dx 1 <£r2 a (sin 0)3' for all values of 0 between 0 and j., is negative, and the curve is concave downwards; and for all values of 0 between 7T 77 — aud w, the curve is convex downwards; and when 6 — -, that is, when y = —, and x = a, there is a point of inflexion; see fig. 41. In a similar manner it may be shewn that the logarithmic curve and the catenary are both convex downwards. 243.] And now let us consider the results of the preceding Articles from a geometrical point of view; and with this object let us examine fig. 56; wherein a curve is drawn, the infinitesimal arcs of which are assumed to be infinitesimal straight lines, and which in the figure are infinitely magnified. The . . d*y curve is imagined and placed so that ^| may be positive. Let y = f(x) be the equation to the curve, Ok = x, Kf = y, and let us consider x to be equicrescent, so that Kl = Lm — Mn = = dx, and d2x -- d3x = = 0; and let p, Q, a, s be points on the curve corresponding to the successive values of x; and if we conceive the successive elements of the curve, Pq, Qr, Rs, to be infinitely magnified, then they are such straight lines as we have imagined ds to be in Art. 218. In the same manner then as in Arfr. 68, f(x) = y = Kp, f(x + dx) = y + dy = Lq, f(x + 2dx) = y + 2dy + d*y = Mr, f(x + 3dx) - y + 3dy + 3d2y + d3y = Ns, and so on; whence, by subtraction, we have dy = LQ —KP = QU = XV = TW; and Mr = Mz + Rz: but Mz = Mv + 2vx = y + 2dy; As we have not deduced any geometrical properties from d3y, it is unnecessary to do more than to shew what line is represented by it. From above we have Ns = y + 3dy + 3d2y + d3y: but Nw — y, Wg = 3dy, Ge = 2d2y, Se = d2y + d3y, - RZ + d3y; d3y = Se —Rz. Hence it is manifest that = — = tan Qpu = tan Pjo dx Pu = the trigonometrical tangent of the angle made with axis of x by the tangent to the curve; and therefore if at any point on du a curve T£ = 0, y does not increase as we pass from one point to the next consecutive point, and therefore the element of the curve Pq is along the line Pu, and is parallel to the axis of x. dy Similarly, whenever = ao , the element Pq is perpendicular to Pu, and therefore parallel to the axis of y. And since d2y = Rz, it is plain that d2y represents the deflexion of the curve from the tangent line: and therefore if d*y = 0, three consecutive points are in the same straight line, and the curve has for those three points become straight; and if d2y is positive, the line Rz is to be measured up from the tangent, and the curve lies above the tangent: but if d2y is negative, it is to be measured downwards, and the curve lies below the tangent; if therefore d2y is positive on both sides of the point P, the curve is convex downwards, and is such as is drawn in fig. 53, and if d2y is negative on both sides of the point (x, y), then the curve is concave downwards as in fig. 54; and if d2y changes its sign at the point by passing through 0 or oc, the curve is above the tangent on one side of the point, and below it on the other, and therefore there is a point of inflexion. Hence we learn the geometrical meaning of the process of differentiation; it implies a passage from one point of a curve to the next consecutive point; aud, as often as we differentiate, we pass to successive points, and obtain expressions which represent deflexions from straight lines, and so on. Thus, by means of one differentiation, we consider the curve with respect to two points on it; by two differentiations we consider the curve at three points, and so on. More will be said hereafter on the properties of curves under this mode of considering them; and especially in Chapters XII and XIII with respect to several successive points being common to two or more curves. 244.] Let us however consider the direction of curvature and the conditions of points of inflexion when the equation to the curve is given in the implicit form, and in the more general forms, (48) and (49), of Arts. 207 and 208. First, let the equation to the curve be given in the form (48) of Art. 207; viz. V(X,y) = «„ + «„-! + ... + Ml + «0 = 0. (108) then by equation (122), Art. 84, x being equicrescent, /d2v\ (dv\2 / d2F \ (dr\ (dr\ (dif \ d2y __ ~ \dx2> \dy) + \dxdyl W \dyl ~\dy~2' \dx> __ _ . . (109) w and therefore at all points at which the right-hand member of this equation is positive, the curve is convex downwards; and at all points for which it is negative, the curve is concave downwards. And if at any point on the curve id2¥\ idT?\2 . / d2r \ /dv\ /dr\ id2v \ /dr\2 „ fe) (dp -2 (d^dy) y w+M y=°< (u°) and changes sign, and if at the same point the curve is not parallel to the axis of y so that does not vanish, then there is a point of inflexion. The condition therefore for a point of inflexion is primarily (110). Now this equation is in its present form evidently of 3re —4 dimensions in terms of x and y: it expresses therefore a curve of that order: the intersections of which with (108) are the points of inflexion: and, as (108) is of a dimensions, it follows that the number of points of inflexion on a curve of the reth degree cannot exceed re (3 re—4). 245.] And we can shew that 2 re of these are at an infinite distance, so that the number of points of inflexion on a curve within a finite distance does not exceed 3re(w —2). Let u„ = 0 be the equation of all the asymptotes of (108) determined by the process of Art. 239; then if u„_2, u„_3,... u0 are homogeneous functions of re — 2, re—3,... 0 dimensions respectively, equation (108) may be expressed in the form the terms of re — 1 dimensions being contained in u„. Let us take the equation in this form to be the subject of the operations which are indicated in (110). As u„ = 0 is the equation of all the rectilinear asymptotes of the curve, u„ is the product of re linear factors, each of which is of the form ax + by + c = 0; so that where ai = a\X + b\y +Cj, and p is the product of the other re —1 linear factors. Now if we take of this the x- and impartial derived-functions, (113) where Q is the symbol for a quantity which it is unnecessary to express at length; but ai is a factor of the result; and for a similar reason every other linear factor of the right-hand member of (112) will enter into the result; and (110) will take the form _ ,,., ., u» v2n_4 + vSn_6 = 0; (114) where v2„_4 and v3n_6 are two functions of x and y of 2n —4 and 3«—6 dimensions respectively. Now the points of inflexion are the points common to the two curves whose equations are (111) and (114); and if we eliminate u„ between these two, the resulting equation will be of 3n — 6 dimensions; that is, the number of points of intersection of this curve with the original curve will be only 3n(n—2), and this will be the number of the points of inflexion at a finite distance. The 2« other points of inflexion are at an infinite distance; for u„ = 0 is the equation of the n rectilinear asymptotes of the curves (108) or (111) and (114): both curves therefore have the same n asymptotes, and therefore there are 2m points at infinity common to the two curves. Hence it appears that in conies there is no point of inflexion at a finite distance, but there may be four points of inflexion at infinity; of these we have instances in the hyperbola, where the asymptotes are intersected by the infinite branches. Again, curves of the third order may have fifteen points of inflexion, of which six are at infinity, and of the remaining nine, one at least must be real. 246.] As the curve whose equation is (110) has many important properties in reference to not only points of inflexion but also other singular points on the original curve, so it has been found necessary to give it a particular name: and as its use was first indicated by M. Otto Hesse of Koenigsberg*, so it has been called by Mr. Sylvester the Hessian of the curve (108); we shall refer to it hereafter by that name. If the equation to the original curve is expressed in terms of trilinear coordinates, in the form (49), Art 207, the equation to the tangent is, see Art. 216, At a point of inflexion the position of the tangent does not change, as we pass from the second to the third consecutive * See Crelle, Vbl. XXVIII, p. 104. |