contact T of the two teeth, and is perpendicular to the surfaces of both at that point, passes also through the point of contact O of the pitch circles of the wheels. Now this is true, whatever be the positions of the wheels, and whatever, therefore, be the points of contact of the teeth. Thus then the condition established in Art. 199. as that necessary and sufficient to the true action of the teeth of wheels, viz. "that a line drawn from the point of contact to the pitch circles to the point of contact of the teeth should be a normal to their surfaces at that point, in all the different positions of the teeth," obtains in regard to involute teeth.*

The point of contact T of the teeth moves along the straight line AB, which is drawn touching the generating circles BII and AG of the involutes; this line is what is called the locus of the different points of contact. Moreover, this property obtains, whatever may be the number of teeth in contact at once, so that all the points of contact of the teeth, if there be more than one tooth in contact at once, lie always in this

line; which is a characteristic, and a most important property of teeth of the involute form. Thus in the above.

*The author proposes the following illustration of the action of involute teeth, which he believes to be new. Conceive AB to represent a band passing round the circles AG and BH, the wheels would evidently be driven by this band precisely as they would by the contact of their pitch circles, since the radii of AG and BII are to one another as the radii of the pitch circles. Conceive, moreover, that the circles BH and AG carry round with them their planes as they revolve, and that a tracer is fixed at any point T of the band, tracing, at the same time, lines mn and m'n', upon both planes, as they revolve beneath it. It is evident that these curves, being traced by the same point, must be in contact in all positions of the circles when driven by the band, and therefore when driven by their mutual contact. The wheels would therefore be driven by the contact of teeth of the forms mn and m'n' thus traced by the point T of the band precisely as they would by the contact of their pitch cir cles. Now it is easily seen, that the curves mn and m'n', thus described by the point T of the band, are involutes of the circles AG and BH.

figure, which represents part of two wheels with involate teeth, it will be seen that the points rs of contact of the teeth are in the same straight line touching the base* of one of the involutes, and passing through the point of contact A of the pitch circles, as also the points A and b in that touching the base of the other.

EPICYCLOIDAL AND HYPOCYCLOIDAL TEETH.

202. If one circle be made to roll externally on the circumference of another, and if, whilst this motion is taking place, a point in the circumterence of the rolling circle be made to trace out a curve upon the plane of the fixed circle, the curve so generated is called an EPICYCLOID, the rolling circle being called the generating circle of the epicycloid, and the circle upon which it rolls its base.

I

If the generating circle, instead of rolling on the outside or convex circumference of its base, roll on its inside or concave circumference, the curve generated is called the HYPOCYCLOID.

Let PQ and PR be respectively an epicycloid and a hypocycloid, having the same generating circle APH, and having for their bases the pitch circles AF and AE of two wheels. If teeth be cut upon these wheels, whose edges coincide with the curves PQ and PR, they will work truly with one another; for let them be in contact at P, and let their common generating circle APH be placed so as to touch the pitch circles of both wheels at A, then will its circumference evidently pass through the point of contact P of the teeth for if it be made to roll through an exceed ingly small angle upon the point A, rolling there upon the circumference of both circles, its generating point will traverse exceedingly small portions of both curves; since then a given point in the circumference of the circle APH is thus shown to be at one and the same time in the perimeters of both the curves PQ and PR, that point must of necessity be the point of contact P of the curves; since,

*The circles from which the involutes are described are called their bases. This cut and that at page 237. are copied from Mr. Hawkins' edition of Camus on the Teeth of Wheels.

moreover, when the circle APII rolls upon the point A, its generating point traverses a small portion of the perimeter of each of the curves PQ and PR at P, it follows that the line AP is a normal to both curves at that point; for whilst the circle API is rolling through an exceedingly small angle upon A, the point P in it, is describing a circle about that point whose radius is AP.* Teeth, therefore, whose edges are of the forms PQ and PR satisfy the condition. that the line AP drawn from the point of contact of the pitch circles to any point of contact of the teeth is a normal to the surfaces of both at that point, which condition has been shown (Art. 199.) to be that necessary and sufficient to the correct working of the teeth.t

Thus then it appears, that if an epicucloid be described

The circle APH may be considered a polygon of an infinite number of sides, on one of the angles of which polygon it may at any instant be conceived to be turning.

