(suppose 8), and let GF be the line of the centres (fig. 5. pl. XXXI.) the letter F being referred to the point of concourse of the converging lines from A, D, X, R, whose continuations are omitted in the figure, to save room upon the plate. Divide the whole line GF into two parts, in the ratio of m to n,

that is, make AF= FG, and AG=

m+n

n

m+ n

FG, so will AF and AG be the required primitive radii of the wheel and lantern. With the primitive radius of the lantern describe the circle AEHIh, and divide its circumference into n (8) equal parts, marking the points of division A, E, H, I, K, &c. for the positions of the assumed indefinitely thin spindles: also, describe with the primitive radius of the wheel the circumference CSTN, and divide it into m (30) equal parts, one of which is CA; and let AL be the small vacuity or interval which it is thought proper to leave between the teeth of the wheel, for the play of the engagement. Then CL will be the foot of one tooth, through the extremities of which describe two opposite epicycloids CEP and LMP, which turn their convexities towards the neighbouring teeth, and which have the circle CAN for base, and Eнкh for generating circle. And proceed in a similar manner to describe the other teeth, as AON, &c. From this construction it is obvious that the distance from F, the centre of the wheel, to P, the point of concourse of the two epicycloids, forming a tooth, is the greatest true radius that the wheel can have, in the proposed case; and since no more of any tooth is wanted, than what reaches from c to E, the preceding spindle, when the bottom L, of the tooth, comes in contact with the next spindle A, the whole quantity EPM may be cut away from the tooth, and it will still continue effectual in the machinery; so that the distance from E, to the centre F, is the shortest radius which the wheel can have, and any radius may be adopted between the limits FE and FP.

Our next business is to reform all these teeth, to make them accord with the spindles of a finite diameter, which the wheel must have. Here, if the radius of the spindles, which we suppose equal, be given, describe with this radius, on the plane of each tooth, as many small arcs as can well be done, having all their centres in the two epicycloids which form the tooth; and trace out two curves, such as RO, so, through all these arcs parallel to the epicycloids, between which the first teeth are contained. These new curves RO, so, or TY, VY, being thus described, will reform the first teeth CPL, AQN, &c. and will comprehend between them and the primitive circle of the wheel, the spaces ROS, TYV, &c, which will be the proper figures of

the teeth of the wheel, to drive the spindles A, E, H, F, K, i, h, e, the diameters of which are given. For, if we imagine that the centre of a spindle is moved by the tooth CPL, the curve RO which is parallel to the epicycloid cr, and is distant from the radius of the spindle E, will always touch the circumference of that spindle; hence the curve RO will move the cylindric spindle, just as the tooth CPL, formed by two epicycloidal portions, would move the axis E of that spindle, consequently, the tooth ROS has the proper form to move a lantern with the proposed cylindric spindles.

When the common radius of the spindles of the lantern is not given, if it be necessary to correct the first teeth of the wheel CPL, AON, so that the new teeth shall leave between them vacuities equal to the breadth of their feet; and if it be proposed, likewise, that the play of the engagement should always be equal to AL; then divide into two equal parts CD and DL the foot of the tooth CL, and having taken on both sides of the point n, two parts DR and DS, equal to 4 of the arc AC, the are RS will be the foot of the new tooth demanded. Then, with a radius equal to the chord of the arc CR, trace out the circles a, e, h, &c. which will represent the magnitude of the spindles of the lantern. Lastly, to complete the correction of the first teeth of the wheel, describe with the same radius, on the plane of them, as many small arcs as may be, having their centres in the epicycloids which comprise the first teeth: and if there be drawn curves through all these small arcs, such as RO and so, or TY and vy, we shall have new teeth ROS, and TVY, which will leave vacuities between them equal to the breadth of their feet; which will have the play demanded, while acting in each other, and which will drive the spindles whose size has been determined, in like manner as the former teeth would have driven spindles infinitely thin.

