Now, in each revolution of the engine, the cut-off eccentric in effect revolves in the same direction about the center of the main eccentric. Consequently, we may let R C S, parallel to L O M, represent the face of the cut-off valve seat, or, in other words, the back of the main valve, in which the port, C N, corresponds to one of those shown in Fig. 4; and the motion of the cut-off valve over this seat will be precisely, the same as though it were driven directly by an eccentric revolving around the center, C. In determining the position of this eccentric, we proceed upon the assumption that the best results will be effected by such an arrangement that when cutting off at the earliest point required, the cut-off valve shall, at the instant of closing the port, be moving over it at its highest speed. And this requires that the center of the eccentric shall at the instant in question lie in the vertical line through C. [Illustration: Figs. 3-12 IMPROVED STEAM ENGINE.--BY PROF MACCORD.] Next, the least distance to be followed being assigned, the angle through which the crank will turn while the piston is traveling that distance is readily found; then, drawing an indefinite line C T, making with the vertical line, G O, an angle, G C T. equal to the one thus determined, any point upon that line may be assumed as the position of the required center of the cut-off eccentric, at the beginning of the stroke. But again, in order that the cut-off may operate in the same manner when backing as when going ahead, this eccentric must be symmetrically situated with respect to both C and G; and since L O M bisects and is perpendicular to G C, it follows that if the cut-off eccentric be fixed on the shaft, its center must be located at H, the intersection of C T with L M. This would require the edge of the cut-off valve at the given instant to be at Q, perpendicularly over H; and the travel over the main valve would be equal to twice C H, the virtual lever arm of the eccentric, the actual traverse in the valve chest being twice O H, the real eccentricity. This being clearly excessive, let us next see what will occur if the lever arm, CH, be reduced as in the diagram to CK. The edge of the cut-off valve will then be at N; it instantly begins to close the port. CN, but not so rapidly as the main valve opens the port, AB. The former motion increases in rapidity, while the latter decreases; therefore at some point they will become equal in velocity, and the openings of the two ports will be the same; and the question is, Will this maximum effective port area give a sufficient supply of steam?