Theory of Diminishing Entropy. 45 
difficulty arises only in those cases in which the motions of 
separate parts of the bodies under consideration are dyna- 
mically reversible. That being the case, is it possible for a 
motion of the aggregate of such separate parts to be dyna- 
mically irreversible? This question cannot be decided by 
the mere use of a mathematical formula which is wholly in- 
dependent of time, such as Willard Gibbs’s Theorem IX. 
We have a choice of two w ays of dealing with the problem. 
7. One way, which Willard Gibbs in the passage quoted 
by Dr. Bumstead recommends us to follow, is to admit that 
our mathematics, in Willard Gibbs’s language mathematical 
fictions, give us no information whether the motion is for- 
wards or backwards, that is towards or from uniformity, but 
that experience, in Willard Gibbs’s language the real world, 
teaches us that it is towards uniformity. We must, then, 
call experience to our aid as the interpreter of our mathe- 
matics. That is one way. It is perfectly logical, and satis- 
factory so far as the teaching of experience is “complete. 
8. I can call to mind only one writer by whom this method 
has been followed expressly and with decision. Prof. Ladislas 
EK dK 
—- + —- =0 for a 
di * di 
fluid, H denoting kinetic energy of visible motion of masses, 
K kinetic energy of those molecular motions which we cannot 
directly measure. And he says our mathematics give us no 
Natanson, namely, obtains the equation 
. ; BVO Dae oe 
information whether in this equation — is positive and 
dt 
A negative, or vice versd, but we must, he says, accept the 
e . . K e e e e 
teaching of experience, that ee negative, as interpreting 
our mathematical result. 
9, Another way is to make an assumption in aid of our 
mathematics. The mathematical theorem, e.g. Theorem IX. 
of Willard Gibbs’s Chapter XI., although it cannot alone 
determine the direction of motion, may with the aid of an 
assumption determine it. This method is really the appeal 
to experience in another form, because the assumption itself, 
unless consistent with experience, is illegitimate. But as this 
method has been followed by Boltzmann in his H Theorem, 
which is the type of theorems of this kind, we must consider 
“it fully. 
10. Boltzmann’s aaneition leads to this result, that the 
number of pairs of molecules which by their mutual en- 
counter pass out of a certain class in time dé is proportional 
to the number of pairs which are in the class for the time 
being. Let fdu,... dwn denote the number per unit of 
