ax2 a2x2 Яexion sought. 323 = la, the ordinate of the point of in EXAM. 2. To find the point of inflexion in a curve defined by the equation ay = a √ ax2 + xx. EXAM. 3. To find the point of inflexion in a curve defined by the equation ay2 = a2x + x3. ABCE D 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 pole of the conchoid, draw any number of right lines PA, PB, FC, PE, &c, cutting the given line FD 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 conchoid; a curve so called by its inventor Nicomedes. P TO FIND THE RADIUS OF CURVATURE OF CURVES. 108. THE Curvature of a Circle is constant, or the same in 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 bu siness of this chapter to find. 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 abciss AD, also de the fluxion of the ordinate DE, and Ee that of the curve AE. Then put x = AD, Y = DE, Z = AE, and r = CE the radius of curvature; then is Ed = x, de = y, and Ee= =2. Now, by sim. triangles, the three lines Ed, de, Ee, are respectively as the three therefore or x, GE, GC, CE; GC GE.y; y and the flux of this eq. is ac x + GĊ • x = GE. y + GE · -BG, it is GC-BG =GE y+GE. y or, because Gċ = But since the two curves AE and BE have the same cur vature 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 ory is the fluxion both of DE and GE. In the equation above therefore substitute for AG, and y for GE, and it becomes Gcx-xx = GFy + yy, 2 or Gcx-GFÿ= x2 + y = 22. Now multiply the three terms of this equation respectively, by these three quantities,,,, which are all equal, the radius of curvature, for all curves whatever, in terms of the fluxions of the absciss and ordinate. 110. Further, as in any case either x or y may be supposed to flow equably, that is, either or y constant quantities, or or 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 is supposed constant, * being then O, the value of r is barely = EXAM. 1. To find the radius of curvature to any point VOL. II. Bb b of of a parabola, whose equation is ax = y2, its vertex being 4, and axis AD. Now, the equation to the curve being ax = y2, the fluxion of it is ar = 2yy; and the fluxion of this again is a = 2y", supposing y constant; hence then r or 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 far, 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 a, the parameter of the axis EXAM. 2. To find the radius of curvature of an ellipse, whose eqation is a2y2 = c2 .ax —x2. EXAM. 3. To find the radius of curvature of an hyperbola, whose equation is a2y2 = c2, ax + x2. EXAM. 4. To find the radius of curvature of the cycloid. Ans. r = 24/aa - ax, where x is the absciss, and a the diameter of the generating circle. OF INVOLUTE AND EVOLUTE CURVES. 11. 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. 112. Thus AE 112. 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 B wound or plied close to the curve, &c. from H 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 evolute. Or, if the thread, fixed H at н, be unwound from the curve, beginning at A, and keeping it always tight, it will describe the same involute ABCD. 113. 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, AEFGH, of the evolute i that is, AE = AE is the radius of curvature to the point a, = 114. 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. Then, by the nature of the radius of curvature, it is T= 3 yx xy = BC = AE + EC; also, by sim. triangles, and FC AD AE + GC = x a+ = 22; yx- xy which are the values of the absciss and ordinate of the evolute curve EC; from which therefore these may be found, when the involute is given. On the contrary, if v and u, or the evolute, be given : then, putting the given curve ECs, since CB = AE + EC, orr = a + s, this gives r the radius of curvature. Also, by similar triangles, there arise these proportions, viz. which are the absciss and ordinate of the involute curve, and which may therefore be found, when the evolute is given. Where it may be noted, that 1 = v2 +ù2, and ¿2 = x2+ y2. Also, either of the quantities x, y, may be supposed to flow equably, in which case the respective second fluxion, ï or y, will be nothing, and the corresponding term in the denominator y- ay will vanish, leaving only the other term in it; which will have the effect of rendering the whole operation simpler. 116. EXAMPLES. EXAM. 1. To determine the nature of the curve by whose evolution the common parabola AB is described. Here |