76 Lord Rayleigh on the Resultant of a large Number of 



If n of these lines be taken at random in the directions ±x, 

 the probability of resultants also along ±#, and of various 

 magnitudes, is given by preceding expressions. We will now 

 suppose that \n are distributed at random along ±#, and ^n 

 along ±y, and inquire into the probabilities of the various 

 resultants, The probability that the end of the representative 

 line, or, as we may consider it, the representative point, lies in 

 the rectangle dx dy is evidently 



— e n dx dy. 

 irn * 



Substituting polar coordinates r, 6 and integrating with 

 respect to 6, we see that the probability of the representative 

 point of the resultant lying between the circles r and r + dr is 



2 — 



— e~~ n r dr. 



n 



This is therefore the probability of a resultant vibration with 

 amplitude between the values r and r + dr. In this case there 

 are n components distributed in four rectangular directions; 

 and we have supposed that \n exactly are distributed along 

 ±x, and \n along ±y. It is important to remove this restric- 

 tion, and to show that the result is the same when the distri- 

 bution is perfectly arbitrary in respect to all four directions. 



In order to see this, let us suppose that \n-\-m are distri- 

 buted along ±x and \n—m along ±y, and imagine how far 

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

 representative point of the resultant lying in the rectangle 

 dx dy is now expressed by 



]_ *2 V 2 



7r^/(n 2 —4:m 2 ) 



q n+2m n—2m q]x dy 





7r*/(n 2 —4:7n 2 ) 



1 «r2 2?nr2 



dxdy 



cos 20 



it*/ (n 2 — Am 2 ) 

 Also 



Q 7l2-4m2 Q n.2—4m2 y> (%f> q]6 



n 2n 2mr2cos20 C vrpnfi ~\ 



as we find on expanding the exponential and integrating. 

 Thus the chance of the representative point lying between the 

 circles r and r + dr is 



2rdr - * r2 -A \ , rri l r* , > 



V(?i 2 — Am 2 ) I (n 2 —Am 2 ) 2 J 



