1 //" 



306 BELL SYSTEM TECHNICAL JOURNAL 



expanding 



00 



exp 21 {iuy\n/n\ 



as a power series in u, integrating termwise using 



— / (m<o-)" exp —iuax — —^ \du = ( — )"o-~' v?'"^(a-), 

 Zir J— 00 |_ 2 J 



1 _^ 



and finally collecting terms according to their order in powers of p~^'^, gives 



r,/T\ -1 (0)/ \ ^3 0" (3). s j^ X4Cr (4) / > , Xs 0" (6) / \ , 



P(/) ~ (T <^ U) - ^y- <p '{x)+\ -j^ if' (X) + - -^ <^^ X^) + • • • 



(1.6-3) 



The first term is 0(v~^'^), the second term is 0(v~^), and the term within 

 brackets is 0(i'~^ ^). This is Edgeworth's series.^ The first term gives the 

 normal distribution and the remaining terms show how this distribution is 

 approached as r — > oc . 



1.7 The Fourier Components of /(/) 

 In some analytical work noise current is represented as 



m = f + t (a. COS '-f' + b. sin f) (1.7-1) 



where at a suitable place in the work T and iV are allowed to become infinite. 

 The coefficients a„ and i„ , 1 < w < iV, are regarded as independent random 

 variables distributed about zero according to a normal law. 



It appears that the association of (1.7-1) with a sequence of disturbances 

 occurring at random goes back many years. Rayleigh and Gouy suggested 

 that black-body radiation and white light might both be regarded as se- 

 quences of irregularly distributed impulses. * 



Einstein and von Laue have discussed the normal distribution of the 

 coefficients in (1.7-1) when it is used to represent black-body radiation, this 

 radiation being the resultant produced by a great many independent os- 



5 See, for example, pp. 86-87, in "Random Variables and Probability Distributions" 

 by H. Cramer, Cambridge Tract No. 36 (1937). 

 '^Phil. Mag. Ser. 5, Vol. 27 (1889) pp. 460-469. 

 7 A. Einstein and L. Hopf, Ann. d. Physik 33 (1910) pp. 1095-1115. 



M. V. Laue, Ann. d. Physik 47 (1915) pp. 853-878. 



A. Einstein, ^nw. d. Physik 47 (1915) pp. 879-885. 



M. V. Laue, Ann. d. Physik 48 (1915) pp. 668-680. 



I am indebted to Prof. Goudsmit for these references. 



