by the motion of the piston _B_, and that the motion of the card will be simultaneous with that of the piston _B_, but of course of shorter stroke. We will assume that the stroke of the card is equal to the length of the atmospheric and vacuum lines _g h_ and _e f_, fig. 31. If now, the piston being at the beginning of the stroke as shown in fig. 30, we admit steam of 85 lbs. effective pressure per square inch (which is equal to 100 lbs. absolute pressure) into the cylinder _A_, it will be conveyed through the pipe _U_ to the cylinder _T_, and will force up the piston ⁸⁵⁄₄₀ or 2¹⁄₈ inches above the atmospheric line, or ¹⁰⁰⁄₄₀ or 2¹⁄₂ inches above the vacuum line, as shown in fig. 32, and the pencil will draw a vertical line, _g i_, on the card, (represented by a dotted line in fig. 32.) We will suppose that steam is admitted during 8 inches of the stroke, and is then cut off. When the piston _B_, fig. 30, has moved that distance, which is one-third of its stroke, the card will also have moved one-third of its stroke, and will stand in relation to the pencil in the position represented in fig. 33, and as the absolute steam pressure in the cylinder was maintained at 100 lbs. while the card was moving that distance, the pencil will have drawn a horizontal line, _i j_. The steam is now cut off and begins to expand, and its pressure is thereby reduced. When the piston of the engine is at half-stroke, the card will also be at half-stroke, and the steam will be expanded from 8 to 12 inches of the stroke. By the rule given in the answer to question 20, its absolute pressure would then be 66²⁄₃ lbs., and the indicator-piston will then be pressed down by the spring, so that the pencil will stand in the position shown in fig. 34, or 66²⁄₃ fortieths of an inch above the atmospheric line. The pencil meanwhile will have drawn the curved line _j k_. When the piston has moved 16 inches, the steam will be expanded to double its volume and its absolute pressure will therefore be 50 lbs., and consequently the pencil will stand 50 fortieths or 1¹⁄₄ inch above the atmospheric line as shown in fig. 35, and the pencil will have continued the curve _j k_ to _l_. At 20 inches the steam will have 40 lbs., and at the completion of the stroke 33²⁄₃ lbs. absolute pressure, and the pencil will have completed the curve _j k l m n_, as shown in figs. 36 and 37. This curve is called the _expansion curve_, and its form is that which mathematicians call a hyperbolic curve. If the steam is exhausted, the indicator-piston will descend and carry the pencil down to the atmospheric line, and the vertical line _n h_, fig. 38, will be drawn. On the return stroke, after the steam is exhausted from the main cylinder _A_, fig. 30, the pencil would draw the atmospheric line _g h_, fig. 38, thus showing that there is no steam pressure under the piston. Such a diagram is called an _indicator diagram_.[16] In practice there are a great many influences which modify it, such as