To gauge an oblate spheroid :-Place two-thirds of the length of the polar axis over the gauge points of the cylinder, and over the equatorial diameter in inches found on D, will be found the contents on the line C. See ¶ 36, page 119. 3. What is the content of an oblate spheroid in ale and wine gallons, its diameters being 24 and 36 inches? Answers, 70.5024 wine, and 57.7517 ale gallons. 49. PROLATE SPHEROID. A prolate spheroid is a solid generated by the revolution of an ellipse about its major axis. Its solidity is equal to that of twothirds of its least circumscribing cylinder. Therefore, To find the solidity of a prolate spheroid: Multiply the square of the minor by the major axis, and that product by .5236, and the result will be the solidity. Ans. 91.63. 2. What is the solidity of a prolate spheroid, whose axes are 18 and 14? Ans. 1847.26. 3. What is the content of a prolate spheroid in feet, bushels, and wine and ale gallons, its axes being 36 and 24 inches? Answers, 6.3 feet; 5.06 bushels; and 47 wine, and 38.5 ale gallons. To gauge a prolate spheroid:-Place two-thirds of the major axis over the gauge points for the cylinder, and over the minor axis found on D, will be found the contents on C. T50. SEGMENTS OF SPHEROIDS. The segment of a spheroid is a portion cut off by a plane perpendicular to one of its axes. When the plane is perpendicular to the fixed axis, (as represented in figures 1 and 2.) 1. the base is a circle; but when the plane is parallel to the fixed axis, the segment has an elliptical base. The former is called a circular segment, and the latter an elliptical segment. Thus, a segment cut from a prolate spheroid by a plane perpendicular to its major axis, (as represented by figure 1,) is a circular segment; and one cut from an oblate spheroid by a plane perpendicular to its minor axis, (as represented by figure 2,) is likewise a circular segment; but when the segment is cut off by a plane perpendicular to the other axes, the base is an ellipse, and the segment is said to be elliptical. The minor axis of the oblate spheroid is called its polar axis; and the major axis of the prolate spheroid is its polar axis. To find the solidity of a circular segment of a spheroid: Multiply the difference between three times the polar axis and twice the height of the segment, by the square of the height, and the product by .5236; then say :-As the square of the polar axis is to the square of the equatorial axis, so is the last product to the solidity of the segment. To find the solidity of an elliptical segment of a spheroid:Multiply the difference between three times the equatorial axis and twice the height of the segment, by the square of the height, and that product by .5236; then say :— As the equatorial axis is to the polar axis, so is the last product to the solidity of the seg ment. EXAMPLES. 1. The axes of an oblate spheroid are 50 and 30, and the height of a circular segment is 6; what is the solidity ? Ans. 4084.08. 2. What is the solidity of a circular segment of a prolate spheroid, the axes being 40 and 24, and the height 4? Ans. 337.7848. 3. The axes of an oblate spheroid are 25 and 15, and the height of a circular segment of it is 3; what is the solidity of the segment? Ans. 510.51. 4. What is the solidity of an elliptic segment of an oblate spheroid, whose height is 10, the axes of the spheroid being 100 and 60 ? Ans. 8796.48. 5. Find the solidity of an elliptic segment of a prolate spheroid, whose axes are 25 and 15, and the height of the segAns. 306.306. ment 3. 51. FRUSTUMS OF SPHEROIDS. The middle zone or frustum of a spheroid is a portion of it contained between two parallel planes at equal distances from the centre, and perpendicular to one of the axes. To find the solidity of the middle frustum of a spheroid, the ends being circular :— 1. To twice the square of the middle diameter, add the square of the diameter of one end; multiply this sum by the length of the frustum, and the product by .2618, and the result will be the solidity. When the ends of the frustum are elliptical : 2. To double the product of the axes of the middle section, add the product of the axes of one end; multiply this sum by the length of the frustum, and the product by .2618, and the result will be the solidity. EXAMPLES. 1. What is the solidity of a circular middle frustum of an oblate spheroid, the middle diameter being 25, the end diameters 20, and the length 9? Ans. 3887.73. 2. Find the solidity of an elliptic middle frustum of an oblate spheroid, the axes of the middle section being 25 and 15, and those of the ends 15 and 9, and the length 20. Ans. 4633.86. 3. Find the solidity of a circular middle frustum of a spheroid, the middle diameter being 100, and those of the ends 80, and the length 36. Ans. 248814.72. 4. What is the solidity of an elliptic middle frustum of a spheroid, the axes of the middle section being 50 and 30, and those of the ends 30 and 18, and the length 40? T52. PARABOLOID, OR PARABOLIC CONOID. The paraboloid is a solid generated by the revolution of a parabola about its axis, which remains fixed. See ¶ 29. The paraboloid is often called the parabolic conoid, from its resemblance to the cone. A frustum of a paraboloid is a portion of it contained between two planes perpendicular to its axis. The solidity of a paraboloid is equal to that of a cylinder of the same base and half the height. Therefore, To find the solidity of a paraboloid :— Multiply the area of the base by half the altitude. Or, Multiply the square of the diameter of the base by the altitude, and that product by .3927, and the result will be the solidity. EXAMPLES. 1. Find the solidity of a paraboloid, whose altitude is 21, and the diameter of its base 12. Ans. 1187.525. 2. What is the capacity of a vessel or receiver, its form being that of a paraboloid, whose base is 20 inches, and its altitude 28 inches, in ale and wine gallons? Answers, 19.014 wine, and 15.632 ale gallons. To gauge a paraboloid:-Place half the altitude over the guage points for the cylinder, and over the diameter of the base found on D, will be found the contents on the line C. To find the solidity of a frustum of a paraboloid :— Multiply the sum of the squares of the diameters of the two ends by .3927, and the product will be the mean area, which being multiplied by the altitude will give the solidity. 3. What is the solidity of the frustum of a paraboloid, whose altitude is 22.5, and the diameters of the ends 20 and 40? Ans. 17671.5. 4. Find the solidity of a frustum of a paraboloid, the diameters of the ends being 29 and 15, and its altitude 18. Ans. 7535.127. |