246 
If, by introducing special assumptions regarding the molecules and 
their mutual forces, one calculates, in the manner here indicated, 
the entropy S in the equilibrium condition for given energy U and 
volume JV, one obtains directly a fundamental equation of state from. 
which both the specific heats and the thermal equation of state can 
be deduced. 
§ 3. Deduction of the virial-coefficient B for rigid, smooth spheres 
of central symmetry and subject to vAN ver WAALS’ forces of 
attraction. 
Although this problem has already been repeatedly treated, first 
by vaN DER Waats himself in the deduction of his equation of state, 
and since then, in particular, by PraNcK *) by a method which is 
essentially the same as that here developed, we may yet utilise 
this simple case as an introduction to our treatment of the succeeding 
more complex cases. The description of these can then be shortened 
by referring to corresponding definitions and operations in the present 
problem. 
Determination of the maecro-complexion: 
Two states which a macro-observer can distinguish as different 
may be regarded as having their differences arise from the presence 
in definite elements of volume of different numbers of molecules in 
the two cases, and also from different distributions of speed in those 
volume-elements. To determine a macro-complexion we therefore 
take the three-dimensional spaces which are available for each 
molecule with respect to its coordinates ,y,z and the velocities §, n, § 
of its centre, and divide them up into equal elements (dv,dy,dz, =) 
dv,, dv, … dor, and (a§,dn,d5, =) dw,, dw, ... dw. 
In this we make dv,... so great that each contains on the whole 
a great number of molecules, and yet sufficiently small for the density 
variations within those elements of volume to escape the notice of 
the maero-observer; the elements dw,... are also chosen so great 
that to each corresponds a large number of molecules in dv,... and 
yet so small that dg, ,dy,,dc,... are small in comparison with the 
mean speed. 
The group macro-complexion is now determined by the conditions that 
n,, unspecified molecules “are present’ in dv, dw, 
(4) 
Nk 53 ” ” DE 55 dvi. dw; . 
Determination of the micro-complexion : 
As far as velocities?) are concerned, the micro-complexion can be 
1) M. PranckK, Berlin Sitz.-Ber. 32 (1908), p. 633. 
2) As the velocities differ from the momenta only by a constant factor, we may 
