But by example 3 to corollary 3, the fame (b + x) * − (b − x) 2, area cdoc is = 2√a × 1.3.5.7.9 x 3 4.6.8.10.12.1466 3 ; confeq. &c). And when b and x are equal to each other in this equation, we obtain the fame as was found, by a different method, in ex. 3.cor. 3. When is the vertex of the curve, then b is x and the feries for the area CDC becomes 4.6 4.6.8.10 fubftituting for this laft feries its value ✔2, as found above, will be = 3√2AZ = 3√2 x the rectangle CFEC. EXAMPLE III. If the curve be a circle, and A were any point between the circumference and the center, the feries would be very complex; but if A be the center, and r the radius, then will 1.32.6 2.3 2.4.5+4 which feries converges very faft when x is fmall in refpect of r. I Hence I - •785398 &c, = 213 + 2.4.5 + &c, as in pa. 104. I 202 + 1.3x6 1.3.5x8 сс 2.4c+ 2.4.6c6 2.4.6.8cs &C); and by fubftituting in the general feries, the area cdDC will be 2tx × (1+ 2.302 1.3.x6 + 2.4.504 2.4.6.7c0&c), the fame with the feries in the laft example for the area of a circle, differing only in the figns of the fecond, fourth, fixth, &c, terms. When x is c, the series becomes 2.302 24 1.3.6 tx, which taken will leave AB X BE X &c) for the area BDE. So that the rectangle ABEC is to the trilineal BDE, as A BEG will be to the trilineal BeQ, as I is to And when = a, or ac = AG, the above expreffion will become 2GA X AB X (1 + ÷ +} +++ &c) for the area GPBQG. E If there be Any Solid, in which all the Parallel Sections, either Right or Oblique to the Axe, are Like and Similar Figures; and if, when the Abfcifs, or Part of the Axe drawn through the Centers of the Parallel Sections, is reprefented by z, the Value of the Section be expreffed by Any Series of Terms involving z, and known quantities, after the fame manner as the Ordinate is expreffed in the laft Propofition: Then will the Solidity be expreffed by the fame Quantity as the Area in the laft Propofition: That is, fuppofing the relation between the Abfcifs and Section to be expressed by Any Equation of which one fide is the Section, after the manner of the relation between the Abfcifs and Ordinate in the laft Propofition; whenever the value of of the Ordinate agrees with that of the Section, then will the value of the Solidity be the fame with that of the Area as found by the faid Propofition. DEMONSTRATION. For, the fluxion of the folid being equal to the fection drawn into the fluxion of the abfcifs, and the fluxion of the area equal to the ordinate drawn into the fame fluxion of the abfcifs, whenever the ordinate and fection agree, the area and folidity must likewife agree, fince equal fluxions have equal fluents. 2. E. D. SCHOLIUM. All the examples that have been given to the corollaries of the laft propofition, may be confidered as examples to this, the quadrature in those being confidered as the cubature here; and it is evident that curves will be cubable not only whenever they are quadrable, but they will often be cubable when the area cannot be expreffed except in an infinite feries, because the fection of a folid, using the area of a given circle as a given number, is often a terminate expreffion, when the ordinate is denoted by an infinite feries; and this is the cafe in the conic fections; for thofe curves, the parabola excepted, are not quadrable, yet all the folids generated by their revolution have finite expreffions, as we have already found in the foregoing part of this work, and as will more generally appear in what follows; which I have fet down not merely to fhew that the fame conclufions may be brought out by different means, but chiefly for the fake of fome eafy, curious, and general rules, with which this method of investigation fo readily fupplies us. COROLLARY. Thus, by applying here what is done in the laft corollary of the laft propofition, to the curves of the fecond kind, we shall obtain general and terminate expreffions for their folidities. For, by prop. 2 fect. 4 part 3, the flowing section being always as A+B+Cxx, if p denote a given quantity, px (a + Bx + cxx) will denote the fection itself; and confequently px × (A + B + cx2) oгp × (AZ + Cz3) will denote the folidity cdpc, generated by the fection of a folid, in flowing from the place cd to the place CD, z being = cc, H F d D B E CKA C LG Since pa reprefents the middle fection AB, the first term paz.in the above expreffion will be equal to the cylinder CFBEC, whofe bafe is equal to the middle fection AB, and altitude cc z; and confequently pcz3 will always be the difference between the folid cd BDC and the cylinder CFBEC. Or if there be taken CG: cc::c: I, and the three areas, viz. the curve çdвDC, the rectangle CFBEC, and the right-angled triangle cGc, be fuppofed to revolve together about the common axe cc; then will the difference between the folid generated by cd BDC, and the cylinder generated by CFBEC, be conftantly equal to one-fourth of the cone generated by CGC, The fruft. generated by cdвDc is greater than equal to or less than according as the an hyperbola a parabola or an ellipfe. the cylinder generated by CFBEC value of c is |