Chat. XI.

MECHANICS.

939

to apply levers or handspikes, and to walk around it without any interruption, the spokesor levers supplying the place of a wheel. "When the axle is horizontal, the weight of themachine presses upon the pivots upon which it turns ; and when its centre of gravity isequally distant from them, its pressure is distributed equally between them. When, how-ever, the axle is vertical, the whole weight rests upon one pivot, the friction in the twocases varying.

In a machine composed of several wheels and axles connected together by ropes, whichpass from the wheel of the one to the axle of the other, the power is to the weight, as theproduct of the radii of all the axles, is to the product of the radii of all the wheels.

When toothed wheels and pinions are used, the power will always be to the resistance,as the product of the radii of all the pinions, is to the product of the radii of all the wheels.

In all the combinations of the simple mechanical powers, the condition of equilibriummay be easily computed by tracing and calculating the effects of all its separate parts, theeffect produced at one part being always considered as the power which is applied to thenext part. The relation between power and effect may be found by considering that if thepoints of the machine to which the power and resistance are applied be made to describean infinitely small space, the power will be to the weight or resistance as the spacedescribed by the point to which the weight or resistance is applied, is to the space describedby the point to which the power is applied; then

P = the power,

W = thc weight,

p = the space described by the power,tv = the space described by the weight.

We shall then have P : W = w : p,

or, what is the same, the power is to the weight, as the velocity of the weight, is to thevelocity of the power ; consequently whatever is gained in power is lost in time.

Elementary l'arts of Machines. —A system of wheelwork is composed of two parts, eachturning on a fixed axis: considering two points in this system, the distance of whose axisof rotation is invariable, and supposing a force applied to the first causes it to describe anarc of a circle in any given time, the machine transmits this rotary motion to the secondpoint, which will describe in the same time another arc of a circle. The form of thewheelwork depends on the several positions of the two axes of rotation, and on the relationof the arcs of the same radius described in the same time by two points invariably fixed onthese axes. We shall consider the case in which this relation is constant, and determinethe form of the wheelwork by two hypotheses, of two parallel axes of rotation of two inter-secting axes, and indicate the researches made on the form of wheelwork in the case inwhich the two axes do not cut.

On Wheelwork having an uniform Rotation round two parallel Axes. —"When a pointdescribes a circle, the measure of its absolute velocity is thearc which it describes in an unit of time, the relation of thisvelocity to the radius of the arc is the measure of its angularvelocity. The two parallel axes of rotation being given, drawa plane perpendicular to them, and cutting them in two pointsas A and d; join these two points by a right line Ad. Therelation of the angular velocities of the two fixed points to theparallel axis, Dd and a being known, divide the right line Adinto two parts, AB and 13 d touching each other in the point13. Whatever is the force of P applied tangentially to thecircle of the radius A 13, it is evident that the transmission ofthe motion of rotation of this first circle to the second lid willbe made by the point of contact B, and that the arcs describedin the same time by two determined points of the two circleswill be of the same length ; whence it follows that the absolutevelocities of these two points are equal, but the arcs of thesame length and different radii; measure on the circum-ferences to which they belong angles which are to each otheras the inverse ratio of the radii of these circles; whence it follow's, that the angular andsimultaneous velocities of any two points taken on the circumferences of the radii Bd,

A B are to each other in the ratio %=- of these radii.

lid

Supposing the axis of rotation A and d, invariably fixed to two given points, and con-sequently to two circles of the radii A B, Bd: then thp rotary motion of these two pointswill be made in the same time as those of the circles, and the arcs of the same radius,similar to those which they describe simultaneously around the parallel axes A and d , willbe to each other in the inverse ratio of the radii A B, 13 r/. Designating the two givenpoints by M M which turn, the one round the axis A, and the other round the axis d, the

Fig. 1284.