Now, coming back to what actually does happen, we shall be able to understand how far A rotates round C relatively to the motion of B, when C is fixed and B rolls (Fig. 225). B advances 1/3 of distance round C; A advances 1/3 + 1/4 = 7/12 of distance round B. The fractions, if reduced to a common denominator, are as 4:7, and this is equivalent to 40 (number of teeth on A): 40 + 30 (teeth on A + teeth on C.) To leave the reader with a very clear idea we will summarize the matter thus:--If T = number of teeth on A, _t_ = number of teeth on C, then movement of A: movement of B:: T + _t_: T. Here is a two-speed hub. Let us count the teeth. The chain-ring (= A) has 64 internal teeth, and the central cog (= C) on the axle has 16 teeth. There are four cogs (= B) equally spaced, running on pins projecting from the hub-shell between A and C. How much faster than B does A run round C? Apply the formula:--Motion of A: motion of B:: 64 + 16: 64. That is, while A revolves once, B and the hub and the driving-wheel will revolve only 64/80 = 4/5 of a turn. To use scientific language, B revolves 20 per cent. slower than A. This is the gearing we use for hill-climbing. On the level we want the driving-wheel to turn as fast as, or faster than, the chain-ring. To make it turn at the same rate, both A and C must revolve together. In one well-known gear this is effected by sliding C along the spindle of the wheel till it disengages itself from the spindle, and one end locks with the plate which carries A. Since B is now being pulled round at the bottom as well as the top, it cannot rotate on its own axis any longer, and the whole train revolves _solidly_--that is, while A turns through a circle B does the same. To get an _increase_ of gearing, matters must be so arranged that the drive is transmitted from the chain-wheel to B, and from A to the hub. While B describes a circle, A and the driving-wheel turn through a circle and a part of a circle--that is, the driving-wheel revolves faster than the hub. Given the same number of teeth as before, the proportional rates will be A = 80, B = 64, so that the gear _rises_ 25 per cent. By means of proper mechanism the power is transmitted in a three-speed gear either (1) from chain-wheel to A, A to B, B to wheel = _low_ gear; or (2) from chain-wheel to A and C simultaneously = solid, normal, or _middle_ gear; or (3) from chain-wheel to B, B to A, A to wheel = _high_ gear. In two-speed gears either 1 or 3 is omitted. * * * * * AGRICULTURAL MACHINES. THE THRESHING-MACHINE. Bread would not be so cheap as it is were the flail still the only means of separating the grain from the straw. What the cream separator has done for the dairy industry (p. 384), the threshing-machine has done for agriculture. A page or two ought therefore to be spared for this useful invention. [Illustration: FIG. 226.--Section of a threshing machine.]