We shall now have this proportion, (see [fig. 4]) 268 (A B) : 1035 (B C) ∷ 2769792 (no. of particles in one inch of circumference of base) : x = 10696771 particles in that part of the line B C, which corresponds with that inch of the circumference. Thus each of the latter particles measured in the direction A B, is equal to the fraction 1⁄10696771ths of an inch. And if that fraction be taken for the arc g d, ([fig. 5]) then to find the length of the line d e, (on which the friction of this and all other geering depends) we must use this analogy; 12 inch (rad. of wheel) : 1⁄10696771 of an inch (chord g d) ∷ 1⁄10696771 of an inch (g d) : d e, the line required = 1⁄1273050917917292 of an inch. This result is still beyond the truth, as we do not know how much smaller the ultimate molecules of gold are.
To advert now to some of the practical effects of this system, I would beg leave to present a form of the teeth, the sole working of which would be a sufficient demonstration of the truth of the foregoing theory. A, B, ([fig. 6]) are two wheels of which the primitive circles or pitch-lines touch each other at o. As all the homologous points of any screw-formed tooth, are at the same distance from the centres of their wheels, I am at liberty to give the teeth a rhomboidal form, o t i; and if the angle o exists all round both wheels, (of which I have attempted graphically to give an idea at D G,) in this case, those particles only which exist in the plane of the tangents f h, &c. and infinitely near that plane passing at right angles to it through the centres A and B, will touch each other; and there, as we have already proved, no sensible motion of the kind producing friction, exists between the points in actual contact. I might add, as the figure evidently indicates, that if any such motion did exist, the angles o would quit each other, and the figure of such teeth become absurd in practice; but on the other hand, if such teeth can exist and work usefully (which I assert they can, nay that all teeth have in this system a tendency to assume that form at the working points;) this circumstance is of itself a practical evidence of the truth of the foregoing theory, and of what I have said concerning it.
It must have been perceived that I have in some degree anticipated the demonstration of my third proposition, namely, that the epicycloidal or any other given form of the teeth, is not essential to this geering. It appears that teeth formed as epicycloids, will become more convex by working; since the base of the curve is the only point where they suffer no diminution by friction; whilst those of every other form, that likewise penetrate beyond the primitive circles of the wheels, will also assume a figure of the same nature, by the rounding off of their points, and the hollowing of the corresponding parts of the teeth they impel; and that operation will continue till an angle similar to that at o, but generally more obtuse, prevails around both wheels; when all sensible change of figure or loss of matter will cease, as the wheels now before you will evince.
On the right of the drawing, ([fig. 6]) the teeth of the wheel B are angular, (suppose square) and those of the wheel C rounded off by any curve s, within an epicycloid. All that is necessary to remark in this case is, that the teeth of the wheel B must not extend beyond its primitive circle, whilst the round parts of those of the wheel C, do more or less extend beyond its primitive circle; whence it becomes evident, that the contact of such teeth, (if infinite in number) can only take place in the plane of the common tangent at right angles to A B; also that if these teeth are sufficiently hard to withstand ordinary pressure, without indentation in these circumstances, there is no perceptible reason for a sensible change of form; since this contact only takes place where the two motions are alike, both in swiftness and direction. A fact I am going to mention may outweigh this reasoning in the minds of some, but cannot invalidate it. I caused two of these wheels made of brass, to be turned with rapidity under a considerable resistance for several weeks together, keeping them always anointed with oil and emery, one of the most destructive mixtures known for rubbing metals; but after this severe trial, the teeth of the wheels, at their primitive circles were found as entire as before the experiment. And why? Certainly for no other reason than that they worked without sensible friction.
Hitherto nothing has been said of wheels in the conical form, usually denominated mitre and bevel geer. But my models will prove, that they are both comprehended in the system. The only condition of this unity of principle is, that the axes of two wheels, instead of being parallel to each other, be always found in the same plane. With this condition, every property above-mentioned, extends to this class of wheels, which my methods of executing also include, as indeed they do every possible case of geering.
Being afraid of trespassing on the time of the society, I have suppressed a part of this paper, perhaps already too long; but I hope I may be indulged with a few remarks on the application of those wheels to practical purposes. And first, as to what I have myself seen; these wheels have been used in several important machines to which they have given much swiftness, softness or precision of motion as the case required. They have done more; they have given birth to machines of no small importance, that could not have existed without them. In rapid motions they do all that band or cord can perform, with the addition of mathematical exactness, and an important saving of power. In spinning factories these properties must be peculiarly interesting; and in calico-printing, where the various delicate operations require great precision of motion. In clock-making also, this property is of great importance in regulating the action of the weight, and thus giving full scope to the equalizing principle whatever it be. I may add, it almost annuls the cause of anomaly in these machines, since a given clock will go with less than 1⁄4 of the weight usually employed to move it. Another useful application may be mentioned; in flatting mills, where one roller is driven by a pinion from the other, there is a constant combat between the effort of the plate to pass equally through the rollers, and the action of the common geering, which is more or less convulsive. Whence the plate is puckered, and the resistance much increased, both which circumstances these wheels completely obviate; and many similar cases might be adduced.
I shall only add, that my ambition will be highly gratified if, through the approbation of this learned society, I may hope to contribute to the improvement and perfection of the manufactures of this county; and if the invention be found of general utility to my much loved country.”