or towards the axis AD, from A to E, but convex from E to F; and on the contrary, the latter figure is convex from a to E, and concave from E to F. 71. From the nature of curvature, as has been remarked before at art. 28, it is evident, that when a curve is concave towards an axis, then the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the absciss; but, on the contrary, that it increases, or is in an increasing ratio to the fluxion of the absciss, when the curve is convex towards the axis; and consequently those two fluxions are in a constant ratio at the point of inflexion, where the curve is neither convex nor concave; that is, a is. j à to in a constant ratio, or is a constant quantity. j But constant quantities have no fluxion, or their fluxion is equal to nothing; so that in this case, the fluxion of or of is equal to nothing. And hence we have this general rule: 72. Put the given equation of the curve into fuxions ; 3 from which find either Then take the fluxion of this ratio, or fraction, and put it equal to 0 or nothing; and j from this last equation find also the value of the same j Then put this latter value equal to the former, which will form an equation; from which, and the first given equation of the curve, x and y will be determined, being the absciss and ordinate answering to the point of inflexion in the curve, as required. . or X EXAMPLES. Exam. 1. To find the point of inflexion in the curve whose equation is ar? = a'y + xy. This equation in fluxions is 2axi = a’j + 2.xyz + x?), * a’ +.x2 which gives j=2ax – 2xy Then the fluxion of this quantity made = 0, gives 2ri (ax-xy) = (a? + x) * (at - öy- xy); a2 + x2 and this again gives j al y * Lastly, this value of being put equal the former, gives Х 22 .2 . $a? a + .22 and hence 2.r2 = a- , ? - za y у a = =fa, the ordinate of the point of in. a2 + x2 flexion sought. Exam. 2. To find the point of inflexion in a curve defined by the equation ay = ax ax? + xr. EXAM. 3. To find the point of inflexion in a curve defined by the equation aya = ar t.. EXAM. 4. To find the point of inflexion in the Conchoid of Nicomedes, which is generated or constructed in this manner: From a fixed point P, which is called the HGH pole of the conchoid, draw any number of right lines PA, PB, PC, PE, &c, cutting P the given line fp in the points F, G, H, I, &c: then make the distances FA, GB, HC, IE, &c, equal to each other, and equal to a given line; then the curve line ABCE &c, will be the conehoid; a curve so called by its inventor Nicomedes. in TO FIND THỂ RADIUS of CURVATURE OF CURVES. 73. The Curvature of a Circle is constant, or the same every point of it, and its radius is the radius of curvature. But the case is different in other curves, every one of which has its curvature continually varying, either increasing or decreasing, and every point having a degree of curvature peculiar to itself; and the radius of a circle which has the same curvature with the curve at any given point, is the radius of curvature at that point; which radius it is the business of this chapter to find. 74. Let Abe be any curve, concave towards its axis Al); draw'an ordinate pe to the point E, where the curvature is to be found; and suppose Ec perpendicular to the curve, and equal to the radius of curvature sought, or equal to the radius of a circle having the same curvature there, and with that radius describe the said equally curyed Then put curved circle Bee; lastly, draw Ed parallel to AD, and de parallel and indefinitely near to DE: thereby making Ed the fluxion or increment of the absciss AD, also de the fluxion of the ordinate DE, and Ee that of the curve AE. X = AD, Y = DE, z = AE, and r = ce the radius of curvature; then is Ed = , de = j, and Ee = ż. Now, by sim. triangles, the three lines ed, de, El, or i, j, 2, are respectively as the three GE, GC, CE; therefore GC. = GE.j; and the flux. of this eq. is Gc.č + Gc.* = GE. y. + GE.), or, because Gc= - BG, it is Gc.i - BG .; =ge.y + Ge.j. But since the two curves AE and be have the same curvature at the point E, their abscisses and ordinates have the same fluxions at that point, that is, Ed or is the fluxion both of AD and BG, and de or j is the fluxion both of DE In the equation above therefore substitute å for BG, and ġ for ge, and it becomes GCZ * = GFŰ + jj, or gcä – GFj = til + y2 = 2. Now multiply the three terms of this equation respectively у * ż by these three quantities, Gc which are all equal, 23 and it becomes jä and GE. GE СЕ or and hence is found r = for the general value of the radius of curvature, for all curves whatever, in terms of the fluxions of the absciss and ordinate. 75. Further, as in any case either x or y may be supposed to flow equably, that is, either å or y constant quantities, or ä or j equal to nothing, it follows that, by this supposition, either of the terms in the denominator, of the value of r, may be made to vanish. . Thus, when i is supposed constant, ä being then = 0, the value of r is barely 23 yx EXAMPLES Exam. 1. To find the radius of curvature to any point of of a parabola, whose equation is ax =y', its vertex being A, and axis AD. Now, the equation to the curve being ax=y', the fluxion of it is ač = 2yj; and the fluxion of this again is aä = 2ją, supposing j constant; hence then r or (*? + ja) or or (a? + 4y2) (a + 4.x) is 2a? 2a for the general value of the radius of curvature at any point E, the ordinate to which cuts off the absciss AD = x. Hence, when the absciss x is nothing, the last expression becomes barely a = r, for the radius of curvature at the vertex of the parabola; that is, the diameter of the circle of curvature at the vertex of a parabola, is equal to aj the parameter of the axis. Exam. 2. To find the radius of curvature of an ellipse, whose equation is a?y* = 62. ax - r. (de* +4 (02 – (2) (az - x2){ Ans. g = 2a4c EXAM. 3. To find the radius of curvature of an hyperbola, whose equation is a'y? = c.ax + x?. Exam. 4. To find the radius of curvature of the cycloid. Ans. r = 2vaa - or, where x is the absciss, and a the diameter of the generating circle. OF INVOLUTE AND EVOLUTE CURVES. 76. An Evolute is any curve supposed to be evolved or opened, which having a thread wrapped close about it, fastened at one end, and beginning to evolve or unwind the thread from the other end, keeping always tight stretched the part which is evolved or wound off: then this end of the thread will describe another curve, called the Involute. Or, the same involute is described in the contrary way, by wrapping the thread about the curve of the evolute, keeping it at the same time always stretched. . 77. Thus 77. Thus, if EFGH be any curve, and AE be either a part of the curve, or a right line : then if a thread be fixed to the curve at H, and be BA wound or plied close to the curve, &c, from # to A, keeping the thread always stretched tight; the other end of the thread will describe a certain curve ABCD, called an Involute; the first curve EFGH being its H evolute. Or, if the thread, fixed at H, be unwound from the curve, beginning at A, and keeping it always tight, it will describe the same involute ABCD. 78. If AE, DF, CG, DH, &c, be any positions of the thread, in evolving or unwinding; it follows, that these parts of the thread are always the radii of curvature, at the corresponding points, A, B, C, D, and also equal to the corresponding lengths AE, AEF, AEFG, A EFGH, of the evolute; that is, BF AE = AE is the radius of curvature to the point a, = AF is the radius of curvature to the point B, = AG is the radius of curvature to the point c, DH = AH is the radius of curvature to the point D. CG 79. It also follows, from the premises, that any radius of curvature, bf, is perpendicular to the involute at the point B, and is a tangent to the evolute curve at the point F. Also, that the evolute is the locus of the centre of curvature of the involute curve. 80. Hence, and from art. 74, it will be easy to find one of these curves, when the other is given. To this purpose, put * = AD, the absciss of the involute, y = DB, an ordinate to the same, % = AB, the involute curve, g = BC, the radius of curvature, the absciss of the evolute EC, 4 = FC, the ordinate of the same, and & = AE, a certain given line. V = EF, Then, |