CHAPTER III.

I shall now proceed to describe a mode of forming the teeth of spur wheels, the first hint of which, I have been informed, was given by Professor Robison, of Edinburgh. In order to understand the description and demonstration, it will be necessary to recollect what was proved in chapter I. No. 2, viz. That if a wheel move uniformly, it is necessary, in order to move another wheel uniformly, that the form of the teeth be such as that the perpendicular from the touching surfaces in all situations, cut the line of centres, in the same point, A, which point divides the line of centres, so that the one part, A, B, shall be to the other, A, G, as the number of teeth in the one wheel, B, is to the number of teeth in the other, G.

This being understood, let B and G be

the centres of the two wheels, and d ef, ghi, the rings upon which the teeth are placed; the diameters of which rings are, to one another, as their number of teeth. If the thread, k d, be lapped round the ring; as it folds up, its extremity, k, will describe the curve k, l, m. It is evident, that the thread, in describing the curve, is perpendicular to it; and the thread, e,

l, f, A, is therefore perpendicular to the curve at 1, or A. In the same manner it may be shown, that the point, n, of the thread, gn, will describe a similar curve, nop, which is perpendicular to the thread gn, ho, i A, at the points n, o and A. If therefore m, A, l, k, be a curve formed by the evolution of the ring B, and n op, be a curve formed by the evolutions of the ring of the wheel G, a line drawn through the point of contact A, perpendicular to the touching surfaces, will touch both rings in the points ƒ and i, and the two curves, supposing them teeth, will act as if the one pulled the other by the thread if. The line, if, will be the line of action; that is, the teeth will always touch each other in a point. of this line, and since this line always passes through the same point of the line of centres, B G, the action will be invariable; so that if the one move uniformly, the other will also move uniformly, and two weights which balance in any one position, will balance in all positions of the teeth.

It is obvious, that though these teeth must work, both before and after passing the line of centres, that they will work with equal truth, whether pitched deep or shallow; a quality peculiar to them, and of very great importance.

SUPPLEMENTARY OBSERVATIONS.

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

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