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 evi

dent, 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 con

ceive the diameter of the circle whose centre is C (see fig. p. 236.) 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 repre

sent 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 API 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 API 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.

215. THE TEETH OF A WHEEL WORKING WITH A LANTERN OR

TRUNDLE.

In some descriptions of mill work the ordinary form of the toothed wheel is replaced by a contrivance called a lantern or trundle, formed by two circular discs, which are connected with one another by cylindrical columns called staves, engaging, like the teeth of a pinion, with the teeth of a wheel which the lantern is intended to drive. This combination is shown in the following figure.

It is evident that the teeth on the wheel which works with the lantern have their shape determined by the cylindrical

shape of the staves. Their forms may readily be found by the method explained in Art. 200.

Having determined upon the dimensions of the staves in reference to the strain they are to be subjected to, and upon the diameters of the pitch circles of the lantern and wheel, and also upon the pitch of the teeth; strike arcs AB and AC of these circles, and set off upon them

B

the pitches Aa and Ab from the point of contact A of the pitch circles (if the teeth are first to come into contact in the line of centres, if not, set them off from the points behind the line of centres where the teeth are first to come into contact). Describe a circle ae, having its centre in AB, passing through a, and having its diameter equal to that of the stave, and divide each of the pitches Aa and Ab into the same number of equal parts (say three). From the points of division A, a, in the pitch Aa, measure the shortest distances to the circle ae, and with these shortest distances, respectively, describe from the points of division 7, d of the pitch Ab, circular arcs intersecting one another; a curve ab touching all these circular arcs will give the true face of the tooth (Art. 200.). The opposite face of the tooth must be struck from similar centres, and the base of the tooth must be cut so far within the pitch circle as to admit one half of the stave ae when that stave passes the line of centres.

216. THE RELATION

BETWEEN TWO PRESSURES P, AND P,

APPLIED TO TWO TOOTHED WHEELS IN THE STATE BORDER

ING UPON MOTION BY THE PREPONDERANCE OF P1.

Let the influence of the weights of the wheels be in the first place neglected. Let B and C represent the centres of the pitch circles of the wheels, A their point of contact, P the point of contact of the driving and driven teeth at any period of the motion, RP the direction of the whole resultant pressure upon the teeth at their point of contact, which resultant pressure is equal and opposite to the resistance R of the follower to the driver, BM and CN perpendiculars from the centres of the axes of the wheels upon RP; and BD and CE upon the directions of P, and P.. BD=a,, CE=a,, BM=m,, CN=m

BA=r,, CA=",

P1, P2 radii of axes of wheels.

9, 9, limiting angles of resistance between the axes of the wheels and their bearings.

Then, since P, and R applied to the wheel whose centre is

H