# An Elementary Treatise on Cubic and Quartic Curves

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1901 edition. Excerpt: ...points of inflexion. We shall now prove that the maximum number of real points of inflexion is eight. Let 0 be a node on a curve; then it follows from 46 and 85 (i) that 0 is a node on the Hessian, (ii) that the nodal tangents at 0 are common to the curve and its Hessian, (iii) that the curve and its Hessian intersect in six coincident points at 0. Hence each nodal tangent is equivalent to three stationary tangents. If 0 is a conjugate point, all six tangents are imaginary; hence a conjugate point reduces the number of imaginary points of inflexion by six. If 0 is a real cusp, the curve and its Hessian intersect in eight coincident points at 0; hence the cuspidal tangent is equivalent to six imaginary and two real stationary tangents. It therefore follows that a cusp reduces the number of imaginary points of inflexion by six and the number of real ones by two. If the cusp becomes a crunode, two of the imaginary stationary tangents move away to some other points on the curve, and each nodal tangent is equivalent to one real and two imaginary stationary tangents. Hence a crunode reduces the number of imaginary points of inflexion by four and the number of real ones by two. If a node or a cusp is imaginary, all the tangents are imaginary; but since imaginary singularities occur in pairs, it follows that a pair of imaginary nodes or cusps reduces the number of imaginary points of inflexion by twelve and sixteen respectively. 177. To prove that a quartic cannot have more than eight real points of inflexion. We have already shown that a crunode reduces the number of real points of inflexion by two; hence a real biflecnode reduces the number by four. Now if it were possible for a quartic to have ten real points of inflexion, the fourteen...