A treatise on the theory of Bessel functions, by G. N. Watson.

13-48]

INFINITE INTEGRALS

419

If z- = V(a2 + 2 - 2ac cos >), the second method gives
(14) J, (bt) J {a V(t+ z)} J {c V(t + Z2)} dt
(t2 + Z2)v
(Wav [f (bt)J t (t2 + )f ct
2 J, (+t) tt-' sin2, Odqdt
(V + ) r () Jo J (b) (t~ + Z2)"v
_2F'- r (a) J, (az) Jv (cz)
by zV zv
if b > a + c and R (2v + ) > R () > 0.
By induction it follows that, if b > Ea,
(15) [J(at) V( 2/-1 r (g) JI J(az,)
(15) J.(t) (t2 + Z2)JV t cit - zV
where the product applies to n values of a, and
R(nv+ 1n + l) >R( k)> 0.
If the induction of the second method is used after applying the first method once, we
find still further generalisations.
The special case of (15) when z ---0 is
21 -I r (nF(a)v
(16) J, (bt)nii[J(at)]t-V- dt= )[i( );
this has been pointed out by Kluyver, Proc. Section of Sci., K. Akad. van Wet. te Amsterdam, xi. (1909), pp. 749-755.
13'48. The problem of random flights.
A problem which was propounded by Pearson* (in the case of two-dimensional displacements) is as follows:
"A man starts from a point 0 and walks a distance a in a straight line;
he then turns through any angle whatever and walks a distance a in a second
straight line. He repeats this process n times.
"I require the probability that after these n stretches he is at a distance
between r and r + $r from his starting point, 0."
The generalised form of the problem, in which the stretches may be taken
to be unequal, say al, a2,..., a., has been solved by Kluyvert with the help of
the discontinuous integrals which were discussed in ~ 13.42; and subsequently
Rayleigh+ gave the full details of the analysis of the problem (which had been
examined somewhat briefly by Kluyver), and then obtained the solution of the
corresponding problem for flights in three dimensions.
If s,, is the resultant of a, a2,..., am (m = 1, 2,..., n - 1), and if 0m is the
* Nature, LXXII. (1905), pp. 294, 342 (see also p. 318); Drapers' Company Research Memoirs,
Biometric Series, III. (1906).
t Proc. Section of Sci., K. Akad. van Wet. te Amsterdam, vIIi. (1906), pp. 341-350.
+ Phil. Mag. (6) xxxvII. (1919), pp. 321-347. [Scientific Papers, vi. (1920), pp. 604-626.]
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