It may be remarked, that when a thread winds off from one circle to another, and these circles touch one another at the circumference, a point in the thread will describe an epicycloid; and the same epicycloid would be described by making the circle from which the thread winds off, the generating circle, the other being the base.

The length of involute teeth may be determined with sufficient accuracy by the rule, Art. 36. (Ed.)

SUPPLEMENTARY OBSERVATIONS.

45. The foregoing Essay was written several years before the publication of the Supplement to the Encyclopædia Britannica. Professor Robison has there (Vol. II. page 103, 106) described and recommended the mode which will be found, Chap. III. of forming teeth of wheels by involutes of circles. Dr. Brewster, however, in his second Edition of Ferguson's Lectures, Vol. II. page 227, observes, that this principle is not new; De la Hire having long ago considered the involute of a circle, as the last of the exterior epicycloids; which it may be proved to be, if we consider the generating straight line as a curve of infinite radius.*

Professor Robison says,† that "this form of teeth admits of several teeth to

* See Art. 53. (ED.)

+ Encyclopædia Britannica, Volume XX. page 104. See also Rees's Cyclopædia, Art. Clock Movement.

be acting at the same time, (twice the number that can be admitted in M. De la Hire's method.) This, by dividing the pressure among several teeth, diminishes its quantity on any one of them, and therefore diminishes the dents or impressions which they unavoidably make on each other. It is not altogether free from sliding and friction, but the whole of it can hardly be said to be sensible. The whole slide of a tooth, three inches long, belonging to a wheel of ten feet diameter, does not amount to th of an inch, a quantity altogether insignificant.*

In the same article, this highly respectable philosopher was mistaken, in supposing, with other eminent authors, that the mutual action of the teeth, (when formed into epicycloids, by the method of M. Camus,) is absolutely without friction, and in saying, "That one tooth only APPLIES itself to the other, and ROLLS on it, but does

* See Art. 51. (Ed.)`

I

not SLIDE or RUB on it in the smallest degree. This makes them last long, or rather does not allow them to wear." A very slight examination of the figures given in various parts of the preceding Essay, will, I hope, show, that the point of contact must slide from the pitch line of the conducting tooth outwards. Dr. Young, in his Natural Philosophy, Vol. II. page 183, says, that "a form [of teeth,] without friction, is perfectly impracticable, although, for a single tooth, possible."

46. In the first volume of the same work, he makes the following judicious observations on our present subject:

"It has been supposed by some of the best authors that the epicycloidal tooth has also the advantage of completely avoiding friction; this is however by no means true, and it is even impracticable to invent any form for the teeth of a wheel, which will enable them to act on other teeth without friction.

"In order to diminish it as much as possible, the teeth must be as small and as numerous as is consistent with strength and durability; for the effect of friction always increases with the distance of the point of contact from the line joining the centres of the wheels.

"In calculating the quantity of the friction, the velocity with which the parts slide over each other has generally been taken for its measure: this is a slight inaccuracy of conception, for, as we have already seen, the actual resistance is not at all increased by increasing the relative velocity; but the effect of that resistance, in retarding the motion of the wheels, may be shown, from the general laws of mechanics, to be proportional to the relative velocity thus ascertained. When it is possible to make one wheel act on teeth fixed in the concave surface of another, the friction may be thus diminished in the proportion of the difference of the diameters to their sum.