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The theory of the mean square variation of a function formed by adding known functions with random phases, and applications to the theories of the shot effect and of light

Published online by Cambridge University Press:  24 October 2008

E. N. Rowland
E. N. Rowland
Affiliation:
Gonville and Caius College
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Extract

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given by

We may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), where

Both the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 4 , December 1936 , pp. 580 - 597
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† DrCampbell, (Proc. Camb. Phil. Soc. 15 (1909), 117–36, 310–28Google Scholar (313)) discussed the special case

with only one type of event.

‡ For the sake of logical clarity we add a D to the equation number when the equation is really a definition of the quantity on the left side and a T when the equation states a principal theorem.

† In this paper there occur several multiple and repeated integrals in bounded spaces of several of the dimensions t, t′, t″, τ, …, n, n′. We write, after Hobson, the multiple integrals in the form

and the repeated integrals in the form

The existence of these integrals cannot be taken for granted, since it is possible for the repeated integrals to exist when the multiple integral does not. (For the definition of a multiple integral, see Hobson's Theory of functions of a real variable, vol. 1, § 338, and for a repeated integral § 362; see also § 365.) Now since f (t), g (t) and f (n, t), g (n, t) are integrable in the Riemann sense, the points of any finite domain at which these functions are discontinuous have measure zero. (See Hobson, loc. cit. § 333.) Hence in this paper, in all the integrals in bounded spaces of several dimensions, the regions of discontinuity of the integrands have measure zero in the space of integration, and the multiple integrals exist. Hence the repeated integrals exist and are equal to the multiple integrals. (See Hobson, loc. cit. § 363.) We may write any of the finite integrals in this paper either as multiple or as repeated integrals, and we may interchange the order of integration in the repeated integrals.

† Loc. cit. § 378.

† Moullin, E. B., Proc. Roy. Soc. A, 147 (1934), 107.CrossRefGoogle Scholar

† Phil. Mag. (5), 10 (1880), 73–8Google Scholar; (5), 27 (1889), 460–469; (6), 37 (1919), 320–347, 498–515.

† Michelson, A. A., Interference methods, Smithsonian Institute, Washington, 1892.Google Scholar

‡ Discussed by Rayleigh, Lord, Phil. May. (5), 27 (1889), 298–304.CrossRefGoogle Scholar