179. Q.—­What will be the pressure of the steam, under such circumstances, at the end of the stroke?

A.—­If the steam be shut off at half stroke, the pressure of the steam, reckoning the total pressure both below and above the atmosphere, will just be one-half of what it was at the beginning of the stroke.  It is a well known law of pneumatics, that the pressure of elastic fluids varies inversely as the spaces into which they are expanded or compressed.  For example, if a cubic foot of air of the atmospheric density be compressed into the compass of half a cubic foot, its elasticity will be increased from 15 lbs. on the square inch to 30 lbs. on the square inch; whereas, if its volume be enlarged to two cubic feet, its elasticity will be reduced to 7-1/2 lbs. on the square inch, being just half its original pressure.  The same law holds in all other proportions, and with all other gases and vapors, provided their temperature remains unchanged; and if the steam valve of an engine be closed, when the piston has descended through one- fourth of the stroke, the steam within the cylinder will, at the end of the stroke, just exert one-fourth of its initial pressure.

180. Q.—­Then by computing the varying pressure at a number of stages, the average or mean pressure throughout the stroke may be approximately determined?

[Illustration:  Fig. 32.  Diagram showing law of expansion of steam in a cylinder.]

A.—­Precisely so.  Thus in the accompanying figure, (fig. 32), let E be a cylinder, J the piston, a the steam pipe, c the upper port, f the lower port, d the steam pipe, prolonged to e the equilibrium valve, g the eduction valve, M the steam jacket, N the cylinder cover, O stuffing box, n piston rod, P cylinder bottom; let the cylinder be supposed to be divided in the direction of its length into any number of equal parts, say twenty, and let the diameter of the cylinder represent the pressure of the steam, which, for the sake of simplicity, we may take at 10 lbs., so that we may divide the cylinder, in the direction of its diameter, into ten equal parts.  If now the piston be supposed to descend through five of the divisions, and the steam valve then be shut, the pressure at each subsequent position of the piston will be represented by a series, computed according to the laws of pneumatics, and which, if the initial pressure be represented by 1, will give a pressure of .5 at the middle of the stroke, and .25 at the end of it.

If this series be set off on the horizontal lines, it will mark out a hyperbolic curve—­the area of the part exterior to which represents the total efficacy of the stroke, and the interior area, therefore, represents the diminution in the power of a stroke, when the steam is cut off at one- fourth of the descent.  If the squares above the point, where the steam is cut off, be counted, they will be found to amount to 50; and if those beneath that point be counted or estimated, they will be found to amount to about 69.  These squares are representative of the power exerted; so that while an amount of power represented by 50 has been obtained by the expenditure of a quarter of a cylinder full of steam, we get an amount of power represented by 69, without any expenditure of steam at all, merely by permitting the steam first used to expand into four times its original volume.