An elementary course of infinitesimal calculus . e diameter is equal to the constant value of the motion is equivalent to the rolling of a circle on theinside of a fixed circle of twice its size. This kind of motion hasbeen considered in Art. 138, Ex. 2, and it has been shewn thatany point P fixed relatively to AB will describe an ellipse, whichin certain cases, viz. when P is on the circumference of the rollingcircle, degenerates into a straight line. Ex. 2. In the linkage known as the crossed parallelogram(see Art. 146, 2°), if the bar AD be held fixed, the instantaneouscentre / for
An elementary course of infinitesimal calculus . e diameter is equal to the constant value of the motion is equivalent to the rolling of a circle on theinside of a fixed circle of twice its size. This kind of motion hasbeen considered in Art. 138, Ex. 2, and it has been shewn thatany point P fixed relatively to AB will describe an ellipse, whichin certain cases, viz. when P is on the circumference of the rollingcircle, degenerates into a straight line. Ex. 2. In the linkage known as the crossed parallelogram(see Art. 146, 2°), if the bar AD be held fixed, the instantaneouscentre / for the opposite bar BC is at the intersection of AB andCD. Also it is evident from symmetry that the sums AI+ID and BI+IC are constant, being each equal to A B or CD. Hence the locus of 448 INFINITESIMAL CALCULUS. L^n- -- I relative io AD is an ellipse with A, B foci, and the locus of/ relative to BG is an equal ellipse with B, G as foci. Themotion of BG relative to AD is therefore represented by therolling of an ellipse on an equal Fig. 143. If the bar AB he supposed fixed, the relative motion ofGD wiU be represented by the rolling of a hyperbola with G, Das foci on an equal hyperbola with A, B as foci. 168. Double Greneration of EpicycUcs as Rou-lettes. As a further example we return to the mechanicalmethod of compounding uniform circular motions, by meansof a jointed parallelogram OQPQ!, referred to in Art. 139. We will suppose for definiteness that the angular velo-cities n, n, of the bars OQ, OQ, have the same sign. The instantaneous centre (7) of the bar QP will be apoiat in QO such that (1). For the velocity of any point rigidly attached to QF will bemade up of a translation n. OQ at right angles to OQ, and arotation with angular velocity n relatively to Q. Henceunder the above condition the velocity of the point attached 167-168] CURVATURE. 449 to QP which at the instant under consideration is at / willbe zero. The two centrodes for the motion of
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