and Random Flights in one, two, or three Dimensions, 337 

 Writing s for hnl 2 , we have 



log J,^)= log(l- ^ + m? ~ m ^ + •••) 



so;" s*x s\ar 



that 



12/ <? 2 r 4 s ? « 6 <?' l r 8 \ 



{0 (Lv)\ =e 2 1— — — -, ■ + ^nj-o ), 



1 v /J V lb?i /2?i 2 0l2nv 



making 



2^„(,'0=J o ^J.(«>--(l-l± - ^ + ^). (40) 



Calling the four integrals on the right I l5 I 2 , I 3 , and I 4 , 

 *we have by (39) 



I X =C xdxJ Q {rx)e-* 83 *=~e- r * l2s , . . (41) 

 Jo s 



2 ^ = fivv-n . . . (42) 



_l_ ffl(l 2/2 A (m 



ds* ~9n 2 dsAs r ■ ' v ; 



U 32,* 2 ds* ^ZZtfdAi* J' ' V ^ 



12 ~4/l </s 2 



__ s^ d 3 I 

 ; '~9?z 2 



,4 



ri 4 l, / d 4 /l -r» 



• 4 ~32^ 



Thus 



2 ^^=v{ i -i( 2 -v + a 



_l_/ r _9r2 9?* 4 r 6 \ 



^2^ 5 + 6- 2 s 3 + 16sVJ 



L 4>zV s + 4sV ^12n 2 \ s 



+ -4?"-12?+T287jj> (45) 



in agreement (so far as it goes) with Pearson, whose a 2 is 

 equal to our 5. The leading term is that given in 1880. 

 Phil. Mag, S. 6. Vol. 37. No. 220. April 1919. 2 B 



e -r*/2s 



s 



