456 - PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



1, what is to make the radius of the wheel for the first revolution will be equal 

 to this in J-s-, or 1 line; but for 2, 3, and 4 revolutions, and so on, we must add 

 ig. of a line to the radius already laid down, for as many revolutions as the pinion 

 must make more than the wheel: a wheel of 12 teeth will consequently have 

 •ff of a line radius or diameter; one of 18 teeth, -fi; one of 24 teeth if, and 

 so on. The same thing holds true in all the other pinions, whatever the 

 number of them may be, where the same rule is to be observed. Thus on 10 

 revolutions of a pinion of 10, we must deduct from the diameter of the wheel 1 

 diameter of the pinion; as many on 8 revolutions of a pinion of 8; on 7 of a 

 pinion of 7j '^s many; on (3 of a pinion of 6, as many; and so on for all the 

 movers acting by revolutions and teeth. 



The diameter of tlie wheel must of necessity increase in a ratio of the actual 

 revolutions of the pinion, and not in that of its apparent diameter; since we are 

 considering the working parts of wheels and pinions, where the angles, relatively 

 to the change of the curves and the circumference, must be reciprocally in the 

 same proportion, to operate constantly the degrees of raising which are required. 

 We may easily conceive, that as the teeth of a wheel become constantly more 

 parallel to each other, and approach to the straiglit line, as their number 

 increases, the depth in that case need not be so great, and the curve being 

 shorter is more favourable to the uniformity of the frictions than on wheels that 

 are few in number: this is what most commonly happens to pinions of 6, the 

 numbers of whose wheels hardly ever exceed 6o or 72 teetli. For if, in the 

 usual method of using a pinion-gage, we were to take, on a wheel of 12 or 18 

 teeth, a little more than the 3 points (which pomts from the nature of their 

 angles would hitch into the 3 teeth at one and the same time) supposed to be in 

 due proportion with a pinion of 6, and that afterwards we were to make use of the 

 same measure on a wheel of 42 teeth, the natural consequence would be, that this 

 pinion would be, by -i, or half its diameter, too large. A pinion of 7, supposed to be 

 of a proper size, with a wheel of 2 1 teeth, would, according to the same measure of 

 3 full teeth taken on a wheel of 70, be too large by its whole diameter, insomuch 

 that the wheel, instead of 70 teeth, ought to have but 63, &c. Though the supe- 

 rior angles, that is, the circumference of the wheels from 12 to 120, and above ad 

 infinitum, relatively to the size of a pinion given according to our rule, are and 

 ought to be invariably the same; the lower angles, that is, the thickness of the 

 teeth, are yet extremely different in every wheel. For instance, all the wheels 

 from 12 to 120 being sloped, and roundid with tiie same file, the teeth of the 

 wheel of 48 are half as thick again as those of 1 2, &c. In [jroportion as 

 the parallelism of the angles increases, tlie teeth of course become larger and 

 fuller, and each tooth consequently always exceeds the intermediate space in the 

 same proportion, and gradually as far as tlie right line. This determines the 



