.510 Lord Rayleigh on the Resultant 



In the case of n — 2, we may go farther and find the 

 expression for the probability of a given amplitude (r) taken 

 always positive, and phase (0). The amplitude of the com- 

 ponents is unity, and the phases, measured from the centre 

 of the arc, 6 l and 2 . The probability that these phases 

 shall lie between dy and Y + d0 1} 2 and 2 + d0 2 is u~ 2 d0 1 d0 2 . 

 We have now to replace the two variables i} 2 by r, 0, 

 where 



7-=2oosi(0i- 2 ), = i(0 1 + 2 ), 



, or 6 1 = 0± cos" 1 ^), 2 = 0T cos-^Jr), 



making 



d0 1 _ d0 1 _ ±1 dJ0 x _ qPl d02_ 



d0 ~ ' dr ~~ v^-r 2 ) ' dr ~ V (k-r*) ' d0 ' 



Accordingly 



d0,d0 2 ±2d0dr 



The interchange of 0! and # 2 makes no difference to r and 

 '#, so that we may take 



±d0dr_ 

 rf^^-r 2 )' { ' 



:as the chance that the amplitude of the resultant shall lie 

 between r and r-\-dv and the phase between and 0+d0. 

 In (29) cc is supposed not to exceed 2tt. 



As a check, we may revert to the case where a = 2ir. 

 The limits for are then independent of the value of r, and 

 are taken to be —it and +rr. And 



Ur f+"d0 2 dr 2irrdr 



C+"d0 =s 2 



r 2 )J-n Ct 2 ~~ TT 



X 



a/ (4 — 7' 2 ; J—tt « 2 i" ^(4 — r 2 ; 7r 2 rv / (4— r*) 



represents the chance that r shall lie between r and r + dr 

 independently of what may be, in agreement with Pearson's 

 expression *. Integrating again with respect to r, we find 



' 2 2dr 



o 7rv/(4-r 2 )~ ' 



as should be, all cases being now covered. 



In the peneral case the limits for r and are inter- 

 dependent. The possible range for is from — \ol to -\-\ot. 

 (a<7r), but we require the range when r is prescribed. In 

 virtue of the symmetry it suffices to consider a positive 0, 



* Compare Phil. Mag. vol. xxxvii. p. 328 (1919), equation (21). 



