The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . of the rotation A. Secondly; let the axes of rotation be parallel to one another, and per-pendicular to the plane of the paper, and let them pass through A and B. Let them be said to be in the samedirection when A and B begin to
The differential and integral calculus, containing differentiation, integration, development, series, differential equations, differences, summation, equations of differences, calculus of variations, definite integrals,--with applications to algebra, plane geometry, solid geometry, and mechanics Also, Elementary illustrations of the differential and integral calculus . of the rotation A. Secondly; let the axes of rotation be parallel to one another, and per-pendicular to the plane of the paper, and let them pass through A and B. Let them be said to be in the samedirection when A and B begin tomove in contrary directions, and A B vice versa. If then the rotations be of equal angular velocity, and contrary in direction, the result of thetwo motions of rotation will be one motion of translation, in the directionperpendicular to AB. For each of the points A and B only moves invirtue of the rotation round the other : but the angular velocities beingequal, and the directions contrary, the initial velocities of A and B areequal and in the same direction, whence AB is carried without changeof direction in the direction perpendicular to AB. In any other case,take infinitely small lines described by A and B in the time dt, each ofwhich is therefore proportional to the angular velocity round the otheraxis. Thus, let Aa=, Bb = , whence a and b will. represent the positions of A and B at the end of the time dt. Thepoint 0, which remains at rest, and is therefore a point in the axis ofthe compound rotation, is determined by OA: OB : AB fidt, AB .adt, orOA.«=OB.£. 486 DIFFERENTIAL AND INTEGRAL CALCULUS. 1. When the rotations are in contrary directions, that round A beingthe greater, the axis of compound rotation is on the side of A, (OA+AB)yS, or OA=AB/3: (a—fi), OB=AB« :(a—fi).The angular velocity gives the angle Aa: OA, or : (: (a—/3)),or (a—yS) dt in the time dt, and is a—fi. 2. By similar reasoning, if the directions be contrary, t
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