from its axis, that circle had been made to roll (carrying its plane with it) on the circumference of N. For conceive O to represent a third plane on which the centres of E and F are fixed. It is evident that if, whilst the circles M and N are revolving by their mutual contact, the plane O, to which their centres are both fixed, be in any way moved, no change will thereby be produced in form of the curve PQ, which the point P in the plane of M is describing upon the plane of N, such a motion being common to both the planes M and N.* Now let the direction in which the circle N is revolving be that shown by the arrow, and its angular velocity uniform; and conceive the plane O to be made to revolve about F with an angular velocity (Art. 74) which is equal to that of N.

but in an opposite direction, communicating this angular velocity to M and N, these revolving meantime in respect to one another, and by their mutual contact, precisely as they did before.+

It is clear that the circle N being carried round by its own proper motion in one direction, and by the motion common to it and the plane O with the same angular velocity in the opposite direction, will, in reality rest in space; whilst the centre E of the circle M, having no motion proper to itself, will revolve with the angular velocity of the plane O, and the various other points in that circle with angular velocities, compounded of their proper velocities, and those which they receive in common with the plane O, these velocities neutralising one another at the point L of the circle, by which point it is in contact with the circle N. So that whilst M revolves round N, the point L. by which the former circle at any time touches the other, is at rest; this quiescent point of the circle M nevertheless continually varying its position on the circumferences of both circles, and the circle M being in fact made to roll on the circle N at rest.

Thus, then, it appears, that by communicating a certain common angular velocity to both the circles M and N about

Thus for instance, if the circles M and N continue to revolve, we may evidently place the whole machine in a ship under sail, in a moving carriage, or upon a revolving wheel, without in the least altering the form of the curve, which the point P, revolving with the plane of the circle M, is made to trace on the plane of N, because the motion we have communicated is common tc both these circles.

M and N may be imagined to be placed upon a horizontal wheel 0, first at rest, and then made to revolve backwards in respect to the motion of N.

the centre F, the former circle is made to roll upon the other at rest; and, moreover, that this common angular velocity does not alter the form of the curve PQ, which a point P in the plane of the one circle is made to trace upon the plane of the other, or, in other words, that the curve traced under these circumstances is the same, whether the circles revolve round fixed centres by their mutual contact, or whether the centre of one circle be released, and it be made to roll upon the circumference of the other at rest.

This lemma being established, the truth of the proposition stated at the head of this article becomes evident; for if M roll on the circumference of N, it is evident that P will, at any instant, be describing a circle about their point of contact L.*

Since then P is describing, at every instant, a circle about L when M rolls upon N, N being fixed, and since the curve described by P upon this supposition is precisely the same as would have been traced by it if the centres of both circles had been fixed, and they had turned by their mutual contact, it follows that in this last case (when the circles revolve about fixed centres by their mutual contact) the point P is at any instant of the revolution describing, during that instant, an exceedingly small circular arc about the point L; whence it follows that PL is always a perpendicu lar to the curve PQ at the point P, or a normal to it. Now let p be a point exceedingly near to T in the curve m'n', which curve is fixed upon the plane of the circle A. It is evident that, as the point p passes through its contact with the curve mn at T (see Art. 198.), it will be made to describe, on the plane of the circle B, an exceedingly small portion of that curve mn. But the curve which it is (under these circumstances) at any instant describing upon the plane of B has been shown to be always perpendicular to the line DT; the curve mn is therefore at the point T perpendicular to the line DT; whence it follows that the curve m'n' is also perpendicular to that line, and that DT is a normal to both those curves at T. This is the characteristic property of the curves mn and m'n', so that they may satisfy the condition of a continual contact with

For either circle may be imagined to be a polygon of an infinite number of sides, on one of the angles of which the rolling circle will, at any instant, be in the act of turning.

one another, whilst the circles revolve by the contact of their circumferences at D, and therefore conversely, so that these curves may, by their mutual contact, give to the cir cles the same motion as they would receive from the contact of their circumferences.

200. To describe, by means of circular arcs, the form of a tooth on one wheel which shall work truly with a tooth of any given form on another wheel.

