and Random Flights in one, two, or three Dimensiom 

 Finally we obtain 



327 



32n 2 \ 3/i 



2 38# 4 

 - + 



12^ 



3w 



9nV.J 



)} 



(15) 



as the probability when n is large of the resultant ampli- 

 tude x. It is to be remembered that x is limited to a series 

 of discrete values with a common difference equal to 2, and 

 that our approximation has proceeded upon the supposition 

 that x is not of higher order than \/n. 



If the component amplitudes or stretches be /, in place of 

 unity, we have merely to write xjl in place of x. 



The special value of the series (15) is realized only when 

 n is very great. But it affords a closer approximation to 

 the true value than might be expected when n is only 

 moderate. I have calculated the case of 7i = 10, both directly 

 from the exact expression (10) and from the series (15) for 

 all the admissible values of x. 



Table I. 

 n = 10. 



X. 



From (10). 



From (15). 







J -24609 



•24608 



'> 



•20508 

 •11719 



•20509 

 •11722 



4 



6 



•04394 



•04392 



8 



•00977 



•00975 



io :. 



•00098 



•00102 



The values for x=0 and twice those belonging to higher 

 values of x should total unity. Those above from (10) give 

 1-00001 and those from (15) give 1-00008. It will be seen 

 that except in the extreme case of x= 10, the agreement 

 between the two formulae is very close. But, even for much 

 higher values of ?i, the actual calculation is simpler from the 

 exact formula (10). 



When I is very small, while n is very great, we may be- 

 able for some purposes to disregard the discontinuous 

 character of the probability as a function of ss 9 replacing the 



