A small anomaly, of form, may be mentioned here to prevent mistakes. The shaded triangle e f g in the Plate, looks higher than the rest: but if higher, it is also longer in the same proportion; and the roller p never reaches the bottom: so that the effect of this Plate is the same as though it resembled the others in every respect. In general the effect of the Plates depends on their length compared with their height: and indeed they must be made higher than the thickness of the wheel to be cut, that the latter may disengage itself from the (fixed) cutter both above and below.

It is proper to observe, that for every pair of wheels there must be a pair of plates; one leaning to the right and the other to the left, (see the [diagram]) but, as before said, the degree of obliquity must be different in each pair, except in the case of equal wheels, when the same plate serves for both; only turning it to the right for one wheel, and to the left for the other. Nor does this offer any difficulty, as the plates are made of common tin plate: which is easily brought to fit the rim, whichever way it is applied. I shall now add another example of the process for finding the length of the plates: and to that end repeat that the plate rim c d, is 22 inches in diameter, or 11 inches radius. Supposing then that we wished to cut a pair of wheels, one of them being 1 inch in diameter and the other 12 inches; both to have teeth inclined 15 degrees to the axes; (as without that they could not work together) to do this we must effect these two proportions:

Both proportions being effected, the first plate is 5896 parts.
And the second491.33do.

The one of course, to be directed toward the right hand, and the other toward the left, on the plate rim; where note, that if the height (1000 parts) is found so numerous as to create confusion, let 100 parts be assumed; when the length of the plate will become 26.8 or 26 and 8⁄10 instead of 268, and the operation will be so much the more simple.

It should be added that this process admits of being further simplified: since the product of 11 inches, radius of the plate rim, multiplied by 268 (tangent of 15 degrees, or length of the plate for a wheel equal in diameter to the plate rim) since this product, I say, is a constant number, namely: 2948—which, divided by the half diameter of any wheel, gives, at once the length of the plate adapted to that operation, in parts of which the height contains 1000; or supposing the height to be 100 only, this constant number becomes (nearly enough for practice) 295. In a word, on a height of plate of 100 parts, when wishing to cut a wheel of 4 inches in diameter, I merely divide 295 by 2, and get for the length of my plate 147.5 parts of which the aforesaid height is 100.

It may possibly be suggested that this method of using plates to determine the obliquity of the teeth is a homely method, giving some trouble in the execution, and leaving a certain degree of roughness in that execution. The fact is allowed; but this method has the advantage of a very general application, which many a better looking apparatus would not present.

Besides, for most uses, these teeth require chiefly that the obliquity should be correct, and not that the surface should be licked like those of a gewgaw. In fine, the principle of this Machine once known, its best form will occur to the reflecting mechanician according to the quality of the work he has in view: And in fact, in the hands of a well known artist, this form has been already varied so as to produce effects much higher wrought than could be drawn from the Machine above described: which latter however in point of generality, still preserves the advantage.