An elementary course of infinitesimal calculus . contact withthe fixed line, and the point Z will move as if it were carried bythe small circle. Its locus is therefore a cycloid. Ex. 2. Similarly if a circle (A) roll on a fixed circle (B), theenvelope of any diameter of A is an epi- or hypo-cycloid whichwould be generated by the rolling of a circle of half the size of Aon the circumference of B. 165. Curvature of a Foint-Roulette. To investigate the curvature of any point P fixedrelatively to the rolling curve, let / be the point of contact,and let / be a consecutive point of contact, P the co
An elementary course of infinitesimal calculus . contact withthe fixed line, and the point Z will move as if it were carried bythe small circle. Its locus is therefore a cycloid. Ex. 2. Similarly if a circle (A) roll on a fixed circle (B), theenvelope of any diameter of A is an epi- or hypo-cycloid whichwould be generated by the rolling of a circle of half the size of Aon the circumference of B. 165. Curvature of a Foint-Roulette. To investigate the curvature of any point P fixedrelatively to the rolling curve, let / be the point of contact,and let / be a consecutive point of contact, P the corre-sponding position of P. Since the displacement of the pointof the rolling curve which comes to I is of the second orderof small quantities, the angle through which the figure hasturned is Se = ^lPIP (1), ultimately. Let the normals to the path of P, viz. PI and 164-165] CUEVATUEE. 441 PT, be produced to meet in G. If h^ be the inclinationof these normals, we have 8^ = ^101 = ^-^ CI (2). if ^ be the angle which IP makes with the normal at Fig. , from the figure Sx^ = z PIT - Z TPF5v, Ss cos d) = ^0—PT- 1 cos ~pr by Art. 164 (1), if R and R be the radii of curvature of thefixed and rolling curves. Equating (2) and (3), we find =Kj R^ R ) (3), cos f[m+ip)=R+R (^>- This gives the limiting position of G, the centre ofcurvature of the path of P. The radius of curvature (p) isthen found from p = GP=GI + IP (5). 442 INFINITESIMAL CALCULUS. [oh. We have taken as our standard case that in which thetwo curves are convex to one another, as in Figs. 135, other case may be included by giving proper signs to Rand R. Ex. 1. In the cycloid, if a be the radius of the generatingcircle, we have iJ = oo, R = a, /P = 2acos^ (6). Substituting in (4), we find CI = 2a cos = IP (7), andtherefore p = 2IP (8). Ux. 2. In the epicycloid (Art. 137) we haveP^a, E = h, IP = 2b cos -2ab , a whence CI= 2{a + h) IP ...(9),(10), P = - .IP (11). a+26We note that if 6 = - Ja, we hav
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