. Differential and integral calculus, an introductory course for colleges and engineering schools. ns sin 0, cos 0, tan 0, cot 6, sec 0, esc 0 can be expressed as a rational function of a single parameter t wheret= tan|0. Problem 2. Point out the curves of Art. 91 that are rational. Problem 3. Observe that in each of the exercises of Art. 91 in whichy = tx, the origin is a multiple-point of order one less than the degreeof the x- and ^/-equation of the curve. What general principle can beinferred from this fact ? * Equations (a), (b), (c) can be derived one from another: thus, the sub-stitutio
. Differential and integral calculus, an introductory course for colleges and engineering schools. ns sin 0, cos 0, tan 0, cot 6, sec 0, esc 0 can be expressed as a rational function of a single parameter t wheret= tan|0. Problem 2. Point out the curves of Art. 91 that are rational. Problem 3. Observe that in each of the exercises of Art. 91 in whichy = tx, the origin is a multiple-point of order one less than the degreeof the x- and ^/-equation of the curve. What general principle can beinferred from this fact ? * Equations (a), (b), (c) can be derived one from another: thus, the sub-stitution of sec 0 for t in (a) or of tan|0 for t in (b) gives (c). CHAPTER XIICYCLOIDAL CURVES 93. The Cycloid. This is the curve described by a fixed pointon the circumference of a circle, as the circle rolls along a straightline. It is obvious that the curve consists of an unlimited numberof equal arches. Let the fixed line be the axis of x, and let the origin be one ofthe points of contact of the generating point, P, with the fixedline. Let a be the radius of the generating circle, and let x and y. be the coordinates of P when the circle has turned through anangle 0. Then x = ON - PM and y = a - ON = arc PN = ad, PM = a sin 0, MC = a cos 6. Therefore, x = a(6 — sin 6), y = a(l — cos 6),and these are the parametric equations of the cycloid. Problem 1. Determine Dxy, and show that 4> = , and thattherefore PN is the normal to the curve at P. From this show that thetangent passes through T, the highest point of the generating circle. a Show also that = \ (t — 6) and that Dx2y = - 128 V1 94 CYCLOIDAL CURVES 129 Problem 2. If the origin be taken at H, the highest point of the cycloid,and if we write d = w+ d, show that the equations of the cycloid arex= a(d +sin0)j y= a( — l+cosd).Problem 3. Obtain the x- and ^/-equation of the cycloid. 94. The Epi- and Hypo-cycloids. When a circle rolls on afixed circle, a fixed point in the circumference of the rolling circledesc
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