Let the wheels be required to revolve by the action of

their teeth, as they would by the contact of the circles ABE and ADF, called their primitive or pitch circles. Let AB represent an are of the pitch circle ABE, included between any two similar points A and B of consecutive teeth, and let AD represent an arc of the pitch circle ADF equal to the arc AB, so that the points D and B may come simultaneously to A, when the cir cles are made to revolve by their mutual contact. AB and AD are called the pitches of the teeth of the two wheels. Divide each of these pitches into the same number of equal parts in the points a, b, &c., a', b', &c.; the points a and a', b and ', &c., will then be brought simultaneously to the point A. Let mn represent the form of the face of a tooth on the wheel, whose centre is C,, with which tooth a corresponding tooth on the other wheel is to work truly; that is to say, the tooth on the other wheel, whose centre is C,, is to be cut, so that, driving the surface mn, or being driven by it, the wheels shall revolve precisely as they would by the contact of their pitch circles ABE and ADF at A. From A measure the least distance Aa to the curve mn, and with radius Aa and centre A describe a circular are a3 on the plane of the circle whose centre is C. From a measure, in like manner, the least distance aa', to the curve mn, and with this distance aa' and the centre a, describe a circular arc Sy, intersecting the arc as in S. From the point b measure similarly the shortest distance ba" to mn, and with

the centre b' and this distance ba" describe a circular arc 76, intersecting By in 7, and so with the other points of division. A curve touching these circular arcs 3, 7, 79, &c., will give the true surface or boundary of the tooth.

*

In order to prove this let it be observed, that the shortest distance aa' from a given point a to a given curve mn is a normal to the curve at the point a' in which it meets it; and therefore, that if a circle be struck from this point a with this least distance as a radius, then this circle must touch the curve in the point a', and the curve and circle have a common normal in that point.

Now the points a and a' will be brought by the revolution of the pitch circles simultaneously to the point of contact A, and the least distance of the curve mn from the point A will then be aa', so that the arc By will then be an arc struck from the centre A, with this last distance for its radius. This circular arc By will therefore touch the curve mn in the point a' and the line aa', which will then be a line drawn from the point of contact A of the two pitch circles to the point of contact a' of the two curves mn and m'n', will also be a normal to both curves at that point. The circles will therefore at that instant drive one another (Art. 196.) by the contact of the surfaces mn and m'n', precisely as they would by the contact of their circumferences. And as every circular arc of the curve m'n' similar to By becomes in its turn the acting surface of the tooth, it will, in like manner, at one point work truly with a corresponding point of mn, so that the circles will thus drive one another truly at as many points of the surfaces of their teeth, as there have been taken points of division a, b, &c. and arcs aß, By. &c.t

*This method of describing, geometrically, the forms of teeth is given, without demonstration, by M. Poncelet in his Mécanique Industrielle, 3me partie, Art. 60. The greater the number of these points of division, the more accurate the form of the tooth. It appears, however, to be sufficient in most cases, to take three points of division, or even two, where no great accuracy is required. M. Poncelet (Méc. Indust. 3 partie, Art. 60.) has given the following, yet easier, method by which the true form of the tooth may be approximated to with sufficient accuracy in most cases. Suppose the given tooth N upon the one wheel to be placed in the position in which it is first to engage or disengage from the required tooth on the other wheel, and let Aa and Ab be equal arcs of the pitch circles of the two wheels whose point of contact is A. Draw Aa the shortest distance between A and the face of the tooth N; join aa; bisect that line in m, and draw mn perpendi cular to aa intersecting the circumference Aa in n. If from the centre n a circular arc be described passing through the points a and a, it will give the required for of the tooth nearly.

N

INVOLUTE TEETH.

201. The teeth of two wheels will work truly together if they be bounded by curves of the form traced out by the extremity of a flexible line, unwinding from the circumference of a circle, and called the involute of a circle, provided that the circles of which these are the involutes be concentric with the pitch circles of the wheels, and have their radii in the same proportion with the radii of the pitch circles.

G

E

Let OE and OF represent the pitch circles of two wheels, AG and BH two circles concentric with them and having their radii C ̧A and C,B in the same proportion with the radii C,O and CO of the pitch circles. Also let mn and m'n' represent the edges of teeth on the two wheels struck by the extremities of flexi ble lines unwinding from the circumferences of the circles AG and BH respectively. Let these teeth be in contact, in any position of the wheels, in the point T, and from the point T draw TA and TB tangents to the generating circles GA and BH in the points A and B. Then does AT evidently represent the position of the flexible line when its extremity was in the act of generating the point T in the curve mn; whence it follows, that AT is a normal to the curve mn at the point T*; and in like manner that BT is 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 Co Co :: CA: C,B. But C,A: CB: C,O: CO; ... C,o: Co :: CO. CO; therefore the points O and o coincide, and the straight line AB, which passes through the point of

a

E

For if the circle be conceived a polygon of an infinite number of sides, it is evident that the line, when in the act of unwinding from it at A, is turning upon one of the angles of that polygon, and therefore that its extremity is through an infinitely small angle, describing a circular are about that point.