Apparatus for Graphically Showing the Acceleration and Retardation of the Reciprocating Parts of an Engine.] Let the motion be in the direction from the crank. The crank now begins insensibly, by pulling through the spring _e_, to arrest the motion of the weight _h_. This pull will increase in intensity to the end of the stroke, when the weight is brought to rest, and the spring will become correspondingly elongated. Then, by a continuance of the same pull, the crank puts the cross-head and this free weight in motion in the reverse direction. This pull gradually relaxes, until at the mid-stroke it has ceased. The weight _h_ has acquired its full velocity again; all stress is off the spring, and the spring and weight are back in the positions in the box _d_ from which they started. This action is repeated during the opposite half of the revolution, but in the reverse direction, the pull being changed to a push, and the spring being compressed instead of elongated. Thus at every point the position of this free weight shows the amount of the accelerating or retarding force that is being exerted upon it at that point, elongating or compressing the spring. This varying accelerating or retarding force is recorded as follows: A paper _b_, Fig. 2, is stretched on the surface _ff_. This surface is the arc of a circle described about the center _j_, and is secured on the lath _B_, so that as this lath vibrates by the motion of the cross-head the different points in the length of the paper pass successively under the pencil. This is set in the end of the long arm _a_ of the right-angled lever-arms 4 to 1 seen in Fig. 2, which is actuated by the rod _e_ passing centrally through the spring and secured in the head _c_. This pencil has thus imparted to it a transverse motion four times as great as the longitudinal motion of the weight _h_ in the box _d_. The pencil is kept lifted from the paper (as permitted by the elasticity of the arm _a_) by the cord _m_. By letting the pencil down and turning the engine by hand, the neutral line _x_, Fig. 2, is drawn. Then when the engine is running, on letting the pencil come in contact with the paper, the diagonal lines are drawn as shown on Fig. 2. [Illustration: EDWIN F. WILLIAMS]