The entire demonstration by which it has been here shown that the curves generated by a point in the circumference of a given generating circle APH rolling upon the convex circumference of one of the pitch circles, and upon the concave circumference of the other are proper to form the edges of contact of the teeth, is evidently applicable if any other generating curve be substituted for APH. It may be shown precisely in the same manner, that the curves PQ and PR generated by the rolling of any such curve (not being a circle) upon the pitch circles, possess this property, that the line PA drawn from any point of their contact to the point of contact of their pitch circles is a normal to both, which property is necessary and sufficient to their correct action as teeth. This was first demonstrated as a general principle of the con struction of the teeth of wheels by Mr. Airy, in the Cambridge Phil. Trans. vol. ii. He has farther shown, that a tooth of any form whatever being cut upon a wheel, it is possible to find a curve which, rolling upon the pitch circle of that wheel, shall by a certain generating point traverse the edge of the given tooth. The curve thus found being made to roll on the circumference of the pitch circle of a second wheel, will therefore trace out the form of a tooth which will work truly with the first. This beautiful property involver

on the plane of one of the wheels with any ger erating circle, and with the pitch circle of that wheel for its base; and if a hypocycloid be described on the plane of the other wheel with the pitch circle of that wheel for its base; and if the faces or acting surfaces of the teeth on the two weeels be cut so as to coincide with this epicycloid and this hypocycloid respectively, then will the wheels be driven correctly by the intervention of these teeth. Parts of two wheels having epicycloidal teeth are represented in the preceding figure.

203. LEMMA.-If the diameter of the generating circle of a hypocycloid equal the radius of its base, the hypocycloid becomes a straight line having the direction of a radius of its base.

Let D and d represent two positions of the centre of such a generating circle, and suppose the generating point to have been at A in the first position, and join AC; then will the generating point be at P in the second position, i. e. at the point where CA intersects the circle in its second position; for join Ca and Pd, then Pda=/PCd+ /CPd=2ACa. Also 2da x Pda=2CA-ACa; .da × Pda--CA× ACa; .arc Aa-arc Pa. Since then the arc aP equals the arc aA, the point P is that which in the first position coincided with A, . e. P is the generating point; and this is true for all positions of the generating circle; the generating point is therefore always in the straight line AC. The edge, therefore, of a hypocycloidal tooth, the diameter of whose generating circle equals half the diameter of the pitch circle of its wheel, is a straight line whose direction. is towards the centre of the wheel.*

2da CA;

the theoretical solution of the problem which Poncelet has solved by the geometrical construction given to Article 200. If the rolling curve be Logarithmic spiral, the involute form of tooth will be generated.

The following very ingenious application has been made of this proper of the hypocycloid to convert a circular into an alternate rectilinear motioL. AB represents a ring of metal, fixed in position, and having teeth cut upon its

TO SET OUT THE TEETH OF WHEELS.

204. All the teeth of the same wheel are constructed of the same form and of equal dimensions: it would, indeed, evidently be impossible to construct two wheels with dif ferent numbers of teeth, which should work truly with one another, if all the teeth on each wheel were not thus alike.

All the teeth of a wheel are therefore set out by the work

man from the same pattern or model, and it is in determining the form and dimensions of this single pattern or model of one or more teeth in reference to the mechanical effects which the wheel is to produce, when all its teeth are cut out upon it and it receives its proper place in the mechanical combination of which it is to form a part, that consists the art of the description of the teeth of wheels.

The mechanical function usually assigned to toothed wheels is the transmission of work under an increased or diminished velocity. If CD, DE, &c., represent arcs of the pitch circle

concave circumference. C is the centre of a wheel, having teeth cut in its circumference to work with those upon the circumference of the ring, and having the diameter of its pitch circle equal to half that of the pitch circle of the teeth of the ring. This being the case, it is evident, that if the pitch circle of the wheel C were made to roll upon that of the ring, any point in its circumference would describe a straight line passing through the centre D of the ring; but the circle C would roll upon the ring by the mutual action of their teeth as it wo d by the contact of their pitch circles; if the circle C then be made to roll upon the ring by the intervention of teeth cut upon both, any point in the circumference of C will describe a straight line passing through D. Now, conceive C to be thus made to roll round the ring by means of a double or forked link CD, between the two branches of which the wheel is received, being perforated at their extremities by circular apertures, which serve as bearings to the solid axis of the wheel. At its other extremity D, this forked link is rigidly connected with an axis passing through the centre of the ring, to which axis is communicated the circular motion to be converted by the instrument into an alternating rectilinear motion. This circular motion will thus be made to carry the centre C of the wheel round the point D, and at the same time, cause it to roll upon the circumference of the ring. Now, conceive the axis C of the wheel, which forms part of the wheel itself, to be prolonged beyond the collar in which it turns, and to have rigidly fixed upon its extremity a bar CP. It is evident that a point P in this bar, whose distance from the axis C of the wheel equals the radius of its pitch circle, will move precisely as a point in the pitch circle of the wheel moves, and therefore that it will describe continually a straight line passing through the centre D of the ring. This point P receives, there fore, the alternating rectilinear motion which it was required to communicate.