and Random Flights in one, two, or three Dimensions. 331 

 We are now prepared to investigate the probability 

 P.(r; k,l»... k) 

 that after n stretches / 1? Z 2 , ... l n taken in directions at random 

 the distance from the starting-point (fig. 2), shall be less- 

 Fig. 2. 



than an assigned magnitude r. The direction of the first 

 stretch l Y is plainly a matter of indifference. On the other 

 hand the probability that the angles 6 lie within the limits 

 l and 0i + rf0i, 2 and Oz + dOt, ... n -i and n .i + dd n -i is 



(2^ dOld0 * d6 *-* ' ' ' (26) 



which is now to be integrated under the condition that the 

 nth radius vector s n shall be less than r. 



Let us commence with the case of two stretches / x and l 2 . 

 Then 



Mdd, 



<ri M^=sj 



the integration being taken within such limits that s 2 <r, 

 where 



t* = l x * + / 2 2- 2l 1 l 2 COS0 1 . 



The required condition as to the limits can be secured by 

 the introduction of the discontinuous function afforded by 

 Weber's integral. For 



i 



J\ {rx) J (s 2 x) da 



vanishes when s 2 >r, and is equal to unity when s 2 <r. After 

 the introduction of this factor, the integration with respect 

 to 0i mav be taken over the complete range from to 2ir. 

 Thus 



PgC 7 * > l uk)=2~\ dd l \ dsc J^ra) J {s 2 ,r) . 



