169 
the prescribed total energy / and dissociation N,, V,,...N; the 
total kinetic energy 
Koa Shy SNe eats” Rye eee 
is also fixed (comp. (10) in $ 3 and the computations further on in § 6. 
Changes of type [B]. By the mutual permutations of simi/ar- 
atoms starting from a given y-point new y-points arise *). In connec- 
tion with the X/Y/Z! possible permutations of the individual atoms 
of the same kind a set of X/Y/Z! different y-points in the y-space 
will be seen to belong together and all these points give the gas 
the same / and the same dissociation (V,, N,,...). 
In order to reach the total y-region which agrees with y, in the 
quantities NM and N,, N,,...N; we must combine the changes of 
the two types [A] and [4], in such a manner, however, that no 
portion of the region is counted twice. 
It may be proved, that including the region (A,) altogether 9 
identical regions (A,), (A,),... Aw, are obtained, in this manner, where 
f ED BVA 
p = - E ke A (1 8) ) 
Mi ! N, (ie Nj If on 6 Ns atd. of 
We shall give a few short indications as regards the proof of this statement. 
For this purpose we introduce the notion of “internal” permutation. 
A permutation of the atoms will be called internal, if the result may also be 
obtained by translations and rotations of the rigid molecules. 
Simple instances. 1. Two molecules of the same kind are made to exchange 
their position and orientation by translation and rotation. 2. A molecule of symme- 
try-number ci (comp. eq. (6) ) is made to pass from one orientation to another 
equivalent one 5). 3. The same operations are carried out at the same time 
with a number of molecules. 
An internal permutation carries the phase-point of the system say from 7’ to 
/; but here the following circumstance must be remembered: y’ is still inside 
1) Since each individual atom has six co-ordinate axes of the y space referring 
to it. Thus when two atoms of the system are exchanged, nearly all co-ordinates 
of the point remain unchanged, only 12 co-ordinates exchanging their values 
two by two. ; 
2) BoLTZMANN in his well-known paper: ‘Ueber das Arbeitsquantum, welches bei 
chemischen Verbindungen gewonnen werden kann,” [Wied. Ann. 22 (1884), p. 39. 
Wisschensch, Abh. III, p. 71] has determined a similar combinatory quantity. But 
in comparing the quantity Z in his equation (3) with our , the difference should 
be noted which is referred to in the next note 3. 
3) In a molecule of the constitution ABA, therefore, the permutation of the two 
A atoms is an internal one, in a molecule of the form AAB it is not. With 
BOLTZMANN the latter permutation would also have to be regarded as internal. 
This difference is due to the fact that with him the changes of type [A] form a 
wider class than with us and contain all exchanges of similar atoms inside the 
same molecule. 
