be the point where the line EF intersects the involute mn; then if the teeth on the two wheels are to be nearly of the same thickness at their bases, bisect the line Ab in c; or if they are to be of different thicknesses, divide the line Ab in c in the same proportion*, and strike through the point c an involute curve hg, similar to ef, but inclined in the opposite direction. If the extremity fg of the tooth be then cut off so that it may just clear the circumference of the circle/L, the true form of the pattern involute tooth will be obtained. There are two remarkable properties of involute teeth, by the combination of which they are distinguished from teeth of all other forms, and cateris paribus rendered greatly preferable to all others. The first of these is, that any two wheels having teeth of the involute form, and of the same pitch f, will work correctly together, since the forms of the teeth on any one such wheel are entirely independent of those on the wheel which is destined to work with it (Art. 201.). Any two wheels with involute teeth so made to work together will revolve precisely as they would by the actual contact of two circles, whose radii may be found by dividing the line joining their centres in the proportion of the radii of the generating circles of the involutes. This property involute teeth possess, however, in common with the epicycloidal teeth of different wheels, all of which are struck with the same generating circle (Art. 210.). The second no less important property of involute teeth — a property which distinguishes them from teeth of all other forms — is this, that they work equally well, however far the centres of the wheels are removed asunder from one another; so that the action of the teeth of two wheels is not impaired when their axes are displaced by * This rule is given by Mr. Hawkins (p. 170.); it can only be an approximation, but may be sufficiently near to the truth for practical purposes. It is to be observed that the teeth may have their bases in any other circles than those, fL and cE, from which the involutes are struck. t The teeth being also of equal thicknesses at their bases, the method of ensuring which condition has been explained above. 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 C2 any distance asunder, a band AB (p. 257., note) passing round both, and a point T in this band generating 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 CaO of the pitch circle OF thus becomes infinite, that C2B 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 (Jig. 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, (i.) W i felt that if from the point of contact P (Jig. 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 {Jig. 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 |