after passing the line of centers is given; it appears then that for every value of N a value of n may be assigned, a less number than which will make the action of the teeth impossible; and it is of some practical importance to determine these limiting values of n in every case, that we may avoid setting out impossible pairs of numbers in wheelwork.

103. A formula may be investigated thus: produce dT towards G, and from A draw AG perpendicular to and meeting it in G ;

Now the angle TBd and the radius BT are given by the conditions, and also the arc Ta, which is the supposed arc of action; whence Tf is known;

But if we attempt to extract the value of AT from the above expression, it will be found to be so involved as to make a direct solution of the equation impossible, although approximations may be obtained.

However, on account of the practical importance of the question, I have arranged in the following Tables the exact required results, which I derived organically from the diagram of fig. 51, by constructing it on a large scale with moveable rulers.

TABLE I. FOR SPUR-WHEELS.

TABLE of the least numbers of teeth that will work with given pinions. (Tooth Space).

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TABLE II. FOR ANNULAR WHEELS.

TABLE of the greatest numbers of teeth that will work with given pinions. (Tooth Space).

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N. B. The case of annular wheels differs from that of spur-wheels in this respect, that, with a given pinion a smallnumbered wheel works with a greater angle of action than a large-numbered one, and therefore we have to assign the greatest number that will work with each given pinion. This will easily appear if a similar diagram to 51 be constructed for the case of annular wheels.

104. In these Tables I have supposed the tooth of the wheel to equal the space throughout, and have given the

whole of the limiting cases, and under three suppositions : first, that the arc of action Ta shall be equal to the pitch, in which case, if required, the teeth of the follower may be cut down to the pitch circle, and the contact of the teeth thus confined to their recess from the line of centers; for since the action of each pair of teeth continues through a space equal to the pitch, it is clear that at the moment one pair quits contact the next will begin. However, as some allowance must be made for errors of workmanship, it is better to allow the teeth to act a little before they come to the line of centers; or else, by selecting numbers removed from the limiting cases in the Table, to enable the teeth to continue in action through a greater space than one pitch. This principle will be examined more at length presently.

The limiting numbers under two other suppositions are inserted in the Tables, namely, that the arc of action Ta, shall equal and of the pitch, and when these are employed it is of course necessary that an arc of action, at least equal to and of the pitch respectively, shall take place between the teeth before they reach the line of centers.

It appears that a smaller pinion may be employed to drive than to follow. Thus, when the action begins at the line of centers the least wheel that can drive a pinion of eleven is 54, but the same pinion can drive a wheel of 21 and upwards; again, nothing less than a rack can drive a pinion of ten, but this pinion can drive a wheel of 23, and upwards. No pinion of less than ten leaves can be driven, but pinions as low as six may be employed to drive any number above those in the Table. And, lastly, the least pair of equal pinions that will work together is sixteen. These limits being geometrically exact, it is better in practice to allow more teeth than the Table assigns.

105.

Other problems of the same nature as those already given might be suggested; as for example, to find the least numbers that can be employed when, without considering the relative action before and after the line of centers, the teeth are supposed to be drawn, as in fig. 48, with entire points both in the driver and follower, and the tooth equal the space; on which suppositions it would be found that the least possible number of teeth in a pair of equal wheels is five, that four will just work with six, and three with about twelve, and that two will not even work with a rack.

106. To adapt the second solution to racks.-If we suppose the lower pitch circle of fig. 48 to become a right line, we shall obtain a rack, and the epicycloidal faces ab of the rack teeth will become cycloids, because their describing circle BT now rolls upon a right line, but the radial flanks hT of the pinion will remain unaltered. On the other hand, when the radius TA is thus increased to an infinite magnitude the describing circle Tfa coincides with the pitch circle whose center is A, and they unite in one straight line, tangent to the upper pitch circle at T'; which line is, as already stated, the pitch line of the rack. But the curved faces Tg... of the upper pitch circle being thus described by the rolling of a tangent upon its circumference, are involutes of the circle, and the straight flanks Tc of the rack-teeth become parallel to each other and perpendicular to its pitch line.

Also, because Tf the locus of contact now coincides with the pitch line of the rack, therefore the action of the faces of the wheel-teeth is confined to that single point of each rack-tooth which lies upon the pitch line.

Fig. 52 represents a pinion and rack constructed upon the above principles, from which it appears, that, supposing the rack to be the driver, and to move in the direction of