iture 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.

181. _Q._--Then by working an engine expansively, the power of the steam is
increased, but the power of the engine is diminished?

_A._--Yes. The efficacy of a given quantity of steam is more than doubled
by expanding the steam four times, while the efficacy of each stroke is
made nearly one-half less. And, therefore, to carry out the expansive
principle in practice, the cylinder requires to be larger than usual, or
the piston faster than usual, in the proportion in which the expansion is
carried out. Every one who is acquainted with simple arithmetic, can
compute the terminal pressure of steam in a cylinder, when he knows the
initial pressure and the point at which the steam is cut off; and he can
also find, by the same process, any pressure intermediate between the first
and the last. By setting down these pressures in a table, and taking their
mean, he can determine the effect, with tolerable accuracy, of any
particular measure of expansion. It is necessary to remark, that it is the
total pressure of the steam that he must take; not the pressure above the
atmosphere, but the pressure above a perfect vacuum.

182. _Q._--Can you give any rule for ascertaining at one operation the
amount of benefit derivable from expansion?

_A._--Divide the length of stroke through which the steam expands, by the
length of stroke performed with full pressure, which last call 1; the
hyperbolic logarithm of the quotient is the increase of efficiency due to
expansion. According to this rule it will be found, that if a given
quantity of steam, the power of which working at full pressure is
represented by 1, be admitted into a cylinder of such a size that its
ingress is concluded when one-half the stroke has been performed, its
efficacy will be raised by expansion to 1.69; if the admission of the steam
be stopped at one-third of the stroke, the efficacy will be 2.10; at
one- fourth, 2.39; at one-fifth, 2.61; at one-sixth, 2.79; at one-seventh,
2.95; at one-eighth, 3.08. The expansion, however, cannot be carried
beneficially so far as one-eighth, unless the pressure of the steam in the
boiler be very considerable, on account of the inconvenient size of
cylinder or speed of piston which would require to be adopted, the