294 
MATHEMATICS: N. WIENER 
Proc. N. A. S. 
3 A detailed paper giving the full spectra of the light atoms with photographs will 
soon be published in the Astrophysical Journal in collaboration with Mr. I. S. Bowen. 
4 Zeit. Physik, 1, 1920 (439). 
5 Ibid., 2, 1920 (470). 
6 This convention is more logical than that used in a former paper (cit. 2 ) in naming 
the L a line. 
7 Astroph. J., 43, 1916 (102). 
THE AVERAGE OF AN ANALYTIC FUNCTIONAL AND THE 
BROWNIAN MOVEMENT 
By Norbkrt Wiener 
Department of Mathematics, Massachusetts Institute of Technology 
Communicated by A. G. Webster, March 22, 1921 
The simplest example of an average is the arithmetical mean. The 
arithmetical mean of a number of quantities is their sum, divided by their 
number. If a result is due to a number of causes whose contribution to 
the result is simply additive, then the result will remain unchanged if for 
each of these causes is substituted their mean. 
Now, the causes contributing to an effect may be infinite in number 
and in this case the ordinary definition of the mean breaks down. In this 
case seme sort of measure may be used to replace number, integration 
to replace summation, and the notion of mean reappears in a generalized 
form. For instance, the distance form one end of a rod to its center of grav- 
ity is the mean of its length with reference to its mass, and may be written 
in the form 
J* ldm -7- J* dm 
where I stands for length and m for mass. It is to be noted that / is a func- 
tion of m, and that the mean we are defining is the mean of a function. 
Furthermore, the quantity, here the mass, in terms of which the mean is 
taken, is a necessary part of its definition. We must assume, that is, a 
normal distribution of some quantity to begin with, in this case of mass. 
The mean just discussed is not confined to functions of one variable; it 
admits of an obvious generalization to functions of several variables. Now, 
there is a very important generalization of the notion of a function of sev- 
eral variables: the function of a line. For example, the attraction of a 
charged wire on a unit charge in a given position depends on its shape. The 
length and area of a curve between two given ordinates depend on its 
shape. As a curve is essentially a function, these functions of lines may 
be regarded as functions of functions, and as such are known as func- 
tional. Since a function is determined when its value is known for all 
arguments, a functional depends on an infinity of numerical determi- 
