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THE PROCEEDINGS is the official organ of the NATIONAL ACADemy of SCIENCES and of the NATIONAL RESEARCH COUNCIL for the publication of brief accounts of important current researches of members of the ACADEMY and of the COUNCIL and of other American investigators, and for reports on the meetings and other activities of the ACADEMY and of the COUNCIL. Publication in the Proceedings will supplement that in journals devoted to the special branches of science. The Proceedings will aim especially to secure prompt publication of original announcements of discoveries and wide circulation of the results of American research among investigators in other countries and in all branches of science.

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Copyright, 1921, by the National Academy of Sciences

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EDITORIAL BOARD

The HOME SECRETARY and the FOREIGN SECRETARY of the ACADEMY

The CHAIRMAN and the PERMANENT SECRETARY of the NATIONAL Research COUNCIL

WILLIAM DUANE, '23 A. L. DAY, '22
R. G. HARRISON, '23
J. C. MERRIAM, '23
E. H. MOORE, '23
F. SCHLESINGER, '23
W. M. WHEELER, '23
F. G. COTTRELL
C. E. MCCLUNG

GANO DUNN, '22
L. J. HENDERSON, '22
W. J. V. OSTERHOUT, '22
R. M. YERKES, '22

AUGUSTUS TROWBRIDGE
E. B. MATHEWS
CLARK WISSLER

J. M. CLARKE, '21
LUDVIG HEKTOEN, '21
H. S. JENNINGS, '21
R. A. MILLIKAN, '21
W. A. NOYES, '21
C. A. ADAMS
G. W. McCoy

F. L. RANSOME

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DEPARTMENT of MathematTICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Communicated by A. G. Webster, March 22, 1921

1. Conceive a particle free to wander along the x-axis.
probability that it wander a given distance independent
(1) of the position from which it starts to wander,
(2) of the time when it starts to wander,

(3) of the direction in which it wanders.

Suppose the

It may be shown that under these circumstances, the probability that after a time, t, it has wandered from the origin to a position lying between x=xo and x=x1 is

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where t is the time and c is a certain constant which we can reduce to 1

by a proper choice of units. This choice we shall make in what follows. The exponential form of this integral needs no comment, while the mode in which enters results from the fact that

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This identity will be presupposed in all that follows.

Let us now consider a particle wandering from the origin for a given period of time, say from t=0 to t=1. Its position will then be a function of the time, say x=f(t). There are certain quantities-functionalswhich may depend on the whole set of values of ƒ from t=0 to t=1. If we take a large number of particles (i.e. a large number of values of ƒ ) at random, it is natural to suppose that the average value of the functional will often approach a limit, which we may call the average value of the functional over its entire range. What will this average be, and how shall we find it?

If F {f} is a functional depending on the values of ƒ for only a finite num

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