324: Lord Rayleigh : Problem of Random Vibrations, 



was interested in the " Problem of the Random Walk," which 

 he thus formulated: — "A man starts from a point and 

 walks I yards in a straight line; lie then turns through any 

 angle whatever and walks another I yards in a second straight 

 line. He repeats this process n times. I require the pro- 

 bability that after these n stretches he is at a distance between 

 r and r -\-dr from his starting point 0." 



" The problem is one of considerable interest, but I have 

 only succeeded in obtaining an integrated solution for two- 

 stretches. I think, however, that a solution ought to be 

 found, if only in the form of a series in powers of 1/n, when 

 n is large "*. In response, I pointed out that this question is 

 mathematically identical with that of the unit vibrations with 

 phases at random, of which I had already given the solution 

 for the case of n infinite f, the identity depending of course 

 upon the vector character of the components. 



In the present paper I propose to consider the question 

 further with extension to three dimensions, and with a com- 

 parison of results for one, two, and three dimensions J. The 

 last case has no application to random vibrations but only to 

 random /lights. 



One Di 



mension. 



In this case the required information for any finite n is 

 afforded by Bernoulli's theorem. There are n + 1 possible 

 resultants, and if we suppose the component amplitudes, or 

 stretches, to be unity, they proceed by intervals of two from 

 -\-n to — n, values which are the largest possible. The pro- 

 babilities of the various resultants are expressed by the cor- 

 responding terms in the expansion of (J + ■£)". For instance 

 the probabilities of the extreme values ±n are (l/2) n . And 

 the probability of a combination of a positive and b negative 

 components is 



»)"«!TT «» 



in which a + b — n, making the resultant a — b. The largest 

 values of (6) occur in the middle of the series, and here a 

 distinction arises according as n is even or odd. In the 



* 'Nature,' vol. lxxii. p. 294 (1905). 



t ' Nature/ vol. lxxii. p. 318 (1905) ; Scientific Papers, vol. v. p. 256. 



X It will be understood that we have nothing here to do with the 

 direction in which the vibrations take place, or are supposed to take 

 place. If that is variable, there must first be a resolution in fixed 

 directions, and it is only after this operation that our present problems 

 arise. 



