249 
After multiplying by the permutability index, a little reduction 
in which use is made of SrirLING’s formula, gives with sufficient 
approximation 
8 
En nn MOGEN ne ee EN 
dv dw U, dw 2 
In this, terms have been omitted which remain constant when 7 
is constant and the division into elements remains the same. > and XE 
dr dw 
indicate summations taken over all the elements dv and dw. Use has 
also been made of the fact that the elements dv are all of the same size. 
The expression which one obtains for BorTzZMANN’s H-function by 
reversing the sign of (6), agrees to the degree of approximation 
here given, with the expression given by Ornstein’) for this case. 
State of equilibrium : 
This is determined by the condition that for constant v and u, 
W is a maximum. The condition v = const. is fulfilled by varying 
only the values of n,,, ete. which occur in (6), and ‘keeping 
ni Nl Nn constant. With regard to the condition w= const. 
the assumption that the molecules behave as if they were rigid 
smooth spheres, of central symmetry (so that their density is constant 
or only a function of the distance from the centre, and therefore 
their mass centres and their geometrical centres coincide) enables us 
to disregard angular speeds about axes through their mass centres. 
To enable us to find an expression for the potential energy we 
shall assume that the maero-volume-elements are great in comparison 
with the sphere of action of a molecule. With reference to the 
potential energy we shall, in conformity with the assumptions under- 
lying the van per Waars attractive forces, further assume that, in 
states of equilibrium and in states closely approximating thereto, 
each sphere of action can be regarded as being uniformly filled 
with the number of molecules which that sphere would contain if 
the molecules were uniformly spread over the whole macro-volume 
element. In making this assumption cover even the molecules which 
lie near the boundaries of the volume-element we neglect the influence 
of capillary forces. Calling the potential energy of. molecules 
; aw ; 
uniformly spread over the volume v, — —, with dy constant, we may 
Dv 
write the whole potential energy contained in the element dv, as 
dwn,” 
n*dv, 
IL. S. Oansren. Diss. 1908, p. 60. 
. The condition for the energy then becomes 
