Chap. XI.

MECHANICS.

SS7

Fig.1450.

The mechanism may also be employed in the generalsolution of the problem, without a counterpoise. Let a 6and cd be two axes, each furnished with a pulley anda toothed wheel; an endless cord embraces the pulleys, andcommunicates the circular motion from one to the other,making them turn in contrary directions, because they crossin the intervals which separate the pulleys. Let ef be anotheraxle furnished with a toothed wheel; this axle may approachthe axle ab or cd. because its gudgeons are enclosed in oblongorifices. It will then suffice to bring it near sometimes to theaxle ab, so as to engage the wheel c with the wheel b; some-times the axle C D, so as to engage the wheel c with the wheelh , to render its motion circular alternate, supposing that ofits axle ab anti cd circular continuous.

Circular continuous motion with a uniform velocity, or vari-able according to a given law, may be changed into a motion ofa given continuous curve re-entering in itself and closed, with avelocity of the same nature as that of the motion which produces

it, constant or variable according to a given law, in the some plane or in different planes*

Let f be a given curve : it is required that a point of a crayonor any other tool may draw this curve on a surface pq bymeans of the circular continuous motion of the wheel d.

Take three toothed wheels o, b, c, of any convenient size:place them as indicated in the figure, that they may receivetheir motion from that of the wheel d : mm,pq are two ruleswhich move alternately, preserving their direction by meansof guides or bridges eeej they are terminated by two per-pendicular rules, each pierced with a groove; the crayon or toolwhich traces the given curve is placed at the intersectionof the two grooves. On the given curve take any number ofpoints, which must be most numerous in the places where thechanges of the curve are most rapid.

Draw on the surfaces a and c the proper curves, which shouldbe such that the extremities of the

Fig. 1451•

rods lean on the curves, drawn in relief by means ofa spring or otherwise : the intersection of the grooves where the crayon is placed goesover the points selected, which operation does not present any difficulty.

Kjdc°

Fig. 1452.

The object proposed by this problem is not to make a point,such as the extremity of a tool, go over a certain curve, but totrace this same curve on a given plane surface; for this effectfix the surface a to the extremity of the rod b. By this meansthe surface apparently immovable will receive an alternate rec-tilinear motion ; having fixed the crayon or tools to the end a ofthe other rod, cd, draw convenient curves on the surfaces e andf and we shall have a machine susceptible of useful application inthe arts.

Second general Solution of the Problem. The points of a givencurve may also be subjected to angular co-ordinates : these curvesare of two kinds, first, those enclosed in the interior, in which wecan find such a point that all right lines passing through it canonly meet two points of the curve: secondly, curves closed inthe interior, in which we cannot find any point which has this pro-perty ; we can without any difficulty draw curves of the first kindby circular continuous motion. We have in this simple ma-chine an auxiliary wheel conveying its movement to two otherson the same axis; the rosette d is attached to the first wheel b,which touches n n, fixed to the regular movement, resting on itsedge by means of a spring, b, which acts on the extremity of therule in the movable branch pq, and at the point q is a pencil,which may be placed at pleasure on any point whatever of thesurface on a circle of paper placed on the surface c.

We must always bear in mind that among the machines ap-plied to useful purposes, there is scarcely one in which there isnot a deviation from a uniformity of action, either from thenature of the impelling power, or the method by which it is ap-plied, besides a desultory motion, which arises from the construc-tion of the machinery; these peculiarities have been ably noticed by Desaguliers .

Let abd (fig. 1456.) be the given curve, and take in its interior any given point c for thecentre of the circular motion of the rule p q, which turns in the same direction, whilst the

Fig.1453.