to trace out a side of the interior face of the same tooth, describe on the surface of the zone, represented by its profile r s b k, a portion of a spherical epicycloid, having for generating circle the base of the cone, the profile of which is the triangle k mr, and for base the circumference of the circle of which rs is the diameter.

The two straight lines A a, Nn, (fig. 201) directed towards the apex C of the cone of the pinion, being the origin of the curved sides of a tooth, and the two portions AI, a i of the epicycloids, which ought to border one side of the exterior and interior faces of that tooth being traced out, describe two other portions NI, ni of epicycloids, equal and similar to the two former, which they will meet in the points I, i; and the spaces AIN, a in will be the models on which must be formed the exterior and interior faces of all the teeth of the wheel: the problem therefore will be completely solved.

Advertisement.

570. In this problem we have determined only the curved parts of the teeth of the wheel, by which the plane sides of the leaves of the pinion ought to be impelled, after they have arrived in the plane of the axes of the wheel and pinion; and we have considered the cone of the pinion only in the case where the sides of its leaves are entirely plane, and cannot be impelled but beyond the plane of the axes. But to avoid the shocks which might occur, if the teeth should meet with any of the leaves be

fore the plane of the axes, it is necessary to make the cone of the pinion a little larger than it has been found.

If the cone of the pinion be enlarged in such a manner, that each diameter shall be increased by a quantity equal to the thickness which the leaves will have at the place of this diameter, it will be sufficient to round, in the form of a demicone, all the parts by which the leaves have been lengthened: by means of this precaution all shocks will be avoided.

If it be required that the parts by which the leaves are lengthened should be curved in such a manner, that the wheel may move the pinion with as much regularity when its teeth meet with the leaves before the plane of the axes, and impel them by their curved parts, as when they impel them by their plane sides, beyond the plane of the axes, it will be necessary to make the curvatures of the ends of these leaves in the form of spherical epicycloids, generated by the rolling of a right cone, having for diameter the radius of the circle which passes through the origin of the curvatures of all the teeth of the wheel. This right cone, in rolling, will have its axis parallel to that of the wheel, and its apex in the axis of the pinion.

These epicycloids, which will have their commencement at the extremities of the principal diameters of the pinion, being traced out, you must then form the curved surfaces of the ends of the leaves, by means of a straight line which will pass through the apex of the cone of the pinion, and

which must be made to glide along these epicycloids.

This construction and the demonstration of it will be easily understood, if the pinion be considered as a crown-wheel and the wheel as a pinion.

Whether the ends of the leaves be rounded into the form of a demi-cylinder, or be curved in the form of a spherical epicycloid; it will be necessary, in order to lodge these augmentations of the leaves between the teeth of the wheel, to sink in the rim of that wheel, below the circumference of the circle which passes through the origin of the curvatures of all the teeth, the spaces by which the teeth are separated, and to form the sides of these depressions according to the planes drawn through the axis of the wheel, and through the commencement of the curvatures of the teeth.

A profile of a crown wheel with the pinion in which it acts, is given in fig. 202. The curved parts of the teeth of the wheel are separated from the straight parts of the same teeth by a straight line RES, which represents the circumference of the circle that passes through the origin of the curvatures of these teeth. The plane sides of the leaves of the pinion are also separated from the curved parts of these leaves, by the convex surface of the cone CABT.

BOOK XI.

OF THE NUMBER OF TEETH WHICH THE WHEELS OF A MACHINE OUGHT TO HAVE, THAT TWO OR

MORE OF THEM MAY PERFORM IN THE SAME

TIME A GIVEN NUMBER OF REVOLUTIONS.

571. In general, there are two kinds of machines those which serve to multiply the moving power, and those the chief object of which is a regularity of the contemporary motion of certain parts. In the former, it is scarcely ever of any importance that a wheel should perform one turn or a certain number of turns exactly while another performs another certain given number of revolutions; and the principal object ought to be, that the moving power may be communicated from one part to another with the least loss possible. In the other machines, the preservation of the force, in its communication from one part to another, is not the only thing to be considered; and it is often essential to them that several parts should perform, in the same time, a certain number of revolutions. As machines proper for measuring time are of this kind, it will be chiefly to the construction of these that we shall apply the art of finding the number of the teeth and of the leaves which wheels and pinions ought to have, to cause the various parts of the machine to perform, in the same time, certain motions or a given number of revolutions.

The methods for finding the numbers of the teeth and leaves which must be given to the wheels

N

and pinions of a machine, to make both these parts perform, in the same time, a certain number of revolutions, being more or less simple, according as it is possible or not possible to give to the wheels and pinions a sufficient number of teeth to produce exactly these revolutions; and as these methods depend on the same principles, order requires that this book should be divided into three chapters. In the first, we shall explain the general principles on which is founded the art of finding the number of teeth and leaves that ought to be given to wheels and pinions. In the second, an application of these principles will be made to finding the numbers of the teeth and pinions, in the case when the product of the wheels and that of the pinions can be decomposed into factors, which may be the numbers of the teeth and leaves of these wheels and pinions. In the third, an application will be made of the same principles to the same research, when the first products found, as those of the wheels and pinions, cannot be decomposed into factors sufficiently small to be the numbers of the teeth and leaves of these wheels and these pinions.

CHAPTER I.

OF THE GENERAL PRINCIPLES FOR FINDING THE NUMBERS OF THE TEETH AND LEAVES OF WHEELS AND PINIONS.

I.

572. The numbers of the teeth of wheels or pinions must not contain fractions. There cannot,