1051 
which we know from thermodynamics as a function of E and v. 
Accordingly nobody has as yet made use of the real fraction of 
probability for the entropy; always either the numerator or the 
denominator was taken’) or a quantity which differed little from 
it. With regard to the latter it is noteworthy that in & log P terms 
without the factor V may be cancelled against terms with the factor 
N, so that for the calculation of P quantities, the ratio of which is 
of lower order than MN, come to the same thing. 
The denominator which represents the whole extension is available 
in the state of equilibrium, the numerator both in the state of 
equilibrium and outside it. In the state of equilibrium they represent 
the same function of £ and v, so that the quotient is independent 
of it. We should further notice that if not the fraction of probability 
itself is taken, but e.g. the numerator it is of no consequence how 
the space is divided into elements, since the extension itself, determined 
by the state, is decisive. We may therefore just as well take the 
N! 
usual 
§ 3. This expression is, however, open to the objection that the 
dimensions are not in order. In the expression //og P P must bea 
number without dimensions. If P has dimensions, the entropy will 
have logarithmic dimensions, i.e. increase or decrease by a definite 
1) In explanation of this fact, which seems so strange at first sight, that both 
numerator and denominator may be taken as measure for the probability, Prof. 
Lorentz was so kind as to make the following remarks : 
Let Q be a comparatively large region in the phase-extension, (either reduced 
by a function of weight or not), Q' the part of Q that is left when a certain 
restriction (a) is introduced (eg. that the energy lies between two closely defined 
limits) and Q" the part of Q’ where besides e.g. the numbers of molecules are in 
the different elements 7,7”... The latter with restriction (a) may define a state S. 
Now the probability of ali the states satisfying (a) conjointly may be represented 
by Pd (1), or if the denominator is disregarded by P= ()' (2). 
But when remaining inside the limits of the restriction (a) we pay attention to 
ui 
the state S, we may write for its probability P= —- (3). This becomes for the 
Q' 
On account of the “sharpness” of the maxima 
" 
m 
qr 
Qm’ differs so little from Q' that (also in connection with the fact that Jog Q is 
required for the calculation) Q' may be replaced by Qm”. Now (2) becomes there- 
fore: P= Qm" (4). 
It is clear: P is determined by the denominator of (3) in (2), and by the 
numerator in (4), 
most probable state: Pin= 
