f the increment of _p_.

For examples of the application of this method, the reader should refer
to some work of exact science. He will find in Deschanel's _Natural
Philosophy_, c. 32, an account of some experiments by which the
connection between heat and mechanical work has been established. It is
there shown that "whenever work is performed by the agency of heat" [as
in driving an engine], "an amount of heat disappears equivalent to the
work performed; and whenever mechanical work is spent in generating
heat" [as in rubbing two sticks together], "the heat generated is
equivalent to the work thus spent." And an experiment of Joule's is
described, which consisted in fixing a rod with paddles in a vessel of
water, and making it revolve and agitate the water by means of a string
wound round the rod, passed over a pulley and attached to a weight that
was allowed to fall. The descent of the weight was measured by a
graduated rule, and the rise of the water's temperature by a
thermometer. "It was found that the heat communicated to the water by
the agitation amounted to one pound-degree Fahrenheit for every 772
foot-pounds of work" expended by the falling weight. As no other
material change seems to take place during such an experiment, it shows
that the progressive expenditure of mechanical energy is the cause of
the progressive heating of the water.

The thermometer itself illustrates this method. It has been found that
the application of heat to mercury expands it according to a law; and
hence the volume of the mercury, measured by a graduated index, is used
to indicate the temperature of the air, water, animal body, etc., in
which the thermometer is immersed, or with which it is brought into
contact. In such cases, if no other change has taken place, the heat of
the air, water, or body is the cause of the rise of the mercury in its
tube. If some other substance (say spirit) be substituted for mercury in
constructing a thermometer, it serves the same purpose, provided the
index be graduated according to the law of the expansion of that
substance by heat, as experimentally determined.

Instances of phenomena that do not vary together indicate the exclusion
of a supposed cause (by Prop. III (c)). The stature of the human race
has been supposed to depend on temperature; but there is no
correspondence. The "not varying together," however, must not be
confused with "varying inversely," which when regular indicates a true
c