1919] RANDOM FLIGHTS IN ONE, TWO, OR THREE DIMENSIONS 615 So much by way of explanation ; but of course we do not really need to discuss the cases n = 1, n = 2, or even n = 3, for which exact solutions can be expressed in terms of functions which may be regarded as known. For higher values of n it would be of interest to know how many differentiations with respect to r may be made under the sign of integration. It may be remarked that since all J's and their derivatives to any order are less than unity, the integral can become infinite only in virtue of that part of the range where x is very great, and that there we may introduce the asymptotic values. We have thus to consider dP . . .. 1 i (35) For the leading term when z is very great, we have 2\ /I 1 ---- COS -7 7T — Z — •= TTZJ \4t 2 .(36) .(37) so that with omission of constant factors our integral becomes l" I J cosn TT- (38) In this cos'1 (£ TT - loo) can be expanded in a series of cosines of multiples of (£TT — lx~), commencing with cos n (|TT — loo) and ending when n is odd with coa (-JTT — loi), and when n is even with a constant term. The various products of cosines are then to be replaced by cosines of sums and differences. The most unfavourable case occurs when this operation leaves a constant term, which can happen only for values of r which are multiples of I. We are then left with The integral is thus finite or infinite according as p< or >£(n-3). If, however, there arise no constant term, we have to consider <jiH 7 dU . dxao cos moo = — sin mx m m where m is finite ; and this is finite if s, that is p -I- £ - £«, be negative. The differentiations are then valid, if We may now consider more especially the cases n = 4, etc. When n = 4, fef \ 4 / V4 /