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XIII.—A Problem in the Random Distribution of Particles

Published online by Cambridge University Press:  14 February 2012

Philip Eggleton  and
William Ogilvie Kermack
Philip Eggleton
Affiliation:
Physiology Department, University of Edinburgh, Royal College of Physicians Laboratory, Edinburgh
William Ogilvie Kermack
Affiliation:
Physiology Department, University of Edinburgh, Royal College of Physicians Laboratory, Edinburgh
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Summary

  1. 1. An n : l is defined as a group of n particles which can be contained within a seeker length l moving around a closed line of length L on which N particles are distributed at random. An expression has been obtained for the average number of distinct n : l's per closed line.

  2. 2. An expression has also been derived for the average numbers of n : l's in the corresponding problem where the line of length L containing the N particles is open and not closed.

  3. 3. Analogous problems in two dimensions are considered, in which the particles are arranged at random on a plane and the place of the seeker line is taken by an orientated rectangle. Exact expressions are given for the desired averages.

  4. 4. The extension of the methods used to analogous problems in three dimensions is discussed. Exact expressions have not been obtained, but approximations are given which hold when n is much greater or much smaller than x, the average number found within the seeker area.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 62 , Issue 1 , 1944 , pp. 103 - 115
Copyright
Copyright © Royal Society of Edinburgh 1944

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