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 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 EFCH being its evolute. Or, if the thread, fixed B H 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, AEFGH, of the evolute; that is, BF AE = AE is the radius of curvature to the point A, 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 x= AD, the absciss of the involute, r = BC, the radius of curvature, and B D Then, 1 Then, by the nature of the radius of curvature, it is = BC = AE + EC; also, by sim. triangles, 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 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 s2 = v2 + u2, and ż2 = *2 + ÿ2. Also, either of the quantities x, y, may be supposed to flow equably, in which case the respective second fluxion, or ÿ, will be nothing, and the corresponding term in the denominator jÿ will vanish, leaving only the other term in its which will have the effect of rendering the whole operation simpler. 81. EXAMPLES. EXAM. 1. To determine the nature of the curve by whose evolution the common parabola AB is described. Here Here the equation of the given involute AB, is cr='y2 where is the parameter of the axis AD. Hence then y = √/cx, and j = {*√√ ——, also ÿ = ✔ - by making 4x * constant. Consequently the general values of v'and u, or of the absciss and ordinate, EF and FC, above given, become, in that case, Hence then, comparing the values of v and u, there is found 3vc 4ux, or 27cv2 16u3; 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. Ans. another cycloid, equal to the former. TO FIND THE CENTRE OF GRAVITY. 82. By referring to prop. 42, &c, in Mechanics, 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 PAQ be a line, or plane, drawn through 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 PQ, and b the whole body or figure ABD; then the distance AC, of the centre of P E G H whether ABD be a line, or a plane surface, or a curve super ficies, or a solid. But 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 rb fluent ri fluent of b AG. b ; putting rd the variable distance Which will divide into the following four cases. 83. CASE 1. When AE is some line, as a curve suppose. b is = ż or √x2 +2, the fluxion of the curve; In this case is the distance of the centre of gravity in a curve. 84. CASE 2. When the figure ABD is a plane; then Z : = y; therefore the general expression becomes AC = fluent of yrx for the distance of the centre of gravity in a fluent of yx plane. 85. 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 c = 3.14159 &c, 2cy will denote the circumference of the generating circle, and 2cyż the fluxion of the fluent of 2cyxz_fluent of yaz surface; therefore AC = fluent of 2cyż fluent of y be the distance of the centre of gravity rated by the rotation of a curve line z. will for a surface gene 86. CASE 4. When the figure is a solid generated by the rotation of a plane ABH, about the axis AH. Then, putting c = 3.14159 &c, it is cy2 the area of the circle whose radius is y, and cy2 = 6, the fluxion of the solid; therefore the distance of the centre of gravity below the vertex in a solid. 87. EXAMPLES. EXAM. 1. Let the figure proposed be the isosceles triangle ABD. It is evident that the centre of gravity C, will be someVOL. II. Ꮓ where 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 CX a:c: x y = ; therefore, by the 2d a = 3 x AH. fluent y AH, when becomes AH: consequently CH = In like manner, the centre of gravity of any other plane triangle, will be found to be at of the altitude of the triangle; the same as it was found in prop. 43, Mechanics. EXAM 2. In a parabola; the distance from the vertex is 3r, or of the axis. cr EXAM. 3. In a circular arc; the distance from the centre of the circle, is ; where a denotes the arc, c its chord, and r the radius. a EXAM. 4. In a circular sector; the distance from the centre 2cr of the circle, is : where a, c, r, are the same as in exam. 3. 3a EXAM 5. In a circular segment; the distance from the. centre of the circle is ; where c is the chord, and a the area, of the segment. 12a EXAM. 6. In a cone, or any other pyramid; the distance from the vertex is 2, or 4 of the altitude. EXAM. 7. In the semisphere, or semispheroid; the distance from the centre is r, or of the radius; and the distance from the vertex of the radius. I EXAM. 8. In the parabolic conoid; the distance from the base is x, or of the axis. And the distance from the vertex of the axis. EXAM. 9. In the segment of a sphere, or of a spheroid; of the segment, and a the whole axis, or diameter of the sphere. EXAM. 10. In the hyperbolic conoid; the distance from the base is 2ax 6a4; where x is the height of the conoid, 4x and a the whole axis or diameter. PRACTICAL |