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 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. Then put x == AD, y = DE, Z = AE, and r = CE the radius of curvature ; then Ed = i, de = ý, and Ee = ż. Now, by sim. triangles, the three lines Ed, de, Ee, which vary as are respectively as the three therefore and the flux. of this eq. is Gc.z+GC.i=GE. Y + GE.ý, or because GC = -BG, it is GC.1-BG.i=GE.Y+GE. . 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 i is the fluxion both of AD and BG, and de or y ∝ the fluxion both of De and GE. In the equation above therefore substitute i for BG, and y for GE, and it becomes GCZ - it = Grÿ + ÿÿ, or GC2 - GFY = 2 + j2 = 22. Now multiply the three terms of this equation respectively ýżż by these three quantities, ' ' ' , which are all equal, and radius of curvature, for all curves whatever, in terms of the fluxions of the absciss and ordinate. 118. Further, as in any case either z or y may be supposed to flow equably, that is, either i or y constant quantities, or or y equal to nothing, it follows that, by this supposition, either of the terms of the denominator, of the value of r, may be made to vanish. Thus, when i is supposed constant, being then = 0, the value of r is barely EXAMPLES. EXAM. 1. To find the radius of curvature to any point of a parabola, whose equation is ax=y', its vertex being a, and axis AD. Here, the equation to the curve being ax = y3, the fluxion of it is aż = 2yy; and the fluxion of this again is az = 2ÿ, supposing y 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 = 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 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 = c2(ax - x2). EXAM. 3. To find the radius of curvature of an hyperbola, whose equation is ary = c(ax + x2). Ans. r = [ac2 + 4(a2 + c2) × (ax + x2)] 2a'c 3 2 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, fas. tened 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. B C D 120. Thus, if EFGH be any curve, and An 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 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 at H, be unwound from the curve, beginning at a, and keeping it always tight, it will describe the same involute ABCD. AE F G 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 ΑΕ, 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, 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. 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 a = AE, a certain given line. B AED F G Then by the nature of the radius of curvature, it is C 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 = 8, 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 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 = v2+ u2, and 2 = 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 denominator ÿä - ży 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 constant. Consequently the general values of v and u, or of the absciss and ordinate, EF and rc, above given, become, 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 = 3. Hence then, comparing the values of v and u, there is found 30/c = 4ux, or 27cv2 = 16u3; which is the equa tion 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. Ans. another cycloid, equal to the former. 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 expressed. It there appears, that if PAQbealine, ora plane, drawnthrough any point, as suppose the vertex of any body, or figure, ABD, and if s denote any section EF of the figure, d = AG, its distance below ro, and P AQ E F G C b = the whole body or figure ABD; then the distance ac, of the centre of B H D sum of all the ds gravity below ra, is universally denoted by b whether ABD be a line, or a plane surface, or a curve super. ficies, 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 ro fluent of b AG. fluent xb b ; putting x = d the variable distance Which will divide into the following four cases. |