. An elementary treatise on the differential and integral calculus. ing string, are said to be parallel; each curve being gotfrom the other by cutting off a constant length on it?normal, measured from the involute. (Williamsons Dif-ferential Calculus, p. 295.) 125. To find the Equation of the Evolute of 3 Given Curve.—The co-ordinates of the centre of curva-ture are the co-ordinates of the evolute (Art. 124). Hence,if we combine (11) and (12) of Art. 119 with the equationof the curve, and eliminate x and y, there will result anequation expressing a relation between m and n, the co-or- 228 EXAM
. An elementary treatise on the differential and integral calculus. ing string, are said to be parallel; each curve being gotfrom the other by cutting off a constant length on it?normal, measured from the involute. (Williamsons Dif-ferential Calculus, p. 295.) 125. To find the Equation of the Evolute of 3 Given Curve.—The co-ordinates of the centre of curva-ture are the co-ordinates of the evolute (Art. 124). Hence,if we combine (11) and (12) of Art. 119 with the equationof the curve, and eliminate x and y, there will result anequation expressing a relation between m and n, the co-or- 228 EXAMPLES, dinates of the required evolute, which is therefore therequired equation; the method can be best illustrated byexamples. The eliminations are often quite difficult; the followingare comparatively simple examples. EXAMPLES. 1. Find the equation of the evolute of the parabola, Here *y=p. y f (i) X = m — p3 y2 4- p2 ysn — y — V—~- • -2. Fig. 39. P 2> 2 i And these values of x and y in (1) give,pin* = %p(m—p); n* = 2^(m ~PY> (2) which is the equation required, and is called the semi-cubicalparabola. Tracing the curve, we find its form as given inFig. 39, where AO = we transfer the origin from O to A, (2) becomes n2 = -=r— mz. 27p EXAMPLES. 229 2. Find the length of the evolute AQ, Fig. 39, in termsof the co-ordinates of its extremities. Let ON = x, NQ = y; ON = m, NQ = by Art. 123, Ex. 2, we have, r = j* Therefore, by Art. 124, we have,Length of AQ = QQ - AO = ^ \P^ -p= (n% -f p%)i — p. (Since y2 = phi%, by Ex. 1.) 3. Find the equation of the evolute of the cycloid,x = r vers-1 - — V%ry - if. (i) Here ^ dy y%ry — y2 dx ~ y dx2 or m = x -f 2 V%ry — y2 and nx = m — 2 a/— 2ni — n2 and : - n; which, in the equation of the cycloid, gives (n\ Y — -) + V—2rn — n2,(2) which is the equation of a cycloidequal to the given cycloid; theorigin being at the highest point, oThis will appear
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