.0054829728 ----------- = .000341, 16.083 which is the well-established coefficient of centrifugal force--the centrifugal force of one pound making one revolution per minute in a circle of one foot radius. So we find that the height _AC_ of this triangle represents the centrifugal force of the reciprocating parts which, in any case, we can ascertain by the formula _WRr²C_, _W_ being the weight of the body; _R_ being the length of the crank; _r_ being the number of revolutions per minute, and _C_ being the coefficient .000341. This accounts for the fact that the reciprocating parts are perfectly balanced by an equal weight revolving opposite the crank. In my treatise on the Richards Indicator and the Development and Application of Force in the Steam-engine, I have given a full exposition of this action here briefly outlined, and to that the reader is referred. I have only to add that this computation is for horizontal engines. In vertical engines the effect of gravity must be considered, adding on the upward stroke and deducting on the downward stroke. Also the counterbalance in the crank-disk of vertical engines must be limited to the horizontal fling of the crank end of the connecting-rod, and all balancing must be as nearly as possible in the same plane. In this respect double-crank engines have this advantage, that one half of the counterweight can be put on each side of the center line. It is evident that the heavier the reciprocating parts and the more rapid the speed the greater the security for smooth and silent running. However loose the brasses and however sudden the impact of the steam on the piston, and however early or late the admission, there can be no sound or jar, if the inertia of the reciprocating parts is sufficient to equal the force of the entering steam, and if this is in excess it can do no harm. It is also evident that under these conditions at any point in the stroke the change of pressure to the opposite side of the crank-pin is made insensibly. Some two or three weeks after this exhibition I received a note from President Barnard asking me to call upon him. On my responding to this invitation, he said to me that he had listened to my exposition of this action before the Polytechnic Club of the Institute, but he did not understand it; he had witnessed the experiments with my shop engine, but while he could not question the action in silencing all knock on the centers, still he did not understand it, and not until he investigated the problem in his own way by the method of the calculus did it become plain to him, and he could not see how I had ever been able to arrive at the exposition of the action without employing that method. This explains why the subject had not been considered in the report of the judges. President Barnard afterward kindly gave me a copy of his demonstration, to insert in my book on the Richards Indicator.