. Applied calculus; principles and applications . The involute of the circle isgiven by the equations, X = a(cos^ + 0sin0),y = a{smd — 0cos^). AP is the arc of an involute of the circle. 3. The cycloid x = a vers~^ (y/ct) ^ ^2 ay — y^. dy _ V2 ay — y^ (Py _ adx y dx^ y^ Substituting these values in (3) of Art. 94: 2/=-ft x = a = 2 V- 2ai8-i32; .-. a = avers-i (-/3/a) db V -2al3- 13^. (1) The locus of (1) is another cycloid equal to the given one, the EQUATION OF THE EVOLUTE 147 highest point being at the origin; that is, the evolute of acycloid is an equal cycloid. Thus, the evolute of the arc
. Applied calculus; principles and applications . The involute of the circle isgiven by the equations, X = a(cos^ + 0sin0),y = a{smd — 0cos^). AP is the arc of an involute of the circle. 3. The cycloid x = a vers~^ (y/ct) ^ ^2 ay — y^. dy _ V2 ay — y^ (Py _ adx y dx^ y^ Substituting these values in (3) of Art. 94: 2/=-ft x = a = 2 V- 2ai8-i32; .-. a = avers-i (-/3/a) db V -2al3- 13^. (1) The locus of (1) is another cycloid equal to the given one, the EQUATION OF THE EVOLUTE 147 highest point being at the origin; that is, the evolute of acycloid is an equal cycloid. Thus, the evolute of the arc OPiis the arc OCi, which equals Pix; and the evolute of Pix isCix, which equals R = 2 V2^ (3, Exercise XII), CiPi = 4 a. ThenOPiX = 2. OCi = 2 . CiPi = 8a. Hence, the length of one branch of the cycloid is eight timesthe radius of the generating circle. (See Example 3, ) If the figure shown be inverted, the principle of the cy-cloidal pendulum may be perceived. A weight, suspendedfrom Ci by a flexible cord, may be made to oscillate in thearc OPiX, by means of some surface shaped hke the arcsCiO and CiX causing a continuous change in the length of thecord as it comes in contact with the surface. The cycloidalpendulum is isochronal, as the time of an oscillation isindependent of the length of the arc. (See Art. 237.) 4. The hyperbola 6V — a^y^ = a^^. (aa)^ - (6/3)t = (a + h)\ 5. The equilateral hyperbola 2xy = d^. (a + ©^- (a-^)3 = 2al 6. Find the length of an arc of the evolute of the parabola^2 _ 4 p^ in terms of the abscissas of its extremities. 148 DIFFERENTIAL CALCULUS Arc AC = CP-AO = ^(^ + ?^) -2p (Example 1, figure) 7. Show that in the catenary y = a/2 \e^ + e °/, a = X — y/a Vy^ — a^, ^ = 2y. 8. Find the equation of the evolute of the hypocycloidX3 -\-y3 = a^, (Q: + i8)3 + (a - ^)i = 2ai CHAPTER VII. CHANGE OF THE INDEPENDENT
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