An elementary course of infinitesimal calculus . be the angle which IZmakes with the normal to therolling curve at /, and // = Ss,we have ultimately Bs cos ^ ??. Fig. 139. (1). 444 INFINITESIMAL CALCULUS. [CH. X Hence, substituting the value of SO from Art. 164 (1),we have cos ^ _ 1 1 , . -W-R^R ^^^- The radius of curvature of the envelope is then given by p = GI + IZ (3)*. If, along the normal to the rolling curve at /, but in thedirection opposite to that chosen in the preceding Art., wemeasure off a length IK such that IK~R^R ^ - and describe a circle on this line as diameter, it appe
An elementary course of infinitesimal calculus . be the angle which IZmakes with the normal to therolling curve at /, and // = Ss,we have ultimately Bs cos ^ ??. Fig. 139. (1). 444 INFINITESIMAL CALCULUS. [CH. X Hence, substituting the value of SO from Art. 164 (1),we have cos ^ _ 1 1 , . -W-R^R ^^^- The radius of curvature of the envelope is then given by p = GI + IZ (3)*. If, along the normal to the rolling curve at /, but in thedirection opposite to that chosen in the preceding Art., wemeasure off a length IK such that IK~R^R ^ - and describe a circle on this line as diameter, it appearsfrom (2) that G lies on this circle ; in other words, the locusof the centres of curvature of all line-roulettes, in any givenposition of the rolling curve, is a circle. Also, when thecarried line passes through K, Z coincides with G, and (7 is a stationary point (Art. 150) on the envelope. The aforesaidcircle is therefore called the circle of cusps. Ex. 1. Regarding a cycloid as the envelope of the diameterof a circle which rolls on a fixed straight line (Art. 164, Ex. 1),we infer that the radius of curvature is double the normal. Ex. 2. If an epicycl
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