and Random Flights in one, two, or three Dimensions. 333 



The modification consists in dealing, not with the chance 

 of a resultant less than r, but with the chance that it lies 

 between r and r-rdr. It may seem easy to pass from the 

 one to the other, as it involves merely a differentiation with 

 respect to r. We have 



£Wi(«r)}*-iW^)} 



= — J '(rx) — r.v J " {rx) = rx J (rx) , 



in virtue of the differential equation satisfied by J . Thus, 

 if the differentiation under the integral sign is legitimate, 



dv r 00 



tp = 2irr<l> n (r>) = r xdxJ (rx)Jo(hx)J (^) . . .J (W • • ■ ,(31) 

 ar jo 



and, if all the Vs are equal, 



4>n(r 2 )=M xdxJ Q (rx){J (la:)}», . . (32) 

 ^ 7r »^ o 



the form employed by Pearson, whose investigation is by a 

 different method. If we put n=l in (32), 



*i(^)=^| xdxJ (rx)J (lx), . . . (33) 



and this is in fact the equation from which Pearson starts. 

 But it should be remarked that the integral (33), as it stands, 

 is not convergent. For when z is very great, 



J ^=V{S> C0S (r- z )> - - - (34) 



so that (r#0) 



ij xdxJ (rx)J (lx) = gpr^jj X ^ 



{sin(r-hl)x+ cos(r—l)x}, 



and this is not convergent when #=oo . 



The criticism does not apply to (29) itself when n = l, but 

 it leads back to the question of differentiation under the 

 sign of integration. It appears at any rate that any number 

 of such operations can be justified, provided that the integrals, 

 resulting from these and the next following operation, are 

 finite for the values of r in question. But this condition is 

 not satisfied in the differentiation under the integral sign 

 of (29) when rc=l. For the next operation upon (32) then 

 yields 



I 



x m 'dx3 x (rx)3 9 {lx). 



