THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SERIES.] 



f 



APRIL 1919. » r 



XXXI. On the Problem of Random Vibrations, and of 

 Random Flights in one, two, or three Dimensions. By 

 Lord Rayleigh, O.M., F.R.S.* 



WHEN a number (n) of isoperiodic vibrations of unit 

 amplitude are combined, the resultant depends upon 

 the values assigned to the individual phases. When the 

 phases are at random, the resultant amplitude is indeter- 

 minate, and all that can be said relates to the probability of 

 various amplitudes (r), or more strictly to the probability 

 that the amplitude lies within the limits r and r-\-dr. The 

 important case where n is very great I considered a long 

 time ago f with tli8 conclusion that the probability in question 

 is simply 



2 



-e~ r% i n rdr (1) 



n ' 



The phase (0) of the resultant is of course indeterminate, and 

 all values are equally probable. 



The method then followed began with the supposition that 

 the phases of the unit components were limited to 0° and 

 180°, taken at random, so that the points (r, 6), representative 

 of the vibrations, lie on the axis = 0, and indifferently on 

 both sides of the origin. The resultant ,v, being the difference 



* Communicated by the Author. 



f Phil. Mag-, vol. x. p. 73 (1880) ; Scientific Papers, vol. i. p. 491. 



Phil. Mag. S. G. Vol. 37. No. 220. April 1919. 2 A 



