( 808 ) 
very fair approximation if we take into account the whole space 
Ean 2) 
§ 2. Before applying the theory of the ensembles to the conside- 
ration of real systems, I shall point out somewhat more accurately 
the idea of mean value and probability. Let us suppose to be 
some quantity relating to a given system, the value will bea function 
of the time. The mean value of ¢ in a time-ensemble between the 
time ¢, and 7, shall be given by the formula: 
eh i 
D= EE |r Wee Se te I ee (3) 
1e 
Ir we determine the value of a quantity g (for example a pressure, 
temperature, or density) it is not the value for a given moment 
that comes to expression in our measurements, but a quantity depend- 
ing on the values taken by it in the course of the time. It is obvious 
to suppose that the mean value as it has been given bv (9), is the 
quantity found by our observations *). 
Making use of this hypothesis we may ask for which of the 
time-ensembles imaginable in the space Moi we have to apply 
1) The fact that in the case mentioned above the quantity M for the greater part 
of the systems of a microcanonical ensemble is 0, shows that we cannot use these 
ensembles for tbose cases in which the moment deviales from O. We can use 
for those cases an ensemble formed in the space (2), but it will be more con- 
venient to use an extension of the canonical ensembles which has been indicated 
by Gipps (p. 38). In this kind of ensembles the number of systems lying in an 
element dp, ..dqn of R2, can be represented by 
2 M 
Ne o Mo 7 dq 
the quantities ©, Mj and A are constants. Without going into details for the 
moment, I will only remark that in the pari of the space in the neighbourhood 
of (c=+9, M=M,) the density surpasses that for all the other parts greatly 
2) To prove in general that it is this quantity which determines the observed 
value of 2 will be difficult, for the pressure the proof has been given. 
In the case of a quantity changing with the time we can still use (3) to define the 
value for an interval of time that is sufficiently small to allow us to neglect the change 
of the ohserved quantity. The formula (3) however may be used only if «| does 
t 
not depend on the length of the interval #,—f. It must be possible for the 
divergences between 9 in g| to compensate each other. In the case that ¢] changes 
t t 
with the time the interval for which e| can be treated as a constant must be 
t 
sufficiently long to allow the compensation of the negative and positive values 
of ol—g. 
t 
