4m 3D d =c, m being the middle diameter.-By corollary 7 to the laft problem. Then proceed as in the last rule. Note. When the generating hyperb. is equilateral, by corollary 8 to the last problem; fo that the content may then be found without having the length given. Another rule might be drawn from corollary 6 to the last problem. PROBLEM XV. To find the Content of the Segment of an Hyperbolic Spindle. Ufe here the rule for the fegment of an elliptic fpindle; only inftead of what is there called the lefs axe, take here the tranfverfe, and inftead of the greater, the conjugate axe; and the content will be obtained. By corollary 2 to problem 13. the fegment MAF. Where a is AE its altitude, and the other letters as in the laft problems. Other rules might be found by fubtracting the fruftum from the femi-fpindle, according to the correfponding rules of the last two problems. PRO PROBLEM XVI. To find the Content of an Univerfal Hyperbolic Spindle; that is, of a Solid Generated from the Revolution of an Hyperbola about an Ordinate to Any Diameter. INVESTIGATION. Put a CB the femi-diameter to which belongs the double ordinate AA, about which the figure ABA revolves, c = CD its femi-conjugate, b = AE, C = CE, Z = EF, y = GF parallel to CE, m = fine of the angle E or F, and n = its cofine, to the radius 1, alfo p = 3.14159; and let GH be perpendicular to AE. Then, by the nature of the hyperbola, c : a : C : but GH is my, and HF ny; = = whofe correct fluent is pmm x [ = ÷n (BE3 — y3) And when z becomes = b, the above fluent 3 appears that pm2 × (= {n BE3 + c2b ́+a2b+ cab cc + bb e 300 +bb) b 6 ex-. preffes the folid generated by ABK; the fign of the firft term +2BE3 being - or, according as к falls between A and E, or without them: and confequently half the diff. of the folids generated by the spaces ABK, KBA, is pnm2 × BE3 = p × BK2 × KE the cone generated by the triangle BKE. And hence the folids generated by the two equal areas ABE, EBA, are alfo equal; the value of each being the half content of the whole folid mentioned above in the investigation. Scholium. The folution of this problem was never but once before attempted, viz. in a periodical performance; but the folution there given is erroneous, the fluxion and fluent of the general fruftum, being both falfely affigned. SECTION VIII. PROMISCUOUS QUESTIONS CON CERNING SOLIDS. QUESTION I. EQUIRED the content of a tub whose REQUIRED greater diameter AB is 60, diagonal вc 66, and the length of the ftave AC 30 inches. Here AB BC + AC :: BC AC: = BE 36 × 96 = 6 × 9:6 = 57·6 AE 60 EA DC. Hence AE AB - CD = 3028.8 1°2. B And EC = √AC2 — AE√302-1.26√. =6√2504 = 6√2496. Then (ABAB X DC + DC2) X÷CE X 785398 (602+60 × 57.6 +57·62) × 22496X785398 = 10373 76 × 439 X 3.14159 81410112 cubic inches 288.688 ale gallons, the content required. QUESTION II. Three perfons having bought a fugar loaf, would divide it equally among them by fections parallel to the bafe; it is required to find the altitude of each perfon's fhare, fuppofing the loaf to be a cone whofe height is 20 inches. 4 Similar Similar cones being as the cubes of their altitudes, we shall have as 3 1: 20: 203 = 13.8672247 the height of the upper part; and as 3: 2 :: 20: 20 = 17:4716107 = that of the upper and middle part together; confequently 17.4716107 13.8672247 = 3.604386 is the height of the middle part, and 20-174716107 2.5283893 is that of the lower part. QUESTION III. A filver cup, in form of the fruftum of a cone, 'whofe top diameter is 3 inches, its bottom diameter 4, and its altitude 6 inches, being filled with liquor, a perfon drank out of it till he could fee the middle of the bottom; it is required to find how much he drank ? By problem 14 of fection I we have D = 4, d = 3, b=6, and BDD = 2; hence D3 — d3 — 64—27 37, P the tabular area whofe verfed fine is or, of 78539816; BD D lar area whofe verfed fine is the tabu quently (37-32n +54Q√2) × 2 = (5n+54Q√2) X 2 = 7·8539816 + 350458624 42.899844 cubic inches •152127 ale gallons, or 1 gill and nearly, the quantity required. QUESTION IV. I have a right cone which coft me £5 13.7, at Ios a cubic foot, the diameter of its base being to its altitude as 5 to 8; and would have its convex furface Gg 2 |