Fig. 12.
In [Fig. 12], let a, b, and c represent the pitch circles of three gears of which a and b are in gear, while c is compounded with b; let e be the shaft of a, and g that for b and c. Let a be 60 inches, b = 30 inches, and c = 40 inches in diameter. Now suppose that shaft e suspends from its perimeter a weight of 50 lbs., the shaft being 4 inches in diameter. Then this weight will be at a leverage of 2 inches from the centre of e and the 50 must be multiplied by 2, making 100 lbs. at the centre of e. Then at the perimeter of a this 100 will become one-thirtieth of one hundred, because from the centre to the perimeter of a is 30. One-thirtieth of 100 is 333⁄100 lbs., which will be the force exerted by a on the perimeter of b. Now from the perimeter of b to its centre (or in other words its radius) is 15 inches, hence the 333⁄100 lbs. at its perimeter will become fifteen times as much at the centre g of b, and 333⁄100 × 15 = 4995⁄100 lbs. From the centre g to the perimeter of c being 20 inches, the 4995⁄100 lbs. at the centre will be only one-twentieth of that amount at the perimeter of c, hence 4995⁄100 ÷ 20 = 249⁄100 lbs., which is the amount of force at the perimeter of c.
Here we have treated the wheels as simple levers, dividing the weight by the length of the levers in all cases where it is transmitted from the shaft to the perimeter, and multiplying it by the length of the lever when it is transmitted from the perimeter of the wheel to the centre of the shaft. The precise same result will be reached if we take the diameter of the wheels or the number of the teeth, providing the pitch of the teeth on all the wheels is alike.
Suppose, for example, that a has 60 teeth, b has 30 teeth, and c has 40 teeth, all being of the same pitch. Suppose the 50 lb. weight be suspended as before, and that the circumference of the shaft be equal to that of a pinion having 4 teeth of the same pitch as the wheels. Then the 50 multiplied by the 4 becomes 200, which divided by 60 (the number of teeth on a) becomes 333⁄100, which multiplied by 30 (the number of teeth on b) becomes 9990⁄100, which divided by 40 (the number of teeth on c) becomes 249⁄100 lbs. as before.
It may now be explained why the shaft was taken as equal to a pinion having 4 teeth. Its diameter was taken as 4 inches and the wheel diameter was taken as being 60 inches, and it was supposed to contain 60 teeth, hence there was 1 tooth to each inch of diameter, and the 4 inches diameter of shaft was therefore equal to a pinion having 4 teeth. From this we may perceive the philosophy of the rule that to obtain the revolutions of wheels we multiply the given revolutions by the teeth in the driving wheels and divide by the teeth in the driven wheels.
Fig. 13.