In the first place it is a well-known fact that the motion of a piston in the cylinder of a steam engine is not a uniform one, but increases in speed from the beginning of the stroke to the middle, and diminishes in speed from the middle to the opposite end. The cause of this is that the crank revolves at a uniform speed during the entire revolution, but the piston moves much less at the beginning of the stroke, with a given amount of revolution of the crank, than it does at the middle. This is shown in fig. 140, in which _A_ is a cylinder and _B_ the piston and _a b c d_ the path of the crank. Now while the crank moves from _a_ to 1, or ¹⁄₁₂ of a revolution, the piston has moved 1³⁄₈ in., or a distance equal to that from _a_ to 1′ or to the base of a perpendicular drawn from 1 to the centre line _a c_. While the crank moves from 1 to 2, or through the second twelfth of a revolution, the piston has moved from 1′ to 2′, or 4³⁄₈ in., or 2³⁄₄ in. further than during the first twelfth of the crank’s revolution. During the third twelfth of the revolution the piston moves from 2′ to 3′, or 6 in., thus showing that it continues to increase in the distance moved during each period of the revolution of the crank until the latter has made a quarter revolution. The speed of the piston then begins to diminish until it reaches the end of the stroke. It is slightly affected by the angularity of the connecting-rod, as already explained, but for the present this is disregarded. It is obvious now that if the momentum, or actual energy stored up in the piston and other reciprocating parts after they have passed the middle of the stroke, added to the pressure behind the piston, is greater than the resistance offered by the crank, the motion of the latter will then be accelerated and thus conveyed to the moving engine and train. If, however, there is any momentum in the piston when it reaches the end of the stroke, evidently it can exert no power to cause the crank to revolve, but must be expended by producing a pressure on the crank-pin and thus on the axle-boxes. Not only will such a pressure not cause the crank to revolve, but it will be more difficult to turn the crank with such a pressure against it than it would be without. The momentum of the piston and other reciprocating parts at the dead point therefore creates a resistance to the movement of the crank instead of helping to turn it. It will also be observed that after the crank has moved slightly from the dead point, any pressure on the piston will exert very little force which will tend to turn the crank. In fact the nearer the piston is to the end of the stroke the greater is the proportion which the friction of the crank-pin and axle bears to the useful effect of the strain in causing the crank to turn. Calculation shows that for about three degrees on either side of the dead points the effect of pressure on the crank-pin is actually to retard the engine. If now the piston reaches