of the circle Y, will touch each other in the point E. For the straight line AE, drawn through the point A, where the generating circle Y touches its bases R and X, will be perpendicular to the two epicycloids; and the straight line b E will touch these two epicycloids in the same point E (540).

Corollary VI.

542. Let us now suppose that the generating circle Y (fig. 183) has for diameter the radius AB of the circle X, within which it is; and that these circles R, X, and Y, touch continually in the point A: the interior epicycloid HE, which will touch the exterior CE, will be a straight line directed towards the centre B of the circle X, and consequently will be a portion of the radius BH, which will always touch the exterior epicycloid CE in the point E, where this radius meets with a perpendicular AE, drawn to it from the point A.

It thence follows, that when two circles R and X continually touch each other, and the one obliges the other to turn around, carrying it along by the point of contact A, if we suppose a radius BH in the circle X, and, having made AC=AH, we describe through the point C, taken at its commencement, an exterior epicycloid CE, having for its generating circle, a circle Y, the diameter of which is equal to the radius BH, this radius BH during the motion of the two circles R and X will always touch the epicycloid CE in the point E, where the epicycloid is intersected by the straight line AE, perpendicular to its curve. Hence, instead of sup

posing that one of the two circles R and X moves the other by the point of contact A, the radius BH of the circle X may be made to be impelled by an epicycloid CE, attached to the circle R, and described by the motion of the circle Y, the diameter of which is equal to the radius BH; reciprocally also, the epicycloid CE attached to the circle R might be made to be impelled by a radius BH of the circle X, and by means of the epicycloid CE and the radius BH, the two circles R and X might move each other as if impelled by the point of contact A.

It is from this corollary chiefly, that we shall deduce the best form which can be given to the teeth of the wheels and pinions of a machine, in which a part of the tooth of the wheel or pinion, or of the teeth of both, ought to be formed in a straight line, tending to the centre of the wheel and the pinion.

Corollary VII.

543. If in the same plane (fig. 182 and 183) there were only two circles R, Y, touching each other in the point A, and if the motion of the one were communicated to the other by the point of contact, any point E of the circumference of the circle Y would describe, on the moveable plane of the circle R, an epicycloid CE; and this epicycloid, which we suppose to be attached to the circle R, would move the circle Y, impelling it by the point E of its circumference, in the same manner as the circle R might move the same circle Y, communicating motion to it by the point of contact A; and

reciprocally the point E of the circumference of the circle Y, turning around its centre G, would cause to revolve the circle R, impelling it by the epicycloid CE, which we suppose to be attached to this circle R; in the same manner Y would move the circle R by communicating its motion in the point of contact A.

From this corollary we shall deduce the best form that can be given to the teeth of a wheel, when the pinion is a lantern composed of rungs; we shall deduce from it also the most advantageous form that can be given to the teeth of a pinion, when the wheel has rungs parallel to each other instead of teeth.

Problem.

544. The number of the teeth of a wheel, and the number of the rungs of a lantern (fig. 184), in which the wheel is to act, being given, with the distance of their centres F and G, to determine the primitive radius and the true radius of the wheel; the size and form of the teeth and the depth of the engagement of the teeth of the wheel in the lantern.

Solution.

As one example will be sufficient to give an idea of this problem, and as one solution for indeterminate numbers of teeth and rungs might render it obscure, without making it easier to be applied to wheels and lanterns, the numbers of the teeth and rungs of which are given; we shall suppose that the wheel ought to have thirty teeth, the lantern eight rungs; and that the centres F and G of the

wheel and the lantern ought to be at the extremities of the given straight line FG.

Since the wheel ought to have thirty teeth, and as a lantern of eight rungs is required, we must divide the line of centres FG into two parts, AF and AG, which shall be to each other as thirty to eight, or as fifteen to four. For this purpose, divide the straight line FG into thirty-eight equal parts; that is to say, into as many parts as there are teeth and rungs together in the wheel and in the lantern; and having taken eight parts for AG, and the remaining thirty for AF, the straight lines AF and AG will be the primitive radii of the wheel and the lantern.

The primitive radii of the wheel and the lantern being thus determined, the next thing is to determine the form of the teeth, and this form will give the true radius of the wheel.

To determine the form of the teeth, which always depends on that of the rungs, we shall first suppose that the rungs are infinitely small, and represented on the circular plane, forming the end of the lantern by eight points, A, E, H, I, K, i, h, e; and when we have found the form of the teeth proper for moving these rungs infinitely small, which cannot be used in practice, we shall correct it; and by its means trace out the real form which must be given to the teeth of wheels, to drive lanterns with cylindric rungs. Hence the solution of the proposed problem will be divided into two parts.

I.

FOR THE FORM OF THE TEETH OF A WHeel when THE RUNGS OF THE LANTERN ARE INFINITELY

SMALL.

545. We have seen (543) that if the circle CA c (fig. 184), which touches the circle EA e were furnished at its circumference with an epicycloid c E, described by the point E of the circumference of the circle EA e, during the revolution of that circle on the circle CA c, the epicycloid c E would move the circle EA e by the point E, in the same manner as the circle CA c would move it by the point of contact A.

It has been seen also, and it is evident, that if the two circles CA c and EA e moved each other by their point of contact A, these circles would both have the same velocity; hence, when the epicycloid c E moves the circle EA e by a point E of its circumference, the circumferences of the two circles CA c and EA e will have the same velocity; consequently the same force (535). The epicycloid c E is, therefore, the best curve that can be given to the tooth of a wheel to drive a lantern, the rungs of which are infinitely small.

Considering only the epicycloid c E, and the property it has of causing the primitive lantern to revolve with the same velocity, and the same force as the primitive wheel, moving the lantern by a point E, which represents a rung infinitely small, it is evident that it is the convex side of the epicycloid