Mensuration of the 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. 14. 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. Here, AB 90 feet: CD=70=4900: hence = B ABX CD x,5236=90x4900 X,5236-230907,6 cubic feet, which is the solidity. 2. What is the solidity of a prolate spheriod, whose fixed axis is 100 and revolving axis 6 feet? Ans. 1884,96. QUEST.-Is the earth an oblate or a prolate syheroid? What is the difference between the two diameters? 14. Give the rule for finding the solidity of an ellipsoid? Mensuration of Cylindrical Rings. 3. What is the solidity of an oblate spheroid, whose fixed axis is 60, and revolving axis 100? Ans. 314160. 4. What is the solidity of a prolate spheroid, whose axes are 40 and 50? 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. OF CYLINDRICAL RINGS. 15. A cylindrical ring is formed by bending a cylinder until the two ends meet each other. Thus, if a cylinder be bent round until the axis takes the position mon, a solid will be formed, which is called a cylindrical ring. A B. The line AB is called the outer, and cd the inner di ameter. PROBLEM XII. 16. To find the convex surface of a cylindrical ring. QUEST.—15. How is a cylindrical ring formed? 16. How do you find the convex surface of a cylindrical ring? Mensuration of Cylindrical Rings. RULE. 1st. To the thickness of the ring add the inner di ameter. 2nd. Multiply this sum by the thickness, and the product by 9,8696-the result will be the area. EXAMPLES. 1. The thickness Ac of a cylindrical ring is 3 inches, and the inner diameter cd, is 12 inches: what is the convex surface? Ac+cd=3+12=15: then 15 × 3 × 9,8696-444,132 square inches the surface. A d 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? Ans. 394,784 sq. in. PROBLEM XIII. 17. To find the solidity of a cylindrical ring. QUEST.-17. How do find the solidity of a cylindrical ring? Mensuration of Cylindrical Rings. RULE. 1st. To the thickness of the ring add the inner di ameter. 2nd. 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, 11 × (3)2 × 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. 3. Required the solidity of a cylindrical ring whose thickness is 2 inches, and inner diameter 12 inches? Ans. 138,1744 cubic inches. 4. What is the solidity of a cylindrical ring, whose thickness is 4 inches, and inner diameter 16 inches? Ans. 789,568 cubic inches. PART IV SECTION I. OF MEASURES. 1. THE CARPENTERS' RULE, sometimes called the sliding rule, is used for the measurement of timber, and artificers' work. By it the dimensions are taken, and by means of certain scales, the superficial and solid contents may be computed. 2. The rule consists of two equal pieces of box wood, each one foot long, and connected together by a folding joint. 3. One face of the rule is divided into inches, half inches, quarter inches, eights of inches and sixteenths of inches. When the rule is opened, the inches are numbered from 1 to 23-the last number 24, at the end, being omitted. QUEST.-1. What is the carpenters' rule used for? 2. Describe the rule? 3. How is the rule divided on one face? When the rule is opened, how are the inches numbered? How long is the rule? |