170 
the plase-region (A’), which is formed from y’ by the changes of type [A] A’) 
(that is what the word “internal” is meant to express). 
Taking any phase-point y as starting point, there are always 
D = Ni! Nyt... Np hoo... oN oe 
internal permutations 3) and all y-points reached in that way lie inside one and the 
same 4-region. 
It will therefore be clear, that, if from the original phase y, by an operation 
[A] we produce the phase-region A,, and if we then apply the X/ Y! Z! operations 
of type [B] to every point of the region (A,), we donot obtain X! Y!Z! regions 
similar to (Aj), but altogether only ‘8 (eq. (18) ), since the X/Y/Z! permutations 
of the atoms divide into § groups of 9. internal permutations each. 
Combining (16) and (18) we obtain for the total y-weight of all 
the phases, which belong to given values of V, Hand N,, N,... Ny 
the expression : 
ALYAZ! MN Na N. 
7 A SNN wot} tb +. tess 7 (20) 
NN SONG Gy Ore 0e 
The expressions {u;} contain the integrals with respect to the 
momenta of all possible motions of translation and rotation of the 
molecules which have still to be computed. 
The total kinetic energy of the molecules is fixed by equation 
(17); the integration is to be taken over all the values of the 
momenta which are compatible with it. Calling these momenta for 
a moment p,,p,...pr, } being given by equation (15), we have 
the following relation between these quantities : 
Pf 
2 Ar 
where A,,A,,..., Ap represent the various molecular masses or 
moments of inertia 
Mis Ee Ordi - ; Mj, BOB. 
according to the index (comp. (5) in § 3). 
The multiple integrals with i to the momenta give together 
the surface of the “ellipsoid” (21). Neglecting numbers of the order 1 
as compared with the large number F’, we a use for it the following 
approximation °). 
== 
‚+ =K zn 
') For the operations [A] include all possible translations and rotations of the 
molecules, hence also those, which may replace our internal permutations 
*) The centres of gravity of the Ni molecules of type i may mutually exchange 
their Nj positions and moreover each of these molecules can choose among the 
ci equivalent orientations. 
*) The volume I of a sphere of radius R in a space of F dimensions and its 
