Problem of Random Flights. 433 



Lord Rayleigh has proved * that the chance that the 

 resultant of n flights (each of length I) may lie between 

 r and r-\-dr is equal to farr* dr . <f> n , where 



-j f* CO 7 



*»=4^sJ -&i sin rv(sin Zy)». . . (1-1) 



On observing this formula it suggested to my mind a 

 corresponding complex integral to which it is at once seen 

 to be equivalent, namely, 



tn=^T +m ~sinhrX( S mhl\y, . (I'll) 



where c > 0. 



In this form the use of Heaviside's symbolical notation 

 is suggested as a convenient abbreviation f. In fact, 

 we have 



where p denotes d/d^ and, after the operations have been 

 carried out, t is put equal to zero. 



It is now an easy matter to evaluate <£ 2 , $& <£ 4 , . . . ., 

 by following the process explained in § 6 of my paper 

 jnst quoted. Thus we find : — 



(i.) n = 2. 



* 2= u^t^ p, ~ e ~'"' ) ^ A - e ~ p 'y 2 > 



and we are simply to count unity for each exponential 

 which has a positive index on expansion (ignoring those 

 with negative indices) J. Hence (since < r\< 21) 



which agrees with (48) of Lord Rayleigh's [taper. 



* Phil. Mag. vol. xxxviii. p. 341 (1919) ; formula (59) gives the 

 integral for dP n /dr, and 4irr*<p n =dP n /dr. The notation' <p n is that 

 adopted by Prof. Karl Pearson. 



t Bromwich, Proc. Loud. Math. Soc. (2) vol. xv. p. 401 (1916) ; 

 see § 4 in particular. 



X Bromwich, Proc. L. M. S. /. c. p. 425. 



Phil. Mag. S. 6. Vol. 42. No. 249. Sept 1921. 2 G 



