and Random Flights in one, two, or three Dimensions. 323 



directions, all the values of m of which there is a finite pro- 

 bability are of order not higher than \/n, n being treated as 

 infinite. But if m is of this order, the above expression 

 becomes the same as if m were zero ; and thus it makes no 

 difference whether the number of components along ±# and 

 along ±y are limited to be equal, or not. The previous 

 result is accordingly applicable to a thoroughly arbitrary 

 distribution along the four rectangular directions. 



The next point to notice is that the result is symmetrical 

 -and independent of the directions of the rectangular axes, 

 from which we may conclude that it has a still higher 

 generality. If a total of n components, to be distributed 

 along one set of rectangular axes, be divided into any number 

 of large groups, it makes no difference whether we first 

 obtain the probabilities of various resultants of the groups 

 separately and afterwards of the final resultants, or whether 

 we regard the whole n as one group. But the probability 

 in each group is the same, notwithstanding a change in the 

 system of rectangular axes ; so that the probabilities of 

 various resultants are unaltered, whether we suppose the 

 whole number of components restricted to one set of rect- 

 angular axes or divided in any manner between any number 

 of sets of axes. This last state of things is equivalent to no 

 restriction at all ; and we conclude that if n unit vibrations 

 of equal pitch and of thoroughly arbitrary phases be com- 

 pounded, then when n is very great the probability of various 

 resultant amplitudes is given by (1). 



If the amplitude of each component be /, instead of unity, 

 as we have hitherto supposed for brevit} r , the probability of 

 a resultant amplitude between r and r+ dr is 



~e- r2 < nl *rdr (5) 



In 'Theory of Sound/ 2nd edition, § 42a (1894), I indi- 

 cated another method depending upon a transition from an 

 equation in finite differences to a partial differential equation 

 and the use of a Fourier solution. This method has the 

 advantage of bringing out an important analogy between 

 the present problems and those of gaseous diffusion, but the 

 demonstration, though somewhat improved later *, was in- 

 complete, especially in respect to the determination of a 

 constant multiplier. At the present time it is hardly 

 worth while to pursue it furl her, in view of the important 

 improvements effected by Kluyver and Pearson. The latter 



* Phil. Mag. toI. xlvii. p. 246 (1899) ; Scientific Papers, vol. v. p. 370. 



2 A2 



