XL.

Given the solidity of a sphere to find the diameter. Multiply the solidity by 21, and divide by 11; then extract the cube root.

Ex. Find the diameter of a sphere whose solidity is 63457.

3. Find the solidity of the segment of a sphere whose diameter is 18 inches, and height 5 inches.

Ans. 681*345.

4. Find the number of gallons contained in a spherical boiler the diameter of whose brim is 4 feet, and depth 2 ft. 6 in. Ans. 149.

5. Find the solidity of a segment of a sphere the diameter of whose base is 6 ft. 10 in., and height 3 ft. 6 in. Ans. 86.663 feet.

6. Find the diameter of a sphere whose solidity is 5946 cubic inches. Ans. 22°474.

7. Find the diameter of a sphere which contains one cubic yard. Ans. 3.721 feet. 8. Find the solidity of a spherical shell whose inner and outer diameters are 12 and 15 inches. Ans. 8625.

9. Find the weight of a spherical shell of cast iron whose inner and outer diameters are 8 and 10 inches.

Ans. 63 lbs.

10. The diameter of a sphere is 27 inches, and the sphere is divided into three parts by two parallel planes which divide the diameter in the ratio of 2, 3, 4; find the content of each part. Ans. 1301, 4299, 47094.

THE SPHEROID.

If an ellipse revolve about the transverse diameter it will generate a solid called the prolate spheroid; if it revolve about the conjugate diameter, the spheroid is oblate.

XLI.

To find the solidity of a prolate spheroid.

Multiply the square of the conjugate by the transverse diameter; then multiply by 11, and divide by 21.

Ex. Find the solidity of a prolate spheroid whose transverse and conjugate diameters are 5 ft. 3 in. and 3 ft 3 in.

XLII.

To find the solidity of the segment of a prolate spheroid.

Find the solidity of a spherical segment of the same height, the diameter of the sphere being equal to the transverse diameter of the spheroid; then

(trans.): (conj.)2 :: spherical seg.: spheroidal seg.

Note. The same rules apply to the oblate spheroid by interchanging the diameters.

Ex. Find the solidity of a segment of a prolate spheroid whose transverse and conjugate diameters are II and 7 inches, and the height of the segment 4 inches.

EXAMPLES.

I. Find the solidity of a prolate spheroid whose transverse and conjugate diameters are 12 and 8.

2. Find the solidity of an oblate transverse and conjugate diameters are 10

3

Ans. 4023.

spheroid whose and 9 feet.

Ans. 5193 feet.

Find the solidity of a segment of a prolate spheroid whose transverse and conjugate diameters are 24 and 14 feet, and the height of the segment 6 feet.

THE PARABOLOID.

Ans. 385 feet.

This solid is formed by the revolution of a parabola about its axis.

XLIII.

To find the solidity of a paraboloid.

Multiply the square of the diameter of the base by the height; then multiply by 11 and divide by 28.