root of their product. Then multiply this sum by of the perpendicular height, and the product will be the solidity. NOTE. If a cone and a pyramid have equal bases and altitudes, they are equal in their solidity. Consequently, the rule already given for the frustrum of a pyramid is equally applicable to the frustrum of a cone. Example.-How many gallons of ale are contained in a cistern in the form of a conic frustrum, ab e f, if the larger diameter, a b, be 9 feet, and the smaller diameter, ef, 7 feet, and the depth, c o d, 9 feet? √244771=49-46 151 55×3=454·65 cubic feet. 454 65 x 1728-785635 cubic inches. 785635-282=2785 gals. Ans. OF SPHERES. PROBLEM VII. To find the Surface of a Sphere or Globe. Rule.-Multiply the diameter of the sphere by its circumference. and the product will be the surface. Or, multiply the square of the diameter by 3.14159. To find the Convex Surface of a Spherical Zone or Segment. Rule-Multiply the height of the zone or segment by the whole circumference of the sphere of which it is a part, and the product will be the convex surface. Example.-If the axis of a sphere, be 42 inches, what is the convex surface of a segment or zone, a b d. whose height, c is 9 inches? cd a d First, 42x3.14159 131 9468-circumference. 9=height. 1187 5212 surface in square inches. PROBLEM IX. To find the Solidity of a Sphere or Globe. Rule.-Multiply the cube of the diameter, c e, by the decimal 5236. Or multiply the square of the diameter by the circumference, and of the product will be the contents. Example.-What is the solidity of a globe whose diameter, c e, is 12 inches? 122×3.14159= 452.38996 = surface of the sphere. Then, 452 38996 × 12÷6=904 78 solidity. Or thus: 12=1728-cube of the diameter. And 1728 x 5236=904·78=solid con tents. PROBLEM X. To find the Solidity of a Spherical Segment. Rule. To three times the square of the radius, a b, of its base, add the square of its height, bc; then multiply the sum by the height, and the product by 5236, for the contents. Example.-What is the solidity of the segment, a d c, (of the sphere e c,) whose height, bc, is 8 feet, and the diameter of whose base, a d, is 14 feet? 211×8 1688 x 5236-883 836. Ans. NOTE. The solidity of a spherical segment is frequently required when the radius of its base is not given; but if the diameter of the sphere and the height of the segment be known, the solidity may be easily found by the following Rule. From three times the diameter of the sphere, subtract twice the height of the segment; then multiply the remainder by the square of the height, and the product by the decimal ·5236. OF SPHEROIDS. PROBLEM XI. To find the Solidity of a Spheroid. Rule.-Multiply the square of the revolving axis by the fixed axis ; and the product, multiplied by 5236, will give the solidity. NOTE. If the generating ellipse revolves about its major axis, the spheroid is late or oblong; if about its minor axis, the spheroid is oblate. OF PARABOLIC CONOIDS AND SPINDLES. PROBLEM XII. To find the Solidity of a Parabolic Conoid. Rule.-Multiply the square of the diameter of the base by the altitude, and the product by 3927 (which is of 7854), and it will give the contents. Example.-What is the solidity of a parabolic conoid, whose height, fg, is 60, and the diameter, c d, of its base 100 inches? 100=10000 And 10000 x 60 x 3927=235620. Ans. PROBLEM XIII. a To find the Solidity of a Frustrum of a Paraboloid Rule.-Multiply the sum of the squares of the diameters of the two ends, a b and c d, by the height of the frustrum, ef, and the product by 3927 (which is of 7854), and it will give the contents. Example. What is the solidity of the frustrum of a paraboloid, a b c d, whose diameter, c d, is 54, a b, 28, and height, f e, 18 inches? a b 542=2916 Then, 3700 x 18 x 3927=26153 82. Ans. 28 784 3700 PROBLEM XIV. To find the Solidity of a Parabolic Spindle. Rule.-Multiply the square of the middle diameter, c d, by the length of the spindle, I m, and the product by 41888 (which is and it will give the solidity. of 7854), To find the Solidity of the Middle Frustrum of a Parabolic Spindle. Rule. Add together 8 times the square of the greatest diameter, c d, 3 times the square of the least diameter, fe, and 4 times the product of these two diameters; multiply the sum by the length, a b, and the product by 05236 (which is of 3 1416); this will give the solidity. Example -What is the solidity of the frustrum of a parabolic spindle, whose dimensions are as follows: a b, 60, e d, 40, fe, 30 inches? Rule. To the square of the radius of the base, a s, add the square of the middle diameter, m r; multiply this sum by the height, sf, and the product by 5236, and it will give the solidity. 1300 And 1300 × 30 x 5236÷1728 11.817 cubic feet. PROBLEM XVII. To find the Solidity of the Frustrum of a Hyperbolic Conoid. (See the foregoing figure.) Rule. Add together the squares of the greatest and least semidiameters, a s and d r, and the square of the whole diameter, m r, in the middle of the two; multiply this sum by the height, r s, and the product by 5236, and it will give the solidity. Example.-Required the solidity of the frustrum of a hyperbola, abd c, whose semidiameter, a s, is 20 inches, and dr, 10 inches; the middle diameter, m r, 30 inches, and whose height is 20 inches? 202=400 1400 Then, 1400×20 × 6236÷1728-8.426 cubic feet. To find the Convex Surface of a Cylindrical Ring. Rule. To the thickness of the ring, a c, add the inner diameter; then multiply this sum by the thickness, and the product by 9.8696 (which is the square of 3·1416), and it will give the convex surface required. Rule. To the thickness of the ring, a c, add the inner diameter, cd; then multiply the sum by the square of the thickness, and the product by 2:4674 (which is of the square of 3.1416), and it will give the solidity. Example.-Required the solidity of an anchor ring, whose inner diameter is 8 inches, and thickness in metal 3 inches. First, 3+8=11 3x3 9 square of thickness. 99×2 4674 244 2726 solidity in inches. GAUGING OF CASKS. Gauging is a practical art, which does not admit of being treated in a very scientific manner. |