Vol. 7, 1921 
MATHEMATICS: N. WIENER 
295 
nations, and may hence be regarded as in some wise a function of infi- 
nitely many variables. 
To determine the average value of a functional, then seems a reasonable 
problem, provided that we have some convention as to what constitutes 
a normal distribution of the functions that form its arguments. Two 
essentially different discussions have been given of this matter: one, by 
Gateaux, being a direct generalization of the ordinary mean in w-space; 1 
the other, by the author of this paper, 2 involving considerations from the 
theory of probabilities. The author assumes that the functions f(t) 
that form the arguments of his functional have as their arguments 
the time, and that in any interval of the small length e as many receive 
increments of value as decrements of equal size. He also assumes that the 
likelihood that a particle receive a given increment or decrement is inde- 
pendent of its entire antecedent history. 
When a particle is acted on by the Brownian movement, it is in a mo- 
tion due to the impacts of the molecules of the fluid in which it is sus- 
pended. While the retardation a particle receives when moving in a fluid 
is of course due to the action of the individual particles of the fluid, it seems 
natural to treat the Brownian movement, in a first approximation, as an 
effect due to two distinguishable causes: (1) a series of impacts received by 
a particle, dependent only on the time during which the particle is exposed 
to collisions; (2) a damping effect, dependent on the velocity of the par- 
ticle. If we consider one component of the total impulse received by a 
particle under heading (1), we see that it may be considered as a function 
of the time, and that it will have the sort of distribution which will make 
our theory of the average of a functional applicable. 
It will result directly from the previous paper of the author that if f(t) 
is the total impulse received by a particle in a given direction when the unit 
of time is so chosen that the probability that f(t) lie between a and b is 
— I e t ax 
V irtJ a 
then the average value of 
A + f b J(t)G(t)dl + faS b af(s)f(t)H(s,t) dsdt [H(s,t) = H(t,s)] 
will be 
A + f\f 1 1 H (s,t) ds dt. (1) 
We now proceed to a more precise and detailed treatment of the question. 
2. Einstein 3 has given as the formula for the mean square displacement 
in a given direction of a spherical particle of radius r in a medium of 
viscosity rj over a time t, under the action of the Brownian movement, 
the formula 
d] = RTt + 3Trrr)N 
