An elementary course of infinitesimal calculus . C Fig. 123. Ex. 2. The normal at any point of the ellipse a: = acos^, y = 6sin<^ (8) i, - _J^ = „»_6^ (9). COS0 sin^ DiflFerentiating with respect to say (10). cos<^ sin^ ^ Substituting in (9), we have \=a-h\ (11). Hence the coordinates of the centre of curvature are x = cos<^, y = 7— sin<^ (12); Of o and the evolute is (ax)i + {by)i = {a^-b)i (13). 159] CURVATURE. 423 This curve, which may be obtained by homogeneous strain fromthe astroid, is shewn in Fig. Pig. 124. The centres of curvature at the points A, B, A, B areE, F, E,
An elementary course of infinitesimal calculus . C Fig. 123. Ex. 2. The normal at any point of the ellipse a: = acos^, y = 6sin<^ (8) i, - _J^ = „»_6^ (9). COS0 sin^ DiflFerentiating with respect to say (10). cos<^ sin^ ^ Substituting in (9), we have \=a-h\ (11). Hence the coordinates of the centre of curvature are x = cos<^, y = 7— sin<^ (12); Of o and the evolute is (ax)i + {by)i = {a^-b)i (13). 159] CURVATURE. 423 This curve, which may be obtained by homogeneous strain fromthe astroid, is shewn in Fig. Pig. 124. The centres of curvature at the points A, B, A, B areE, F, E, F, respectively. Ex. 3. To find the evolute of a cycloid. At any point P on the cycloid APB (Fig. 125), we have, byArt. 151, Ex. 2, p = 2PI (14). Let the axis AB be produced to D, so that BD = AB; andproduce TI to meet a parallel to BI, drawn through D, in /.If a circle be described on // as diameter, and PI be producedto meet the circumference in P, we have PI=PI, so that P isthe centre of curvature of the cycloid at P. And since the arcPi is equal to the arc TP, and therefore to BI or Dl, thelocus of P is evidently the cycloid generated by the circle IPI,supposed to roll on the under side of DT, the tracing pointstarting from D. That is, the evolute is a cycloid equal to theoriginal cycloid, and having a cusp at D. It appears, further, from Art. 136 (4), that the cycloidal arcPD is equal to 2/P, or PP. Hence arc i>P + PP = const (15). 424 INFINITESIMAL CALCULUS. [CH. X The lower cycloid iu Fig. 125 is therefo
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