the vacuum, or separate the gases preparatory to diffusion, requires an expenditure of energy at least equal to the mechanical effect to he derived.

Since the reversible engine is as efficient as any heat-engine, and since all reversible engines of whatever construction and whatever the working substance have the same efficiency, it is allowable, in discussing the question as to the amount of mechanical effect derivable under given conditions from a given amount of heat, to assume any form of reversible engine, using any working substance which may be most convenient. And it makes no difference whether the engine assumed be practically possible, so long as we know the properties of the working substance well enough to determine its action under the assumed conditions. Sir W. Thomson, before 1851, assuming Carnot's engine with air as the working substance, furnished us with a very complete discussion of this question. The properties of air in relation to heat are very simple. Heat expands and cold contracts it with great uniformity. Compression heats and expansion cools it according to a well-known law. The effects of any change of volume or of temperature in the cylinder of an engine can, therefore, be exactly predicted.

Suppose a given mass of air to be compressed and the heat developed by compression removed, so that its temperature remains constant. The pressure exerted by it will increase, as shown graphically in the annexed diagram, where O a, measured on the horizontal axisFig. 1. O x, represents the initial volume, and a e perpendicular to O x represents the pressure exerted at that volume. O b, O c, and O d, represent other volumes, and b f, c g, d h, the corresponding pressures. The curved line e f g h, drawn through the extremities of the perpendiculars, represents to the eye the relation between volume and pressure when temperature is constant. It is called an isothermal line. Now, suppose the air to be compressed without loss or gain of heat. It is warmed by compression, and the rise of temperature causes it to exert a greater pressure. If, then, the substance be at the same initial volume, pressure, and temperature as before, and it be compressed to the volume O b without loss of heat, the pressure exerted will be b m greater than b f. Similarly the pressure c n will correspond to the volume O c, and d o to d. The line l m n o, which shows the relation between volume and pressure when no heat enters nor escapes, is called an adiabatic line.