EXAMPLES. EXAM. 1. To find the radius of curvature to any point of a parabola, whose equation is ax=y3, its vertex being a, and axis AD. Here, the equation to the curve being ax = y; the fluxion of it is aż = 2yy; and the fluxion of this again is aï = 2ÿ, supposing ý constant; hence then ror for the general value of the radius of curvature at any point E, the ordinate to which cuts off the absciss AD = 1. 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 a, the parameter of the axis. See, also, pa. 535, vol. i. EXAM. 2. To find the radius of curvature of an ellipse, whose equation is a2y2 = c(ax - x2). X [ac2+4(a2-c2) × (ax-x)] Ans. r = 2ac EXAM. 3. To find the radius of curvature of an hyperbola, whose equation is ay2 = c2 (ax + x2). Ans. r = [ac2 + 4(a2+c2) × (ax + x2)] 2a°c EXAM. 4. To find the radius of curvature of the cycloid. Ans. r = 2√(aa - ax), where x is the absciss, and a the diameter of the generating circle. OF INVOLUTE AND EVOLUTE CURVES. 119. 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 wrap ping the thread about the curve of the evolute, keeping it at the same time always stretched. 120. 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 wound or plied close to the curve, &c. from 1 to A, keeping the thread A E D C B always stretched tight; the other end of the thread will describe a F G certain curve ABCD, called an Invo lute; the first curve EFGH being its 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. H 121. If AE, BF, 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; that is, AE = AE is the radius of curvature to the point a, BF = AEF is the radius of curvature to the point B, CG = AEG is the radius of curvature to the point c, DH = AEH is the radius of curvature to the point D. 122. It also follows, from the premises, that any radius of curvature, Br, 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. 123. Hence, and from art. 117, it will be easy to find one of these curves, when the other is given. To this purpose, put x = AD, the absciss of the involute, y = DB, an ordinate to the same, z = AB, the involute curve, r = BC, the radius of curvature, v = EF, the absciss of the evolute EC, u = Fc, the ordinate of the same, and B AED a = AE, a certain given line. F G Then by the nature of the radius of curvature, it is r= C = BC = AE + EC; also, by sim. triangles, ri :::r:GB = iż 23 ; -y 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 EC = s, since CB = AE + EC, or r = a + s, this gives r the radius of curvature. Also, by similar triangles, there arise these proportions, viz. theref. AD = AE + FC-GC = a + u and DB GB - GD = GB-EF= ats. S 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 s2 = v3 + u2, and z2 = i + y2. Also, either of the quantities x, y, may be supposed to flow equably, in which case the respective second fluxion, a or y, will be nothing, and the corresponding term in the denomi.. nator jë - iÿ will vanish, leaving only the other term in it; which will have the effect of rendering the whole operation simpler. EXAMPLES. EXAM. 1. To determine the nature of the curve by whose evolution the common parabola AB is described. Here the equation of the given evolute AB, is cx = y where c is the parameter of the axis AD. Hence then く by making i C constant. Consequently the general values of v and u, or of the absciss and ordinate, EF and Fc, above given, become, -ży But the value of the quantity a or AE, by exam. 1 to art. 118, was found to be c; consequently the last quantity, Fc or u, is barely = 3x. Hence then, comparing the values of v and u, there is found 30/c = 4ux, or 27cv2 = 16u2; which is the equation between the absciss and ordinate of the evolute curve EC, showing it to be the semicubical parabola. EXAM. 2. To determine the evolute of the common cycloid. TO FIND THE CENTRE OF GRAVITY. 124. By referring to art. 108, &c. in the Statics, it is seen what are the principles and nature of the Centre of Gravity in any figure, and how it is generally b = the whole body or figure ABD; then the distance ac, of the centre of gravity below ro, is universally denoted by sum of all the ds b whether ABD be a line, or a plane surface, or a curve superficies, or a solid. . But the sum of all the ds, is the same as the fluent of db, and b is the same as the fluent of b; therefore the general expression for the distance of the centre of gravity, is ac = fluent of ri fluent of b AG. fluent b b ; putting x = d the variable distance Which will divide into the following four cases. 125. CASE 1. When AE is some line, as a curve suppose. In this case 6 is = żori + y2, the fluxion of the curve; and bz: theref. AC = = ż fit is the distance of the centre of gravity in a curve. 126. CASE 2. When the figure ABD is a plane; then b = y; therefore the general expression becomes Ac fyri for the distance of the centre of gravity in fyi a plane. 127. CASE 3. When the figure is the superficies of a body generated by the rotation of a line AEB, about the axis AH. Then putting = 3.14159, &c. 2ry will denote the circum. ference of the generating circle, and 2ryż the fluxion of the surface: therefore AC fluent of 2xyxz_fyxz will be the fyż = fluent of 2ryż distance of the centre of gravity for a surface generated by the rotation of a curve line z. 128. CASE 4. When the figure is a solid generated by the rotation of a plane ABH, about the axis AH. Then is my = the area of the circle whose radius is y, and wyx = b, the fluxion of the solid; therefore of the centre of gravity below the vertex in a solid. 1 EXAMPLES. EXAM. 1. Let the figure proposed be the isosceles triangle ABD. It is evident that the centre of gravity o, will be some. where in the perpendicular AH. Now, if a denote AH, C = BD, X = AG, and y = EF any line parallel to the base BD: then as a: c::x: y = ; therefore, by the 2d Case, AC = fluent yxi fluent xi 23 G = = AH, when z becomes = AH : consequently CH = AH. |