This he did not tell me at the time, so that I was at a loss to understand his reluctance to admit what was really the great value of his invention. However, I told him I would be willing to attempt its introduction, provided he would allow me to apply it at once to a high-speed engine; that being a field into which the liberating system could not enter. We had quite an argument on this point. I told him his invention interested me only because it would enable two or three times the power to be obtained from a given engine without additional stress on any part, the fly-wheel to be reduced in size, and the means for getting up the speed of machinery to be largely dispensed with. I represented to him also that a high-speed engine ought to be more economical and to give a more nearly uniform motion. He finally agreed to my condition, and I took him directly to the office of Mr. Richards and engaged him to make an analysis and drawing of Mr. Allen’s system under his direction, and soon afterwards gave him an order for the plans for an experimental engine, 6×15 inches, to make 160 revolutions per minute. As the diagram of the link motion was at first drawn, the center of the trunnions vibrated in an arc which terminated at points _on_ the line connecting the center of the engine shaft with the ends of the rocker arms, and which in the diagram on page 48 is named “radius of link.” I determined to work out this link motion myself on a large scale. For this purpose I drew a diagram in which the throw of the eccentric was 4 inches, and the distance from the center of the shaft to that of the trunnions of the link in their mid-position was 12 inches. I made a three-point beam compass. Two of these points were secured permanently on the beam, 12 inches apart. As one of these points traversed the path of the center of the eccentric, the other could be made to traverse the arc of vibration of the trunnions of the link. I divided the former into 40 equal divisions measured from its dead points, making needle-holes in the circle, in which the taper compass-points would center themselves accurately. The paper was firm and the points of division were fixed with extreme care; and they lasted through all my experiments. I then set out 20 corresponding divisions in the arc of vibration of the center of the trunnions. These showed distinctly the modification of the motion at the opposite ends of this vibration as already described. The third point was adjustable on a hinged beam which could be secured in any position. I drew two arcs representing the lead lines of the link, or the lines on which the link would stand when the eccentric was on its dead points. The third point was now secured on its beam at any point on one of the lead lines, when the other points stood, one on the dead point of the eccentric and the other at the end of the trunnion vibration.