a normal to the curve m'n' at the same point T. Now the

two curves have a common tangent at T; therefore their normals TA and TB at that point are in the same straight line, being both perpendicular to their tangent there. Since then ATB is a straight line, and that the vertical angles at the point o where AB and C,C, intersect are equal, as also the right angles at A and B, it follows that the triangles AoC, and BoC, are similar, and that C10: C20:: C1A: C2B. But CA: CB:: C1O : С2O; .·.С1 : C20 :: C2O : C2O; therefore the points O and o coincide, and the straight line AB, which passes through the point of 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 of 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.*

turning upon one of the angles of that polygon, and therefore that its extremity is, through an infinitely small angle, describing a circular arc about that point.

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 BH 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

S

The point of contact T of the teeth moves along the straight line AB, which is drawn touching the generating circles BH 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 figure, which represents part of two wheels with involute 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.

202. If one circle be made to roll externally on the cir

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 circles. 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.

*The circles from which the involutes are described are called their bases. This cut and that at page 260. are copied from Mr. Hawkins' edition of Camus on the Teeth of Wheels.

cumference of another, and if, whilst this motion is taking place, a point in the circumference 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.

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 exceedingly 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, moreover, when the circle APH 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 APH 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,

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.

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.*

Thus then it appears, that if an epicycloid be described on the plane of one of the wheels with any generating 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 wheels be cut so as to coincide with this epicycloid and this hypocycloid

* 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 construction 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.

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 + L CPd=2ACa. Also 2da CA; ... 2da × Pda=2CA × ACa;

.. da x 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, i. 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.*

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

The following very ingenious application has been made of this property of the hypocycloid to convert a circular into an alternate rectilinear