( 538 ) 
the properties of the thermodynamical.free energy. I shall therefore 
determine the quantity W, which may properly be called the statis- 
tical free energy. 
Taking the sum of the numbers 5, obtained by giving to the num- 
bers 2 all possible values, we get the total number N of the systems 
in the ensemble. I shall represent this sum by >,, so that we have 
the identity 
3 LY Zi 4 Ny &, 
—n — UL I 9 
Rk De (Om nl Noo at _T (w, dzz) OQ 
N= ) S—(227 0m)" New nl y LT 5 e A 
e e PN 
This equation enables us to determine W. In order to find the 
value of 2, 5, we may by means of (VID express the frequency 
S of an arbitrary system in that ¢, of the system of maximum fre- 
quency. From (VID it follows that 
NS dn: th d , adlogw, 
Ss — — —— nN’, = SS 
eee OTs dn, dn, =F 
Se 
x E en 
SE Ak Jd n, % W(vde) (On, + dr) de | 
GO ease 8 
Introducing this into the sum ,, we obtain 
rens 
En En: 1 d log w, 
NS ok So an 1 ieee, ee dd E 
Ee sm | 27, + dn, dn r 
ae EN 
ke 2 
T NX 
+- = , dn, s 43 (dz) (dn, + dn,,) de | 
In my dissertation *) I have shown, that this may be replaced in 
a fair approximation by 
ye = sae En nice dl vol En 
The quantity §, is given by the equation 
WW 3 NzEx 
Ne (220m) 2 Waan 5 ws Me 2 oO 
= = Nes REP 
1 x 
“Ot; (2ar)Flo(n, ...m,...mg)'l2 
1) p.p. 111 and 126. 
