and JRandom Flights in one, tivo, or three Dimensions. 34") 



terms written down in (66) thus suffice for an approximation 

 to the order n~ 2 inclusive. 



The evaluation of the auxiliary terms in (66) can be 

 effected by differentiating the principal term with respect 

 to n. Eiich such differentiation brings in — x 2 \fi as a factor, 

 and thus four operations suffice for the inclusion of the term 

 We get 



containing of 



dr 



-*} — a + nlu • 0- 



v / 7^.^ , L 



dn 2 



die re 

 Finally 





N= 



-3 2 - 



n e 



d? n 



3V 6 • 



f*e-****J 



dr 



.y/TT 



.l 3 .n s < 2 





1 

 + 40n 



/29 



2 U 



69r 2 

 " nl 2 + 



981r 4 

 10n 2 / 4 



3r2/2n^ 



w / t6 .6 3 C^H 



1341? 



10r 2 



ao/^/° ' 20 



+ - 



4- 



3r 4 \ 



Sir 8 \ 1 



(68) 

 (69) 



70) 



Here dP n jdr . dr is the chance that the resultant of a large 

 number n of flights shall lie between r and r + dr. In 

 Pearson's notation, 



±Trr 2 <f> n =d? n \dr. 



The maximum value of the principal term (67) occurs 

 when r/Z=v/(2n/3). 



It is some check upon the formulae to compare the exact 

 results for ?i = 6 in (64) with those derived for the case of n 

 great in (70), although with such a moderate value of n no 

 precise agreement could be expected. The following Table 

 gives the numerical results for ldV G jdr in the two cases . — 



>■:'■ 



From (64). 



From (70). 







•2500 r 2 /P 



•05900 



•2005 



•4167 



•2080 



•0833 



•00652 



•00000 



•2483 r 2 /Z 2 



•05886 



•2007 



•4169 



•2922 



•1055 



•00716 







'5 



1 







3 



4 



i 

 5 



« 





