CURVATURE AND EVOLUTES. 117. To find the radius R of curvature to the common cycloid, whose equation is y = sin." x + ✓ 2rx – x2 dy? dir? R = ( (1 + 24 = 1) a which reduces to R = 2 x 2 a x 2a – x. 118. Required the chord of curvature parallel to the axis, of the common parabola, whose equation is yo = px. dır? dy? X 1+ dx2 R = -doy ( But dy R= 119. By the preceding problem we learn that 1 (4.x + p). 2 p Again, in the expressions 2 – 0 + (y B) dy = 0, dx (See Appendix to dy? døy d.ru Simpson's Fluxions or Lacroix). a, 8 are the co-ordinates of the centre of curvature, and .. of the evolute of a curve; and if we can eliminate from them y the co-ordinates of the curve, the result will be the equation to the evolute. But since y? = px, by substituting for dy, dạy hence obtained, we easily get dx dx2 4y y – B = 4x + piva р • y, and .. x р P. and X, 2 .tp 2.1 2 120. By the question, the equation to the curve is dr = r sin. y dy dox and = q cos. y. dy? 121. bola is By problem 118 the radius of curvature of the para R = 2NP (4 x + p), and the equation to the parabola :. at the vertex of the parabola, where x = 0, 123. To find the evolute of the common cycloid, whose equation is y = vers. * * + N 2rx 2r x ✓ 2rx- x" . 4r): the B + 2 y 2r3 za = vers."x + N 2rx :: B = vers." (a 4r) ✓ 2r (« - 4r) equation to the evolute, which is therefore a cycloid equal to the curve itself, but having its bake I base of the given cycloid at either extremity. CONTRARY FLEXURE. 00000000000000000 124. To find the points of contrary flexure of a curve, whose equation is x = (ly)', we have 3ly (2– ly) = 0 or 6, by the rule, which gives dy? ly= 2, or x= (ly) = 8, and y = e', which are the co-ordinates of the point required. 125. If R be the radius of the wheel, r the distance of the generating point from its centre, x the abscissa of the trochoid, measured from the vertex or highest point of it; then the equation to the curve is R x + 2rx -- x? (1) vers, dy R + g - 2rx 22 (2rx — 2) Rx (R + r) r = 0 or 2rx X2 and we :. have two points of contrary flexure, whose abscissæ are (R + r) Ř and 2r, R vers. * 2r reR spectively, the latter point being a ceratoid, as we learn by substituting in equation 2. |