other; the entire depth or rather length of a tooth is made up, therefore, of the sum of the addenda of the driver and follower, added to this allowance for clearing, which in practice is made of the pitch and termed freedom;

1

10

It is essentially necessary that each pair of teeth should continue in action until the next pair have come into contact, therefore the sum of the arcs of approaching and receding action, must be at least equal to the pitch, that is, F+f=1. But it is better that they should continue in action longer than this, in order to divide the working pressure between more teeth, as well as to prevent the chance of one tooth escaping before the next begins. It is therefore unnecessary to proportion the addendum so accurately as to give the entire arc of action a constant length. It is merely required to find a value that will be sufficient in all cases to prevent the teeth from escaping too soon. Now the expression (1) shows at once that the greatest addendum is required for the smallest numbers of teeth when the arc of action is given; and hence a rule assigned for the small numbers will serve for all cases.

If equal wheels of 15 work together with an arc of receding action of pitch, the expression (1) will give K=28 for the necessary addendum; therefore the millwrights' value (K=·3) is sufficient for all cases of higher numbers than 15. But for smaller numbers the addendum will be greater and must be calculated. For example, the limiting cases in the Table, (page 108) will all be found to require a much greater addendum, varying from about 63 to 5, in the different examples.

153. The arc through which the action of the teeth is continued is governed by the magnitude of the addendum; and as the arc of approach depends on the addendum of the follower, and the arc of recess on the addendum of the driver, we are at liberty to give these arcs any required proportion by properly adjusting these addenda.

Now, considering merely that the friction which takes place before the line of centers is of a different and more injurious character than that which happens after passing that line,* it would seem that the best method would be to exclude altogether any action between the teeth until the line of centers is passed, by giving no addendum to the follower whatever; thereby * Vide Chapter on Friction below.

On

making its true diameter equal to its geometrical diameter. the other hand, it has been shown (Art. 32), that the quantity of friction in both cases increases rapidly with the distance of the point of contact from the line of centers. If the action be entirely confined to one side of the line of centers, it must be continued to a proportionably greater distance from that line, and so the teeth at the extremity of their action may incur greater abrasion and friction than they have lost by avoiding contact before the line of centers.

The best method, then, is to adjust the addenda so that there shall be less action before coming to the line of centers than after it; but the exact proportion between these arcs of action cannot be assigned for want of proper data; for although the fact is certain, no experiments have been hitherto made to compare these two kinds of friction.

154. To examine the effect of a constant addendum upon the ratio of the arcs of approach and recess, put E=e in (2);

When equal wheels work together, or N=n, then f= F, or the arcs of action before and after the line of centers are equal. When a wheel drives a pinion, N is greater than n, and ƒ greater than F; but if a pinion drive a wheel, then n is greater than N, and F than f. In the first case, there is more action before the line of centers than after it, and in the second, the reverse. It appears, then, that the constant addendum of the millwrights produces an effect exactly contrary to the principles just laid down, in every case except that of a pinion driving a wheel; and this is one reason why the action in this case is so much smoother than when a wheel drives a pinion. In fact, any rule that fixes the proportion of the addenda will make the ratio of the two arcs of action vary exceedingly. However, it appears from the expression E F2 2N+n == X

e f2 2n + N'

that the ratio of the addenda is constant when the ratio of the arcs of action and also of the number of teeth is constant; if, therefore, the ratio of the arcs of action is determined, a small table will give the ratio of the addenda corresponding to the principal ratios of numbers of teeth.

The following table of values of

E

e

is calculated for three

different ratios of the two arcs of action; namely, supposing them to be equal, double, or in the proportion of about 2 to 3.

Example. In clocks and watches the wheels always drive the pinions, and the ratio of their numbers varies from 8 to 10. In

225

Mr. Reid's rule (Art. 149) the ratio of the addenda is = 1·5;

150

but from the third column of the Table it appears that this is scarcely enough even to give an equal action before and after the line of centers, and that it would be better to take a ratio of three, which would give the simpler rule,

This rule gives an addendum of about the pitch to the

driver, and to the follower; and may safely be adopted when the wheels drive, or if the wheels be equal; but when the pinion drives, then

To apply the third solution (Art. 119) to the formation of the teeth of wheels.

155. Teeth whose forms are derived from the previous solutions, and especially the latter, are the most commonly adopted in practice; but they are subject to this inconvenience: a wheel of a given pitch and number of teeth, for example 40, if it be made to work correctly with a wheel of 50 teeth, will not suit a wheel of any other number, as 100. This is obvious, for the diameter of the describing circle by which the epicycloid is traced must be made equal to the radius of the pitch circle of the wheel with which the teeth are to work, and will therefore be, in this example, twice as large in the second case as in the first, producing different epicycloids.

In the modern practice of making cast-iron wheels this objection is a very serious one, as it compels the founder to make a new pattern of a wheel of a given pitch with 40 teeth, for every combination that it may be required to make of such a wheel with others; and so on for wheels of every other number.

Besides, it often happens in machinery that one wheel is required to drive at the same time two or more wheels whose numbers of teeth are different, and in this case the teeth cannot be correctly formed at all on the principles hitherto explained.

In cast wheels, then, it is especially essential that the teeth should be shaped so as to allow a given wheel to work correctly with any other wheel of the same pitch; and this may be done by employing the following corollary from the third solution.*

156. If for a set of wheels of the same pitch a constant describing circle be taken and employed to trace those portions of the teeth which project beyond each pitch line by rolling on the exterior circumference, and those which lie within it by rolling on its interior circumference, then any two wheels of this set will work correctly together.

157. Fig. 102 represents a pair of wheels of such a set.

Here A, B are the centers of motion as usual. TdD or TgG the constant describing circle. This is employed to trace the

* Transactions of Civil Engineers, vol. ii. p. 91, in which I stated this principle for the first time.

faces or portions of the teeth that lie beyond the pitch circle FTƒ of the driver, as qr, by rolling upon it, and the flanks or portions that lie within the pitch circle E Te of the follower, as pm, by rolling within it; consequently, by the third solution, these curves will work together with a constant velocity ratio, and the describing circle TdD will be the locus of contact; which beginning upon the line of centers between the point r of the driving tooth, and the point m of the following tooth, will gradually Fig. 102.

recede from the driver's center A, and approach the follower's centre B; the teeth finally quitting contact at the point q of the driver, and the root p of the follower, their action being confined to their recess from the line of centers.

In the same manner, the same constant describing circle at TgG is employed to trace the flanks rs which lie within the pitch circle FTf of the driver, and the faces mn which lie without the pitch circle ETe of the follower; TGg will be the locus of contact which begins between the root s of the driver and the