Steam, entering the cylinder at _a_, is admitted until one-fourth the stroke has been made, when the steam-valve is closed, and the remainder of the stroke is performed without further addition of steam. The variation of steam-pressure is approximately inversely proportional to the variation of its volume. Thus, at half-stroke, the pressure becomes one-half that at which the steam was supplied to the cylinder. At the end of the stroke it has fallen to one-fourth the initial pressure. The pressure is always nearly equal to the product of the initial pressure and volume divided by the volume at the given instant. In symbols, _PV_ _P´_ = ----. _V´_ It is true that the condensation of steam doing work changes this law in a marked manner; but the condensation and reëvaporation of steam, due to the transfer of heat to and from the metal of the cylinder, tends to compensate the first variation by a reverse change of pressure with change of volume. [Illustration: FIG. 28.--Expansion of Steam.] The sketch shows this progressive variation of pressure as expansion proceeds. It is seen that the work done per unit of volume of steam as taken from the boiler is much greater than when working without expansion. The product of the mean pressure by the volume of the cylinder is less, but the quotient obtained by dividing this quantity by the volume or weight of steam taken from the boiler, is much greater with than without expansion. For the case assumed and illustrated, the work done during expansion is one and two-fifths times that done previous to cutting off the steam, and the work done per pound of steam is 2.4 times that done without expansion. Were there no losses to be met with and to be exaggerated by the use of steam expansively, the gain would become very great with moderate expansion, amounting to twice the work done when "following" full stroke, when the steam is cut off at one-seventh. The estimated gain is, however, never realized. Losses by friction, by conduction and radiation of heat, and by condensation and reëvaporation in the cylinder--of which losses the latter are most serious--after passing a point which is variable, and which is determined by the special conditions in each case, augment with greater rapidity than the gain by expansion. In actual practice, it is rarely found, except where special precautions are taken to reduce these losses, that economy follows expansion to a greater number of volumes than about one-half the square root of the steam-pressure; i. e., about twice for 15 or 20 pounds pressure, three times for about 30 pounds, and four and five times for 60 or 65 and for 100 to 125 pounds respectively. Watt very soon learned this general principle; but neither he, nor even many modern engineers, seem to have learned that too great expansion often gives greatly-reduced economy.