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2. The diameter of a sphere is 7957.8: what is its solidity? Ans. 263863122758.4778.

3. The diameter of a sphere is 24 yards: what is its solid Ans. 7238.2464 cubic yds.

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4. The diameter of a sphere is 8: what is its solidity?

Ans. 268.0832.

5. The diameter of a sphere is 16: what is its solidity?

Ans. 2144.6656.

RULE II.

Cube the diameter and multiply the number thus found, by the decimal .5236, and the product will be the solidity.

EXAMPLES.

1. What is the solidity of a sphere whose diameter is 20?

Ans. 4188.8.

2. What is the solidity of a sphere whose diameter is 6? Ans. 113.0976.

3. What is the solidity of a sphere whose diameter is 10?

Ans. 523.6

Mensuration of the Round Bodies.

PROBLEM X.

To find the solidity of a spherical segment with one base.

RULE.

I. To three times the square of the radius of the base, add the square of the height.

II. Multiply this sum by the height, and the product by the decimal .5236, the result will be the solidity of the segment.

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Then, 163 x 4 x .5236-341.3872 solid feet, which is the solidity of the segment.

2. What is the solidity of the segment of a sphere whose height is 4, and the radius of its base 8? Ans. 435.6352.

3. What is the solidity of a spherical segment, the diam. eter of its base being 17.23368, and its height 4.5?

Ans. 572.5566.

4. What is the solidity of a spherical segment, the diam eter f the sphere being 8, and the height of the segment 2 feet? Ans. 41.888 cubic ft.

5. What is the solidity of a segment, when the diameter of the sphere is 20, and the altitude of the segment 9 feet? Ans. 1781.2872 cubic ft.

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A spheroid is a solid described by the revolution of an ellipse about either of its axes.

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If an ellipse ACBD, be revolved about the transverse or longer axis AB, the solid described is called a prolate

spheroid: and if it be revolved

about the shorter axis CD, the solid described is called an oblate spheroid.

The earth is an oblate spheroid, the axis about which it revolves being about 34 miles shorter than the diameter perpendicular to it.

PROBLEM XI.

To find the solidity of an ellipsoid

RULE.

Multiply the fixed axis by the square of the revolving axis, and the product by the decimal .5236, the result will be the required solidity.

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Ilere, AB=90 feet: CD2-702-4900: hence

B

ABX CDX.5236-90 x 4900 X.5236=230907.6 cubic feet, which is the solidity.

Mensuration of Cylindrical Rings.

2. What is the solidity of a prolate spheriod, whose fixed axisis 100, and revolving axis 6 feet?

3. What is the solidity of an oblate axis is 60, and revolving axis 100?

4. What is the solidity of a prolate are 40 and 50?

Ans. 1884.96.

spheroid, whose fixed

Ans. 314160.

spheroid, whose axes Ans. 41888.

5. What is the solidity of an oblate spheroid, whose axes are 20 and 10?

Ans. 2094.4.

6. What is the solidity of a prolate spheroid, whose axes are 55 and 33 ? Ans. 31361.022.

7. What is the solidity of an oblate spheroid, whose axes are 85 and 75?

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Ans.

RINGS.

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The line AB is called the outer, and cd the inner diameter.

PROBLEM XII.

To find the convex surface of a cylindrical ring.

RULE.

I. To the thickness of the ring add the inner diameter.

II. Multiply this sum by the thickness, and the product by 9,8696, the result will be the area.

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2. The thickness of a cylindrical ring is 4 inches, and the inner diameter 18 inches: what is the convex surface?

Ans. 868.52 sq. in.

3. The thickness of a cylindrical ring is 2 inches, and the inner diameter 18 inches: what is the convex surface?

PROBLEM XIII.

Ans. 394.784 sq. in.

To find the solidity of a cylindrical ring.

RULE.

I. To the thickness of a ring add the inner diameter

II. Multiply this sum by the square of half the thickness, and the product by 9.8696, the result will be the required solidity.

EXAMPLES.

1. What is the solidity of an anchor ring, whose inner diameter is 8 inches, and thickness in metal 3 inches? 8+3=11: then, 11x()×9.8696-244.2726, which expresses the solidity in cubic inches.

2. The inner diameter of a cylindrical ring is 18 inches, and the thickness 4 inches: what is the solidity of the ring? Ans. 868.5248 cubic inches

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