74 Lord Rayleigh on the Resultcuit of a large Number of 



pectation of intensity, these different degrees of probability 

 must be taken into account. By well-known rules the expres- 

 sion for the expectation is 



1 f i s . / »\s . w ( w ~-l)/ i\a 



-|l.n- + ».(n-2)-+ ~ (n-±y 



+ n(n-lXn- 2) (w _ 6) ^ | 



The value of the series, which is to be continued so long as 

 the terms are finite, is simply n. as may be proved by compa- 

 rison of coefficients of >r in the equivalent forms 



(^ +€ -*)» = 2"(l+i^+...)" 



= e nx + ne' n - 2x + ?~ 9 ' e '»-*:* + .... 



The expectation of intensity is therefore n. and this whether n 

 be great or small. 



In the more general problem, where the phases are distri- 

 buted at random over the complete period, the expression for 

 the expectation of intensity is 



p.p. p. dOdffdjB" 



\ \ ] ...:v-^^...[(cos£+cos£ +cos£ +...;- 



+ (sin0+ sin#' + sin0" + ...) 2 ]. 

 If we effect the integration with respect to 6. we get 



(*2- (*27T ,7/0' JO'f 



...£■%- ....n + (co,d'+co i e-' + ...y 



Jo Jo "* *"" 



+ (sin#'+sin0 v + ...) 2 ]. 



Continuing the process by successive integrations with respect 

 to 6' . 6' r . .... we see that, as before, the expectation of inten- 

 sity is n. 



So far there is no difficulty : but a complete investigation 

 of this subject involves an estimate of the relative probabilities 

 of resultants lying between assigned limits of magnitude. For 

 example, we ought to be able to say what is the probabilitv 

 that the intensity due to a large number (») of equal compo- 

 nents is less than \n. It will be convenient to begin be- 

 taking the problem under the restriction that the phases are of 

 two opposite kinds only. When this has been dealt with, we 

 shall uot find much difficulty in extending our investigation 

 to phases entirely arbitrary. 



By Bernoulli's theorem* we find that the probability that 



* Todrmnter's ' History of the Theory of Probability/ § 993. 



