mean between those areas; add these three numbers together; multiply the sum by one-third of the perpendicular depth; and the product will be the content. Note. If the ends be squares, their areas may be found by Prob. I.; and if they be regular polygons, their areas may be obtained by Prob. IX. or X., Part IV. A geometrical mean proportional between the areas of the ends, may be found by Prob. I., Part I., or Prob., X., Part II. 325 32 f11.12 bottom area. As11.12 on C:11.12 on D:: 3.63 on C: 6.35 on D, mean area. }: 1,56 2323 J 13.57 bottom area. As 13.57 on C:13.57 on D:: 4.43 on C: 7.75 mean area on D. Note. In finding the areas of the ends, in ale gallons, the square divisor 282 is used; but in finding their areas in wine gallons, the square gauge-point 15.19 is used. It may also be observed that the content in malt bushels may be ob-. tained by either of the above methods. 2. What is the content of a hexagonal wine-vat; each side of the greater end being 92 inches, each side of the less end 54 inches, and the perpendicular depth 126 inches? Ans. 7722.370 wine gallons." 3. Required the content of an octagonal vessel, in ale and wine gallons, and malt bushels; each side of the greater base being 35 inches, each side of the less base 28 inches, and the perpendicular depth 94 inches? Ans. 1603.566 ale gallons, 1957.583 wine gallons, and 210.256 malt bushels. PROBLEM VII. To find the content of a vessel in the form of a cone. RULE I. By the Pen. Multiply the square of the diameter of the top of the vessel, by of the perpendicular depth; divide the product by 359.05, 294.12, and 2738; and the respective quotients will be the content in ale and wine gallons, and malt bushels. RULE II. By Problem XIII., Part IV., find the area of the top of the vessel, in ale and wine gallons, and malt bushels; multiply these areas by of the perpendicular depth; and the respective products will be the content in ale and wine gallons, and malt bushels. Note 1. If the square of the diameter of the base of a cone be multiplied by .7854, the product will be the area of the base in square inches. Multiply this area by of the cone's perpendicular height, and the product will be the content in cubic inches; which being divided by 1728, will give the content in cubic feet. (See Nesbit's Mensuration, Prob. VII., Part IV.) 2. The area of the top of a vessel in the form of an elliptical cone may be found by Prob. XVI., Part IV.; and hence the content. may be obtained by Rule II. EXAMPLES. 1. The diameter A B, of the top of a conical vessel, is 46 inches, and the perpendicular depth DC, 78 inches; what is the content in ale and wine gallons, and malt bushels ? Again, 55016-294.12187.052, the content in wine gallons; and 55016-2738-20.093, the content in malt bushels. By Rule II. Here 46 x 46 x .002785-2116x 002785-5.893060, the area of the base in ale gallons; 2116x.003399-7.192284, the area of the base in wine gallons; and 2116x.000365 -.772340, the area of the base in malt bushels. Hence, 5.893060 × 26 = 153.219560, the content in ale gallons; 7.192284× 26=186.999384, the content in wine gallons; and .772340×26=20.080840, the content in malt bushels. By the Sliding Rule. As the circular gauge-point on D, is to of the perpendicular depth on C; so is the diameter on D, to the content on C. 2. The diameter of the top of a conical vessel measures 84 inches, and the length of its slant side 112 inches; what is the content in ale gallons? Ans. 680.1115 ale gallons. 3. The perpendicular depth of a conical vessel is 96 inches, and the circumference of the top 216 inches; required the content in malt bushels? Ans. 55.232 malt bushels. PROBLEM VIII. To find the content of a vessel in the form of the frus tum of a cone. RULE I. By the Pen. To three times the product of the top and bottom dia. meters, add the square of their difference; multiply the sum by the perpendicular depth; divide the product by 1077.15, 882.36, and 8214; and the respective quotients will be the content in ale and wine gallons, and malt bushels. RULE II. From the square of the sum of the top and bottom diameters, subtract the product of those diameters; multiply the remainder by the perpendicular depth; divide the product by the foregoing divisors; and you will obtain the content in ale and wine gallons, and malt bushels. Note 1. The General Rule given in Prob. VI., will also give the content of a vessel in the form of the frustum of a cone, whether the ends be circular or elliptical. 2. When the ends are elliptical, their areas must be found by Prob. XVI., Part IV. 3. The equal frustums of two similar cones, joined together at their greater ends, form a figure, which, by Gaugers, is called a cask of the fourth variety. This variety, however, is seldom, or perhaps never met with in Practice. EXAMPLES. 1. What is the content in ale and wine gallons, and malt bushels, of the vessel ABCD, in the form of the frustum of a cone; the diameter A B of the bottom being 72 inches, the diameter C D of the top 40 inches, and the perpendicular depth En 86 inches? |