TRANSACTIONS OF SECTION A. 315 
Consider, for instance, the case of a mono-atomic gas. If the number of its mole 
cules is , one finds 
C being a determinate constant factor. Hence, if we omit the corresponding term 
in the entropy, 
R ce 
Ss =n_— 1 B2 x : 
mB og (v1) 
for which we may write, since ” is a very great number, 
8 
7 = log (y E2), 
an expression which will lead to a number that is neither very small nor very great, 
when the mass considered is comparable with a gramme-molecule. 
Now, if the value of P is multiplied by the number of molecules n, or even by a 
high power of this number, such as n'”, this does not produce any appreciable differ- 
ence in the value of S. Indeed, S is increased in these cases by N log n, or 100 N log n, 
and this is very small in comparison with the above value, because for large numbers 
the logarithm is very much smaller than the number itself. Boltzmann’s formula 
is therefore wholly insensible to such factors as m or m' in the value of the 
probability. 
The way in which, in the case of a gas, P depends on the volume v, may be 
understood by a very simple reasoning. If the m molecules are distributed at random 
over a volume’, the probability that they shall all lie in one half of it is P’= oa , whereas 
P=! if all possible distributions are taken together. The difference between the 
two values is enormous. Yet the corresponding difference in the entropy is no 
more than n = log 2, an expression really corresponding to the change in the entropy 
when the volume is reduced to half its original value. 
In virtue of the property of Boltzmann’s formula here pointed out, one is free to a 
great extent in the choice of the value of P. The probability, for instance, that 
exactly > molecules lie in one half of the volume v and the remaining ones in the 
other half—which is the most probable distribution—is given by 
n! 
— 1 ’ 
"G!) 
praa/?, 
n 
This is much smaller than unity, showing that the exact realisation of the most 
probable distribution is very improbable. Yet it does not make any difference worth 
considering in the value of S, whether this small value or the value 1 is substituted in 
the formula. This exemplifies that, in order to determine the value of the entropy, 
one may as well take for P the probability of the most probable state of things, as the 
much higher value that is obtained if all possible states are included. 
a 
or, with a sufficient approximation 
4. The Structure of the Atom. By Professor E. Ruruerrorp, F.R.S. 
5. The Electrical Resistance of Thin Metallic Films. 
By W. F. G. Swann, A.RB.C.S., D.Sc. 
If the specific resistance of a film is plotted against its time of deposit, 
the curve shows, as is well known, a sharp bend. The theory usually given 