Now, as the two curved sides of each of the new teeth mutually terminate in the point or edge of concourse, it is clear that the distance of of the point of one of these new teeth from the centre F of the wheel will be the greatest true radius the wheel will admit of. And, when a spindle E has been moved till the centre A of the following spindle is in the line GF of the centres, the spindle A may, in its turn, be moved by the succeeding tooth Tyv; and then it will be no longer necessary for the tooth ROS, to move the cylindric spindle E. The tooth ROS, therefore, may be terminated at the point x where it touches the spindle E, when the centre of the succeeding spindle is in the line of the centres; and the distance XF of that point of contact from the centre of the wheel will be the least true

radius the wheel will admit of. It will be proper to give the true radius of the wheel a mean length between OF and XF, and to file or round off the point of the tooth.

The teeth of the wheel being thus constructed, it is obvious that they will not move the spindles till their centres have arrived at the line of centres; and that the spindles, on the contrary, will move these teeth by impelling them towards the line of the centres GF, and until their centres have arrived at that line.

The sides Tz and s&, of the vacuities sunk in the primitive wheel, being directed towards the centre F of the wheel, the rounding of the spindle which proceeds beyond the primitive circle of the lantern, ought to have the shape of an epicycloid, which has for its base the primitive circle of the lantern, and is generated by a circle having a diameter equal to the radius AF of the wheel. Hence a circular spindle does not appear proper to be carried towards the line of the centres, by the side Tz of the vacuity sunk into the primitive wheel. But the preceding spindle E, being conducted by the preceding tooth of the wheel, until the centre of the spindle a has arrived in the line of the centres, and the space TA which the right line rz ought to make the spindle pass over, before it attains the line of the centres, being very short; the arc of the spindle on which the side TZ will slide, in driving that spindle will be so minute that it may be taken for a small arc of an epicycloid, and of consequence, if there be any want of uniformity in the movement of the lantern by the wheel, during the little time the part Tz of the tooth moves the spindle, this deviation from uniformity will be too small to be sensible.

It may not be amiss to observe that, since the teeth of a wheel must by impelling the spindles of a lantern, remove them from the line of the centres, and since no shocks need be feared in this method of moving a lantern, a lantern may, without any inconvenience, be caused to be moved by a wheel. But, since, on the other hand, the spindle of a lantern ought by impelling the teeth of a wheel to bring them nearer to the line of the centres, and since shocks may occur in this method of driving a wheel, it seems reasonable to conclude that a pinion is preferable to a lantern when a wheel is to be driven.

We may next proceed to consider the case, in which, knowing the number of teeth in a wheel, and the number of the leaves of the pinion upon which it is to act, with the distance of their centres, we wish to determine their primitive and true radii, as well as the form of the teeth of the wheel, and the leaves of the pinion. Here, having, as in the former instance, divided the distance FB of the centres, (fig. 13. pl. XXXII.) into two parts

AF and AB, proportional to the number of the teeth of the wheel, and leaves of the pinion respectively, these parts will be the primitive radii of the wheel and the pinion; and if there be described with these two parts as radii, from the points F and B as centres, two circumferences AQR, ATX, touching each other in the point A, they will be those of the primitive wheel and pinion.

