PROBLEM XXVII. To find the Solidity of the Elliptic Hoofs of the Fruftum of a Cone made by a Plane cutting off a Part of the Bafe. 1. From the verfed fine, or height of the base of the hoof, fubtract the difference of the diameters of the base and end of the fruftum, and divide the remainder by the diameter of the faid end; find the tabular fegment whofe verfed fine is equal to the quotient; find alfo the tabular fegment whose versed fine is expreffed by the quotient of the verfed fine of the base of the hoof divided by the diameter of the bafe of the fruftum; multiply the former fegment by the cube of the diameter of the end, and by the quotient of the verfed fine of the base of the hoof divided by the difference between the faid versed fine and the difference of the diameters, and by the root of the faid quotient; and multiply the latter fegment = Q2 the elliptic hoof EFCB; putting b for the height of the hoof, and D and d for the diameters of the bafe and end, or top, of the fruftum, refpectively. Or if we would reduce the circular fegments in the last expreffion to those of a circle whofe diameter is 1, to be found in the table of circular fegments at the end of the book, that expreffion will be D + d d the elliptic hoof EFCB. And this is the rule in prob. 27. And segment by the cube of the greater diameter; multiply the difference of these two products by of the height of the hoof, and the product divided by the difference of the diameters will give the content of the hoof required. 2. Subtract And if this value of the hoof be taken from bn × that of the whole conic fruftum, the remainder b D3 d3 D d × : [D3 — ď3) ×n - D3× tab. feg. whofe height is Corol. 2. If the points D and a coincide, the fection E F C will be a whole ellipfe, and the rules in corollary 1 will become - dod D-d cone by н, the laft theor. will become "HX (D2-d/Dd) = ACB, Coro!. 4. But HD2 and HX (D/Dd-dd) = n ACG. the whole cone VA B, and therefore HD2 H X (D2 dDd) in HdDd the top part VAC. Which top part, confequently, is to the whole cone VAR, as in Hd√ Dd to of the whole cone VA B as D3 to . nн D2, or as to D; or the fquare is Corol 2. Subtract the measure of the hoof, above found, from that of the whole fruftum, and the remainder will be the measure of the complemental hoof. That is, if D and d be the extreme diameters of the fruftum, and b its height; alfo the tabular fegment whofe verfed fine is , that whofe ver BD d BD D sed fine is 1D−D+4, and n = 1785398 &c. d Then (PXD3-Q X ď3 × X -d BD-D+d ED-D+ď the content of the elliptic hoof EFCB. And BD BD [n (D3 — d3) — P × D3 + Qxd3× BD-D+d√ BD-DFd] BI = to that of the complemental hoof These are proved in corollary 1 to prob. 26. = D3 D d G V B HI F N d(D-d) DC√(D − d) × d) DC D-d x tab. feg. whofe height is d(D) x d) the parabolic hoof EBC. EXAMPLE I. If a vcffel in the form of the fruftum of a cone, whose bottom diameter is 30 inches, be inclined to the horizon till the furface of the liquor in it cut the bottom, leaving 10 inches of its diameter dry, and meeting the fide in c at the distance c1 of 18 inches from the bafe: how many Winchefter gallons are in it, fuppofing the diameter G C of the vellel at of the liquor to be 19 inches? the top Here x tab. feg. whofe height is -], D × [(D − d) ‡d/d ( v − d) — nd3 + 3x tab. fegment whose height is] will exprefs the meafure of the complemental area of the hyperbolic fection being found by prob. 6 fect. 5 part 3, and substituted in the general theorem, will give the folidity of the hyperbolic ungula. Corol. 7. If D coincide with 1, or CDE be a right angle, the tranfverfe and conjugate axes of the hyperbolic fection will become 2 db D- d and d respectively. Here = 30 D = 30, d = 19, b = 18, and BD = Whence D IO 20. 6666, the tab. verf. = 6 = 2.875 = 4793, D = 20 30 3 the tabular area for which is 55622573; and BD-D+d_20-30+19 46 23 d = 193 96 48 the tabular area anfwering to which is 37187178. Again, D327000, which multiplied by 55622573, the former area, produces 15018.09471. ×√46 = 22686-470698, which multiplied by 37187178, the latter area, gives 8436.45824. Then 6581.63647, the difference of these two pro6 30-5 ducts, being multiplied by = the quotient of of the height divided by the difference of the diameters, will produce 3656.4647 the folidity in inches: which being divided by 2684, the inches in a corn gallon, give 13.6029 corn or Winchester gallons, for the quantity of liquor in the veffel. EXAMPLE II. If a veffel, in form of the fruftum of a cone, clofe at both ends, be placed in fuch a pofition, that the liquor may juft cover the lefs end, and 10 inches of the diameter of the greater; what number of wine gallons are in it, fuppofing the diameters of the two ends to be 30 and 19 inches, and their diftance 18 inches? |