ng a certain velocity depending on
the temperature. This velocity is distributed according to the law of
probabilities, and furnishes a quantity of _vis viva_ proportional to
the temperatures. We must attribute extension to the molecules, and they
will attract one another with a force which quickly decreases with the
distance. Even those suppositions which reduce molecules to centra of
forces, like that of Maxwell, lead us to the result that the molecules
behave in mutual collisions as if they had extension--an extension which
in this case is not constant, but determined by the law of repulsion in
the collision, the law of the distribution, and the value of the
velocities. In order to explain capillary phenomena it was assumed so
early as Laplace, that between the molecules of the same substance an
attraction exists which quickly decreases with the distance. That this
attraction is found in gases too is proved by the fall which occurs in
the temperature of a gas that is expanded without performing external
work. We are still perfectly in the dark as to the cause of this
attraction, and opinion differs greatly as to its dependence on the
distance. Nor is this knowledge necessary in order to find the influence
of the attraction, for a homogeneous state, on the value of the external
pressure which is required to keep the moving molecules at a certain
volume (T being given). We may, viz., assume either in the strict sense,
or as a first approximation, that the influence of the attraction is
quite equal to a pressure which is proportional to the square of the
density. Though this molecular pressure is small for gases, yet it will
be considerable for the great densities of liquids, and calculation
shows that we may estimate it at more than 1000 atmos., possibly
increasing up to 10,000. We may now make the same supposition for a
liquid as for a gas, and imagine it to consist of molecules, which for
non-associating substances are the same as those of the rarefied vapour;
these, if T is the same, have the same mean _vis viva_ as the vapour
molecules, but are more closely massed together. Starting from this
supposition and all its consequences, van der Waals derived the
following formula which would hold both for the liquid state and for the
gaseous state:--

/ a \
(p + --- )(v - b) = RT.
\ v^2/

It follows from this deduction that for the rarefied gaseous state b
would be four times the volume of the molecules, b