296 MATHEMATICS: N. WIENER Proc. N. A. S. 
where R is the gas-constant, T the absolute temperature of the medium, 
and N the number of molecules per gram-molecule. In the deduction of 
this formula, Einstein makes two important assumptions. The first is 
that Stokes' law holds concerning forces of diffusion. Stokes' law states 
that a force F will carry particle of radius r through a fluid of viscosity r\ 
with velocity F-f- Qirrrj. Einstein's second assumption is that the displace- 
ment of a particle in some interval of time small in comparison with those 
which we can observe is independent, to all intents and purposes, of its 
entire antecedent history. It is the purpose of this paper to show that 
even without this assumption, under some very natural further hypoth- 
eses, the departure of d\/t from constancy will be far too small to observe. 
In this connection, it is well to take note of just what the Brownian 
movement is, and of the precise sense in which Stokes' law holds of parti- 
cles undergoing a Brownian movement. In the study of the Brownian 
movement, our attention is first attracted by the enormous discrepancy 
between the apparent velocity of the particles and that which must animate 
them if, as seems probable, the mean kinetic energy of each particle is the 
same as that of a molecule of the gas. This discrepancy is of course due 
to the fact that the actual path of each particle is of the most extreme sinu- 
osity, so that the observed velocity is almost in no relation to the true ve- 
locity. Now, Stokes' law is always applied with reference to movements 
at least as slow as the microscopically observable motions of a particle. It 
hence turns out that Stokes' law must be treated as a sort of average effect, 
or in the words of Perrin, 4 "When a force, constant in magnitude and di- 
rection, acts in a fluid on a granule agitated by the Brownian move- 
ment, the displacement of the granule, which is perfectly irregular at right 
angles to the force, takes in the direction of the force a component pro- 
gressively increasing with the time and in the mean equal to Ft 6ir£a, F in- 
dicating the force, t the time, f the viscosity of the fluid, and a the radius 
of the granule." 
It is a not unnatural interpretation of this statement to suppose that we 
may assume the validity of Stokes' law for the slower motions which are 
all that we see directly of the Brownian motion, so that we may regard the 
Brownian movement as made up (1) of a large number of very brief, inde- 
pendent impulses acting on each particle and (2) of a continual damping 
action on the resulting velocity in accordance with Stokes' law. It is to be 
noted that the processes which we treat as impulsive forces need not be the 
simple results of the collision of individual molecules with the particle, but 
may be highly complicated processes, involving intricate interactions be- 
tween the particle and the surrounding molecules. It may readily be 
shown by a numerical computation that this is the case. 
It follows from considerations discussed at the beginning of my paper 
on The Average of an Analytic Functional that after a time the probabil- 
