and Random Flights in one, two, or three Dimensions. 335 



leaves a constant term, which can happen only for values 

 of r which are multiples of I. We are then left with 



J P+|— i w 



The integral is thus finite or infinite according as 



^>< or >i{n— 3). 



If, however, there arise no constant term, we have to 

 consider 



COO 00 /"*00 



da? # s cos ??i^' = —sin mx S — q \ dxx'" 1 sin w?a?, 



J m m?J 



where m is finite ; and this is finite if s, that isp + j[ — Jn, 

 be negative. The differentiations are then valid, if 



/><i(n-^l). 



We may now consider more especially the cases n = 4, &c. 

 When n = <±, s = p -f f — |«=/? — §. 



If jp = 1, s=— \, and the cosine factors in (38) become 



COS (1-77+ ?\r) COS 4 (J7T— 7ff), 



yielding finally 



cos f -7- + ry»— 4Z«?j J , cos ( *-r rx — 4/^* j , 



cos( -^r -\-rx— 21 x\, cos(j—i*x — 2lx\ cos(— + rx\, 



so that there is no constant term unless r = 4Z, or 2l. With 

 these exceptions, the original differentiation under the 

 integral sign is justified. 



We fall back upon (/> 4 itself by putting p = 0, making 

 s= — § . The integral is then finite in all cases (V=£0), in 

 agreement with Pearson's curve. 



Next for n = 5, s = p — 2. 



When p = l, s=— 1, and we find that the cosine factors 

 yield a constant term only when r = Zl. Pearson's curve 

 does not suggest anything special at r = 3Z ; it may be 

 remarked that the integral with ^> = 1 is there only logarith- 

 mically infinite. 



If n = 5, p==0, 5=— 2; and the integral for <j> 5 is finite 

 for all values of r. 



When n = 6, s = p— 2\. In this case, whether p = l, or 0, 

 no question can arise. The integrals are finite for all values 

 of r. 



A fortiori is this so, when n>6. 



