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To find the Solidity of an Ungula, fig. 28, when the section passes obliquely through the cylinder, abcd..

RULE.-Multiply the area of the base of the cylinder by half the sum of the greater and less heights a e, cf of the ungula, and the product is the solidity.

When the Section passes through the base of the Cylinder and one of its sides fig. 29, abc.

RULE. From of the cube of the right sine a d, of half the arc ag of the base, subtract the product of the area of the base, and the cosine df of said half arc.* Multiply the difference thus found by the quotient of the height, divided by the versed sine, and the prod uct is the solidity.

When the Section passes obliquely through both ends of the Cylinder,

adce-fig. 30.

RULE. Find the solidities of the ungulas a dce and dbc, and the difference is the solidity required (conceiving the section to be con tinued till it meets the side of the cylinder).

NOTE. For rules to ascertain the solidity of conical ungulas, see Ryan's Bonny castle's Mensuration, page 136 (1839).

OF CONES AND PYRAMIDS.

31.

32.

33.

34.

To find the Solidity of a Cone or Pyramid-figs. 31 and 33. RULE.-Multiply the area of the base by the height cd, and the product will be the content.

To find the Solidity of the Frustrum of a Cone-fig. 32.

RULE.-Divide the difference of the cubes of the diameters ab, cd of the two ends by the difference of the diameters; this quotient, multiplied by .7854, and again by of the height, will give the solidity.

To find the Solidity of the Frustrum of a Pyramid—fig. 34. RULE.--Add to the areas of the two ends of the frustrum the square root of their product, and this sum, multiplied by of the height a b, will give the solidity.

*If the height of the base be less than radius, otherwise add them.

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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 a b; multiply this sum by the neight of the wedge # If, 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 abc, 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.

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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.

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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.

RULE.

To find the Solidity of the Segment of a Spheroid-figs. 41 and 42. -When the base ef 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 cà, 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 e 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 cd, 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 yan ellipse revolving about its conjugate diameter.

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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.

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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 Spindlefig. 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.

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To find the Solidity of a Parabolic Conoid*-fig. 49. RULE.-Multiply the area of the base dc 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 a band 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 de and the product by .4188, and it will give the solidity.

To find the Solidity of the Middle Frustrum of a Parabolic Spindlefig. 51.

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

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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.

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