322 Lord Rayleigh : Problem of Random Vibrations, 



between the number of positive and negative components, is 

 found from Bernoulli's theorem to have the probability 



e-***dx (2)* 



4/(2tth) 



The next step was to admit also phases of 90° and 270°, 

 the choice between these two being again at random. If 

 we suppose hi components at random along ±#, and \n also 

 at random along ± y, the chance of the representative point 

 of: the resultant lying within the area dxdy is evidently 



or in terms of r, 0, 



l-e-&+m*d9dy, .... (3) 



7771 



— e- r2 f n rdrd0 (4) 



irn 



Thus all phases are equally probable, and the chance that 

 the resultant amplitude lies between r and r-\-dr is 



-e~ r2n rdr (1) 



n 



This is the same as was before stated, but at present the con- 

 ditions are limited to a distribution of precisely J n components 

 along x and a like number along?/. It concerns us to remove 

 this restriction, and to show that the result is the same when 

 the distribution is perfectly arbitrary in respect to all four 

 directions. 



For this purpose let us suppose that \n + m are distributed 

 along ±x and \n — m along ±y, and inquire how far the 

 result is influenced by the value of m. The chance of the 

 representative point lying in r dr dd is now expressed by 



£-nr*/(n2_4»t2) e ~2mr* cos 20/(?i2-4m2) r J r £Q ^ 



7T\/(n 2 -4m 2 ) 



Since r is of order \/n, and m/n is small, the exponential 

 containing 6 may be expanded. Retaining the first four 

 terms, we have on integration with respect to 6, 



%rdr irW _ 4m2) / 1 _ mV 1 



^(n 2 -4m 2 ) V (w 2 -4m 2 ) 2 r ' ' 'J ' 



as the chance of the amplitude lying between r and r -f dr. 

 Now if the distribution be entirely at random along the four 



* See below. 



