An elementary course of infinitesimal calculus . Fig. 94. 139. Superposition of Circular Motions. Epl-cyclics. The cycloidal and trochoidal curves discussed in —138 present themselves in another manner, as thepaths of points whose motion is compounded of two uniformcircular motions. If an arm OQ revolve about a fixed point 0 with constantangular velocity n, its projections on rectangular axes through0 may be taken to be 00=0 cosnt, y = csm nt (1), where c = OQ, provided the origin of t be suitably another arm OQ revolve about 0 with constant angularvelocity n, starting simult
An elementary course of infinitesimal calculus . Fig. 94. 139. Superposition of Circular Motions. Epl-cyclics. The cycloidal and trochoidal curves discussed in —138 present themselves in another manner, as thepaths of points whose motion is compounded of two uniformcircular motions. If an arm OQ revolve about a fixed point 0 with constantangular velocity n, its projections on rectangular axes through0 may be taken to be 00=0 cosnt, y = csm nt (1), where c = OQ, provided the origin of t be suitably another arm OQ revolve about 0 with constant angularvelocity n, starting simultaneously with OQ from coincidencewith the axis of x, the projections of OQ will be x = ccosnt, y = csmnt (2), where c = OQ. If we complete the parallelogram OQPQ, 360 INFINITESIMAL CALCULUS. [CH. IX the vector OP will represent the geometric sum of OQ andOQ, and the coordinates of P will be a; = ccosn< + ccos nt, y=camnt-\-csin (3)*,. Fig. 95. Ex. 1. If the angular velocities of the component circularmotions are equal and opposite (n = — n), we have x = (c + c) cosnt, y = {G — c)& (4), so that the resultant motion is elliptic-harmonic. In the particularcase of c = c, the ellipse degenerates into a straight line. This example is of importance in Physical Optics. Since, in the figure, QP is always equal and parallel toOQ, the path of P is that of a point describing uniformly acircular orbit relatively to a point Q which itself has auniform circular motion about 0. Curves described in thismanner are called epicyclics. If the angular velocities n, nhave the same sign, the epicyclic is said to be direct; ifthey have opposite signs it is said to be retrograde. * If the parallelogram OQPQ consist of four jointed rods, and if OQ, OQbe made to revolve at the proper rates about 0, the distance of P from anyfixed line through O will represent the sum of two simple-harmonic motionsof periods iirjn, 27r/n. This is
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