It is common to shape a wheel so that the breadth of the teeth is equal to that of the vacuities, which is called by the French-Fendre une roue tant plein que vuide. In this case divide the primitive circumference of the wheel into twice as many equal parts as it ought to have teeth, in order to fix the feet CA, LO, &c. of these teeth, and the vacuities AL, GQ, &c. which ought to be interposed. But if it be proposed that the teeth should fill more space than the vacuities, as is proper in certain circumstances, we must first divide the primitive circumference into as many equal parts CL, LG, &c. as it ought to have teeth; and afterwards divide each part, such as CL, into two other parts, CA, AL, one of them equal to the breadth which we would give to each tooth, and the other equal to the interval proposed to be put between two teeth. The feet CA, LO, &c. of all the teeth being determined upon the primitive circumference of the wheel, draw through the extremities of these teeth, towards the centre of the wheel, right lines cc, aa, Ll, oq, &c. nearly equal to the breadth CA, LQ, of these feet, to mark out the straight flanks of the teeth; and through the extremities of each foot, as CA, let there be drawn two equal epicycloids CP, AP, whose generating circle AEY has the radius AB, of the pinion, for its diameter, and both of which have the primitive circumference of the wheel for the base. These epicycloids, when traced out, will include those parts of the teeth which project beyond the primitive circle of the wheel, in such manner, that the right line FP, drawn from the centre of the wheel to the point P, of concourse, of the two epicycloids of one tooth, will be the greatest true radius which the wheel admits, relatively to the spaces given to the teeth, and to the intervening vacuities.

Having divided the primitive circumference of the pinion into as many equal parts OH, HS, &c. as it ought to have leaves, each part as он, must again be divided into two other parts oo, OH, one equal to the thickness we would give to the leaf, and the other to the breadth of the vacuity proposed to be left between two leaves; giving to оH a breadth rather exceeding that of a tooth of the wheel, to furnish suitable play to the engagement. All the breadths oo, нh, &c. being thus determined, draw right lines a little longer than the projection pp, of

the teeth of the wheel, beyond their primitive circle, through their extremities, towards the centre B of the pinion; and these lines will serve as flanks to the leaves of the pinion, and will determine the vacuities in which the teeth of the wheel will act with the proper play. Then describe through the extremities of the straight sides of each leaf two epicycloids, as om, om, whose generating circle AVF has for diameter the radius of the wheel, and both of which have for base the primitive. circumference of the pinion: these epicycloids being traced out, will contain between them the parts of the leaves which project beyond the primitive circle of the pinion, so that the right line Bm drawn from the centre of the pinion to the point of concourse m of the two epicycloids of the same leaf will be the greatest true radius that the pinion will admit of, relatively to the thickness of its leaves. The parts of the teeth of both wheel and pinion which are left unshaded in the figure may be rounded off, being never exposed to mutual action upon one another.

The preceding directions may suffice to convey a tolerably distinct notion of the scientific method of forming teeth of wheels, to act with either pinions or lanterns, when the motion is communicated in the same plane: our next business is to speak of the teeth of crown wheels, when driving either a lantern, or another wheel, their axles being in different planes. The space assigned to this article compels us to confine ourselves to the case of a crown wheel driving a lantern, whose spindles are ranged in the surface of a cone, in which case the teeth must be shaped by means of a spherical epicycloid*. Here the first thing is to trace out the teeth of a wheel as though it had to drive a lantern with spindles infinitely thin, observing to leave for the play of the engagement small void places between the feet of all the teeth. Then, having made a lantern with conical spindles, all the summits of which con

* Let there be a right cone, the summit of which c remains immoveable: if the base of this cone be made to revolve on any plane RAS (figs. 1, 2. pl. XXXV.) placed at pleasure in respect of the point c, and if we imagine a style or tracer situated in the point A of the circumference of the revolving circle, this style a will describe during its motion a curve called a spherical epicycloid. This curve has not, that we are aware of, been treated at large in any English work; but the curious reader may consult the following papers in foreign publications. Jacobi Hermanni de Epicycloidibus sphoricis. Comment acad. Petropol. tom. 1. an. 1726;-De la Hire Traité des Epicycloides, et de leurs usages dans les Méchaniques. Hist. acad. roy. Paris. 1730, tom. 9;-Probleme sur les Epicycloides sphériques par. M. Bernoulli, Mem. Acad. Roy. Par. 1732;-Des Epicycloides sphériques, par M.Clairaut, Do. an. 1732;-a. J. Lexell, de epicycloidibus in superficie sphærica descriptis. Act. Acad. Imp. Petropol. 1779. P. 2. p. 49.