This section is from the book "Applied Science For Metal Workers", by William H. Dooley. Also available from Amazon: Applied Science For Metal Workers.

The number of turns or revolutions which a gear makes is always proportional to the number of its teeth. It makes no difference how many gears there are :n a train, the gears between the first and last gear have nothing to do with the speed of either of these two. That is to say, the ratio between the speeds of the first and last gear is not changed by putting any number of gears between them. The continued product of the revolutions of the first driver and the teeth of all the driving gears is equal to the continued product of the revolutions of the last follower and the teeth of all the driven gears. The formula for this is RDd=rFf. This principle is true for any number of driving and driven gears. The position of a driver does not affect the speed of the last follower. Thus, either driver in Fig. 154 can be placed at D or at d. Either follower can go on at F or at f without affecting the speed of the last follower.

Fig. 153. - Ratio of Gears.

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