Chap. XI.

MECHANICS.

977

motion to a circular and variable

Fig. 1410.

To transform a circular , continuous , and uniformmotion , whose velocity changes according toa given law. This problem may be re-solved in a general manner: if we requirethat an axle D should make a numbern of revolutions, whilst another, C, makesanother of variable velocities, it is easy toforesee that two points of the two axesshould return to the same position aftera number n of revolutions of the axle D,or one of the axle C. To simplify theapplication we shall suppose that the twoaxles turn in the same time; this casebeing well understood, the others will notoffer the least difficulty.

Let CQ,DMbe the axles of two wheelsa,b; Ca,dD two segments, such that thearc a&is equal to ad, both toothed andplaced at the height of the line 12; d' n,wiD two segments of equal radius, sothat the arc bef equals dnm, both toothed and placed at the height of the line 2 2 ; qCp ,q Dr, two segments equal to the segments aDD and a b C, placed at the height of theline 3 3. "We see that by this arrangement the velocities may vary by finite intervals, andin any manner we please, provided the condition is fulfilled of making the points a, a meetat the end of n revolutions of the axles m,n.

The construction of this machine presents some difficulty, on account of the change ofgear when it passes from one rate of velocity to another; but this difficulty diminishes bymaking the teeth very small. In case they cannot be rendered sufficiently short, we mustaid the passage by an exterior force, as that of a spring or weight.

We may resolve the same problem by means of two truncated cones, a and b, of equaldimensions, placed as indicated in the figure, that is to say at some distance from eachother, their axes parallel, and the small base uppermost, g e

and at the same height as the great base b. A channelis cut, in the form of a helix, in the convex surfaces of thetwo truncated cones, and one of the extremities of a cord,mn, is attached to the point where the spiral channel drawnon the surface b begins at the great base, and after havingenveloped the whole of the same spiral, its other extremity isattached to the corresponding point of the great base at a.

It is evident if the truncated cone a turns in its properdirection with a uniform velocity, the other, b, will turnalso, but with a variable and decreasing velocity; at firstgreater than that of a, afterwards equal, and at the end as much less as it was greaterat the commencement; the cord ram will then be found enveloping the spiral channel drawnon the surface of a. The motion cannot continue longer in the same direction. If, instead ofcutting grooves in the forms of a helix on the convex surfaces of the two truncated cones aand b , we leave these surfaces plain, and substitute instead of the cord nm another endlesscord which embraces both, it is evident that this cord may occupy any required place inthe height of the cones without its being necessary to change its length.

Suppose, first, that the endless cord is placed at the lowest part of the cone; uniformmotion of rotation a will communicate to b another of the same kind, but whose velocitywill be greater than that of a in a known ratio, a motion which may be continued as longas we please; according as we make the endless cord advance towards the upper part of thecones, the ratio of velocities will diminish ; will become equal to unity when the cord shallhave attained the middle of the height of the cones, apd will follow an inverse course inproportion as it approaches the upper bases.

This mechanism is used with the greatest success to modify the velocities of circularcontinuous motion, particularly in the lathes used in the manufactures of earthenware

Fig. 1411.

and porcelain; the endless cordplaced in the midst of the intervalwhich separates the two cones; therack of the jack is placed parallel totheir axes, and directs the course of theendless cord; the mechanism is verysimple, and gives instantly all thechanges of velocity which can be re-quired.

made to ascend or descend by means of a jack

1412.