An elementary course of infinitesimal calculus . cs, viz. to the determinationof the proper shapes to be given to the teeth of wheels. * This proposition is due to Eulcr (1781). 168-169] CUEVATUEE. 451 The kinematical problem is to determine the relationsbetween the forms of two cylindrical surfaces il, Of, whichare free to rotate about fixed parallel axes, so that, wheneither drives the other by sliding contact, the rotations maybe in a constant ratio. There are two methods of solving this problem; theymay be distinguished as the method of envelopes and themethod of roulettes. Considering the
An elementary course of infinitesimal calculus . cs, viz. to the determinationof the proper shapes to be given to the teeth of wheels. * This proposition is due to Eulcr (1781). 168-169] CUEVATUEE. 451 The kinematical problem is to determine the relationsbetween the forms of two cylindrical surfaces il, Of, whichare free to rotate about fixed parallel axes, so that, wheneither drives the other by sliding contact, the rotations maybe in a constant ratio. There are two methods of solving this problem; theymay be distinguished as the method of envelopes and themethod of roulettes. Considering the section made by a plane perpendicularto the generating lines, let 0, 0 represent the fixed take as the standard case that in which the directionsof the angular velocities (n, n) are opposite, as indicated inFig. 146. If we divide 00 in /, so that ^, the point / will have the same velocity whether we regardit as belonging to the first cylinder or the second. The circles(C, C) described through /, with 0, 0 as centres, are called. Fig. 146. the pitch-circles of the respective cylinders. It is evidentthat in the desired motion these circles will roll on oneanother without slipping. The relative motion of the cylinders will be unalteredif we impress on the whole system an angular velocity about0, equal and opposite to that of the first cylinder. The 29—2 452 INFINITESIMAL CALCULUS. [CH. X pitch-circle G will then roll with angular velocity n + n onthe pitch-circle G, which is at rest. Evidently, the requiredcondition is that, in the motion as thus modified, the sectionof the cylinder li shall constantly touch that of the cylinderfl, which is fixed. In other words, the section of XI mustbe the envelope of the successive positions of Q,. Hence,subject to limitations of a practical kind, we may adopt anycurve we please as the section of CI; the corresponding formof the section of 12 is then determined as the envelopeobtained when the circle C rolls on G, carrying th
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