ut that for greater
densities the factor 4 would decrease. If we represent the volume of the
molecules by [beta], the quantity b will be found to have the following
form:--

{ /4[beta]\ /4[beta]\^2 }
b = 4[beta]{ 1 - [gamma]1( ------- ) + [gamma]2( ------- ) &c.}
{ \ v / \ v / }

Only two of the successive coefficients [gamma]1, [gamma]2, &c., have
been worked out, for the determination requires very lengthy
calculations, and has not even led to definitive results (L. Boltzmann,
_Proc. Royal Acad. Amsterdam_, March 1899). The latter formula supposes
the molecules to be rigid spheres of invariable size. If the molecules
are things which are compressible, another formula for b is found, which
is different according to the number of atoms in the molecule (_Proc.
Royal Acad. Amsterdam_, 1900-1901). If we keep the value of a and b
constant, the given equation will not completely represent the net of
isothermals of a substance. Yet even in this form it is sufficient as to
the principal features. From it we may argue to the existence of a
critical temperature, to a minimum value of the product pv, to the law
of corresponding states, &c. Some of the numerical results to which it
leads, however, have not been confirmed by experience. Thus it would
follow from the given equation that p_c.v_c/Tc = 3/8.pv/T, if the value
of v is taken so great that the gaseous laws may be applied, whereas
Sydney Young has found 1/3.77 for a number of substances instead of the
factor 3/8. Again it follows from the given equation, that if a is
thought to be independent of the temperature, Tc/p_c.(dp/dT)_c = 4
whereas for a number of substances a value is found for it which is near
7. If we assume with Clausius that a depends on the temperature, and has
a value a'.273/T, we find Tc/p_c.(dp/dT)_c = 7 That the accurate
knowledge of the equation of state is of the highest importance is
universally acknowledged, because, in connexion with the results of
thermodynamics, it will enable us to explain all phenomena relating to
ponderable matter. This general conviction is shown by the numerous
efforts made to complete or modify the given equation, or to replace it
by another, for instance, by R. Clausius, P. G. Tait, E. H. Amagat, L.
Boltzmann, T. G. Jager, C. Dieterici, B. Galitzine, T. Rose Innes and M.
Reinganum.

If we hold to the supposition that the molecules