and Random Flights in one, two, or three Dimensions. 341 

 we obtain three positive and one negative term. Finally 



f/P 2 = r 

 dr 2Zi/ 2 ' 



in agreement with (47). The expression is applicable only 

 when the triangle is possible. In the contrary case we find 

 dP/dr equal to zero when r is less than the difference and 

 greater than the sum of / x and l 2 . 



This argument must appear very roundabout, if the object 

 were merely to obtain the result for n = 2. The advantage 

 is that it admits of easy extension to the general value of n. 

 To this end we take the last stretch l n and the immediately 

 preceding radius s„_i in place of l 2 and l x respectively, and 

 then repeat the operation with l n -i, s n - 2 , and so on, until 

 we reach l 2 and s, (=/i). The result is evidently 



-n . 7 , ,, 2 f °° 7 sin rx — rxcosrx 



r»(r ; l u l 2i 4) = - \ ax 



7T 1 x 



sin 1-iX sin Lx .sin l n x ,.,_. 



—, j— ~, , .... (o7) 



IxX l 2 X l n x 



or if we suppose, as for the future we shall do, that the Vs 



are all equal, 



_> , 7N 2 f * , smrx—rxcosrz/sinlx\ n , eo x 

 P.(r ; l)= -j o ,./,• v -^-) . (08) 



This is the chance that the resultant is less than r. For 

 the chance that the resultant lies between r and r + dr, we 

 have, as the coefficient of dr, 



dP, 



di 



? " = ~ I'-f" ' —,sinrx sin" lx ... (59) 

 r 7r/"J x n ~ l V 



Let us now consider the particular case of n=3, when 



dP z _ 2r C~di 



dr ~tt/ 3 



, I ~ sin rx sin 3 lx (60) 



In this we have 



sin rx s i n 3 lx = £{ 3 cos (r — l)x — 3 cos (r + /) a? 



And 



- cosO~3/)#+ cos {r+3l)x}. 



C* dx c , , ,.7,i 



I --., (COS (/' — /J./'— COS i/''-f /j.t'j 



Jo x ~ 



(^ dx f . 2 (r + l)x . 2 ( r-l)x \ 



= ^{ r + Z- /— / }; 

 and in like manner for the second pair of cosines. 



