that wearing of their brasses or collars, which soon results from a continued and a considerable strain. The existence of this property will readily be admitted, if we conceive AG and BH to represent the generating circles or bases of the teeth, and these to be placed with their centres C, and C, any distance asunder, a band AB (p. 257., note) passing round both, and a point T in this band gene

rating a curve mn, m'n' on the plane of each of the circles as they are made to revolve under it. It has been shown that these curves mn and m'n' will represent the faces of two teeth which will work truly with one another; moreover, that these curves are respectively involutes of the two circles AG and BH, and are therefore wholly independent in respect to their forms of the distances of the centres of the circles from one another, depending only on the dimensions of the circles. Since then the circles would drive at any distance correctly by means of the band; since, moreover, at every such distance they would be driven by the curves mn and m'n' precisely as by the band; and since these curves would in every such position be the same curves, viz. involutes of the two circles, it follows that the same involute curves mn and m'n' would drive the circles correctly at whatever distances their centres were placed; and, therefore, that involute teeth would drive these wheels correctly at whatever distances the axes of those wheels were placed.

THE TEETH OF A RACK AND PINION.

212. To determine the pitch circle of the pinion. Let H represent the distance through which the rack is to be moved by each tooth of the pinion, and let these teeth be N in number; then will the rack be moved through the space N. H during one complete revolution of the wheel. Now the rack and pinion are to be driven by the action of their teeth, as they would by the contact of the circumference of

213. To describe the teeth of the pinion, those of the rack being straight. The properties which have been shown. to belong to involute teeth (Art. 201.) manifestly obtain, however great may be the dimensions of the pitch circle of their wheels, or whatever disproportion may exist between them. Of two wheels OF and OE with involute teeth which work together, let then the radius of the pitch circle of one OF become infinite, its circumference will then become a straight line represented by the face of a rack. Whilst the radius C2O of the pitch circle OF thus becomes infinite, that C,B of the circle from which its involute teeth are struck (bearing a constant ratio to the first) will also become infinite, so that the involute m'n' will become a straight line* perpendicular to the line AB given in position. The involute teeth on the wheel OF will thus become straight teeth (see fig. 1.), having their faces perpendicular to the line AB de

*For it is evident that the extremity of a line of infinite length unwinding itself from the circumference of a circle of infinite diameter will describe, through a finite space, a straight line perpendicular to the circumference of the circle. The idea of giving an oblique position to the straight faces of the teeth of a rack appears first to have occurred to Professor Willis.

termined by drawing through the point O a tangent to the circle AC, from which the involute teeth of the pinion are struck. If the circle AC from which the involute teeth of the pinion are struck coincide with its pitch circle, the line AB becomes parallel to the face of the rack, and the edges of the teeth of the rack perpendicular to its face (fig. 2.).

Now, the involute teeth of the one wheel have remained unaltered, and the truth of their action with teeth of the other wheel has not been influenced by that change in the dimensions of the pitch circle of the last, which has converted it into a rack, and its curved into straight teeth. Thus, then, it follows, that straight teeth upon a rack, work truly with involute teeth upon a pinion. Indeed it is evident,

BEY

that if from the point of contact P (fig. 2.) of such an involute tooth of the pinion with the straight tooth of a rack we draw a straight line PQ parallel to the face ab of the rack, that straight line will be perpendicular to the surfaces of both the teeth at their point of contact P, and that being perpendicular to the face of the involute tooth, it will also touch the circle of which this tooth is the involute in the point A, at which the face ab of the rack would touch that circle if they revolved by mutual contact. Thus, then, the

condition shown in Art. 199. to be necessary and sufficient to the correct action of the teeth, namely, that a line drawn from their point of contact, at any time, to the point of contact of their pitch circles, is satisfied in respect to these teeth. Divide, then, the circumference of the pitch circle, determined as above (Art. 212.), into N equal parts, and describe (Art. 211.) a pattern involute tooth from the circumference of the pitch circle, limiting the length of the face of the tooth to a little more than the length BP of the involute curve generated by unwinding a length AP of the flexible line equal to the distance H through which the rack is to be moved by each tooth of the pinion. The straight teeth of the rack are to be cut of the same length, and the circumference of the pitch circle and the face ab of the rack placed apart from one another by a little more than this length.

It is an objection to this last application of the involute form of tooth for a pinion working with a rack, that the point P of the straight tooth of the rack upon which it acts is always the same, being determined by its intersection with a line AP touching the pitch circle, and parallel to the face of the rack. The objection does not apply to the preceding, the case (fig. 1.) in which the straight faces of each tooth of the rack are inclined to one another. By the continual action upon a single point of the tooth of the rack, it is liable to an excessive wearing away of its surface.

214. To describe the teeth of the pinion, the teeth of the rack being curved.

This may be done by giving to the face of the tooth of

the rack a cycloidal form, and making the face of the tooth of the pinion an epicycloid, as will be apparent if we conceive the diameter of the circle whose centre is

B

R

P

C (see fig. p. 259.) to become infinite, the other two circles remaining unaltered. Any finite portion of the circumference of this infinite circle will then become a straight line. Let AE in the accompanying figure represent such a portion, and let PQ and PR represent, as before, curves generated by a point P in the circle whose centre is D, when all three circles revolve by their mutual contact at A. Then are PR and PQ the true forms of the teeth which would drive the circles as they are driven by their mutual contact at A (Art. 202). Moreover, the curve PQ is the same (Art. 199.) as would be generated by the point P in the circumference of APH; if that circle rolled upon the circumference AQF, it is therefore an epicycloid; and the curve PR is the same as would be generated by the point P, if the circle APH rolled upon the circumference or straight line AE, it is therefore a cycloid. Thus then it appears, that after the teeth. have passed the line of centres, when the face of the tooth of the pinion is driving the flank of the tooth of the rack, the former must have an epicycloidal, and the latter a cycloidal form. In like manner, by transferring the circle APH to the opposite side of AE, it may be shown, that before the teeth have passed the line of centres when the flank of the tooth of the pinion is driving the face of the tooth of the wheel, the former must have a hypocycloidal, and the latter a cycloidal form, the cycloid having its curvature in opposite directions on the flank and the face of the tooth. The generating circle will be of the most convenient dimensions for the description of the teeth when its diameter equals the radius of the pitch circle of the pinion. The hypocycloidal flank of the tooth of the pinion will then pass into a straight flank. The radius of the pitch circle of the pinion is determined as in Art. 212., and the method of describing its teeth is explained in Art. 208.