( 342 ) 



In a turning point the rambler takes his new direction at random ; 

 hence for anj angle tpk, all values between and Iti have an 

 equal chance, and the probability that those angles are respectively 

 included within the intervals, <p];, (pk + d(pk, is equal to the product 



1 



If we integrate this product over a region, determined by the 

 condition that the ?i^'' radius vector Sa-\ remains less than a given 

 distance c, the result will be the required probability lF„(c ; aa^a^...an—\ ), 

 that the ending point of the path lies within the distance c from 

 the starting point 0. ^) 



The integration becomes less complicated, if we introduce in the 

 usual way a discontinuous factor. Choosing a function T{(p,(p^, . ,(pn—^) 

 such, that it vanishes when Sn—\ ]> c, and that it is equal to unity 

 for Sn—\ <[ c, to each of the variables tpt ^ve may give the whole 

 range from to 2.-T, and we have 



Wn{c ;««! . . a„_i) = I 1 • • • I d<fd(p^ . . d(fn-2 '^<f,<Pv • • , (fiu-2)' 







For the function T we may take Weber's discontinuous integral, 

 that is, we may put 



00 



T{<f,<f^, . , (pn—2) = C I J^{llc)Jg{uSn-\)du, 

 



the integral being equal to zero or to unity according to Sn—i being 

 larger or smaller than c. 



This choice of the factor T makes a good deal of reduction possible. 



If we consider the side c of a triangle as a function of the sides 

 a and b and of the inclosed angle C, the relation holds 



2:r 



Jq {ua) Jq (ub) m — j ƒ J [uc) dC, 







and this formula can be repeatedly used in reducing the integral 

 So we get successively 



1) In the case n = 2, we have, supposing a \- ai> c> a — Ui, Wr,{c;aai) = 



1 ^ I 2 ,,3 



— arccos ^ . Of course for oa + fti Wo becomes equal to unity and 



yt Zaa^ 



it is zero for a — «i > c. 



