solidity.

To find the Solidity of the Segment of a Spheroid-figs. 41 and 42. RULE-When the base e f is circular, or parallel to the revolving axis cd, fig. 41. Multiply the fixed axis ab by 3, the height of the segment ag by 2, and subtract the one product from the other; then multiply the remainder by the square of the height of the segment, and the product by .5236. Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment.

RULE. When the base ef is perpendicular to the revolving axis cd, fig. 42. Multiply the revolving axis by 3, and the height of the segment cg by 2, and subtract the one from the other; then multiply the remainder by the square of the height of the segment, and the product by .5236. Then, as the revolving axis is to the fixed axis, so is the last product to the content.

To find the Solidity of the Middle Frustrum of a Spheroid-figs. 43

and 44.

RULE. When the ends ef and g h are circular, or parallel to the revolving axis c d, fig. 43. To twice the square of the revolving axis c d, add the square of the diameter of either end, ef or gh; then multiply this sum by the length ab of the frustrum, and the product again by .2618, and this will give the solidity.

RULE.-When the ends ef and g h are elliptical, or perpendicular to the revolving axis c d, fig. 44. To twice the product of the transverse and conjugate diameters of the middle section a b, add the product of the transverse and conjugate of either end; multiply this sum by the length lk of the frustrum, and the product by .2618, and this will give the solidity.

* Spheroids are either Prolate or Oblate. They are prolate when produced by the revolution of a semi-ellipse about its transverse diameter, and oblate when produced van ellipse revolving about its conjugate diameter.

To find the Solidity of a Circular Spindle-fig. 45 RULE.-Multiply the central distance oe by half the area of the revolving segment a cef. Subtract the product from of the cube fe of half the length; then multiply the remainder by 12.5664 (or four times 3.1416), and the product is the solidity.

To find the Solidity of the Frustrum, or Zone of a Circular Spindle

fig. 46.

RULE. From the square of half the length hi of the whole spindle, take of the square of half the length ni of the frustrum, and multiply the remainder by the said half-length of the frustrum; multiply the central distance oi by the revolving area* which generates the frustrum; subtract the last product from the former, and the remainder, multiplied by 6.2832 (or twice 3.1416), will give the solidity.

To find the Solidity of an Elliptic Spindle-fig. 47.

RULE.-To the square of the greatest diameter a b, add the square of twice the diameter e f at of its length; multiply the sum by the length, and the product by .1309, and it will give the solidity nearly. To find the Solidity of a Frustrum or Segment of an Elliptic Spindle— fig. 48.

RULE.-Proceed as in the last rule for this or any other solid formed by the revolution of a conic section about an axis, viz.: Add together the squares of the greatest and least diameters, a b, c d, and the square of double the diameter in the middle, between the two; multiply the sum by the length ef, and the product by .1309, and it will give the solidity.

NOTE. For all such solids, this rule is exact when the body is formed by the conic section, or a part of it, revolving about the axis of the section, and will always be very near when the figure revolves about another line.

* The area of the frustrum can be obtained by dividing its central plane into segments of a circle, and triangles or parallelograms. H

OF PARABOLIC CONOIDS AND SPINDLES.

49.

50.

51.

a

To find the Solidity of a Parabolic Conoid*—fig. 49. RULE.-Multiply the area of the base de by half the altitude ƒg, and the product will be the solidity.

NOTE. This rule will hold for any segment of the paraboloid, whether the base be perpendicular or oblique to the axis of the solid.

To find the Solidity of a Frustrum of a Paraboloid-fig. 49. RULE.-Multiply the sum of the squares of the diameters ab and de by the height ef, and the product by .3927.

To find the Solidity of a Parabolic Spindle-fig, 50,

RULE.-Multiply the square of the diameter ab by the length da, and the product by .4188, and it will give the solidity.

To find the Solidity of the Middle Frustrum of a Parabolic Spindle

fig. 51.

RULE. Add together 8 times the square of the greatest diameter cd, 3 times the square of the least diaineter ef, and 4 times the product of these two diameters; multiply the sum by the length a b, and the product by .05236, and it will give the solidity.

To find the Solidity of a Hyperboloid-fig. 52.

RULE. To the square of the radius of the base a b, add the square of the middle diameter n m; multiply this sum by the height cr, and the product again by .5236, and it will give the solidity.

* The parabolic conoid is its circumscribing cylinder.

To find the Solidity of the Frustrum of a Hyperbolic Conoid-fig. 53. RULE.-Add together the squares of the greatest and least semidiameters as and dr, and the square of the whole diameter n m in the middle of the two; multiply this sum by the height rs, and the product by .5236, and it will give the solidity.

To find the Solidity of a Cylindrical Ring—fig. 54. RULE.-To the thickness of the ring a b, add the inner diameter bc; then multiply the sum by the square of the thickness, and the product by 2.4674, and it will give the solidity.

BY MATHEMATICAL FORMULE.

FRUSTRUM OF A RIGHT TRIANGULAR PRISM.

The base (h+h‍+h"), h being the heights.

FRUSTRUM OF ANY RIGHT PRISM.

The base X its distance from the centre of gravity of the section.

CYLINDRICAL SEGMENT.

Contained between the base and an oblique plane passing through a diameter of the base, twice the height X the quotient of the square of the radius -3; or hr2, r being the radius and h the height.

SPHERICAL SEGMENT.

·(3r2+h2), r being the radius of the base, and h the height of the

segment.

SPHERICAL ZONE.

(3R2+3r2+h2), Rr being the radii of the bases.

SPHERICAL SECTOR.

rx the surface of the segment or zone.

ELLIPSOID.

a being the revolving diameter, and b the axis of revolution.