614 ON THE PROBLEM OF RANDOM VIBRATIONS, AND [441
seem easy to pass from the one to the other, as it involves merely a differentiation with respect to r. We have
£ K, («)}--1 {r /«'(«)}
=*-Jt'(ris) — ra;J"(ras)=*raiJ0(ra;))
in virtue of the differential equation satisfied by J0. Thus, if the differentiation under the integral sign is legitimate,
^ = 27rr0n(r2) = r[°°xdxJ0(rx) J0&x) J0(l,x)...JQ(lnx)..., ...(31) cLr J o
and, if all the I's are equal,
' ^l(^) = ^-f«^/0(^){J0(fe)h..................(32)
^vr.'o
the form employed by Pearson, whose investigation is by a different method. If we put n = 1 in (32),
^(Vs)* f xdxJ0(rx)J0(lx), ..................(33)
ZTT.'O
and this is in fact the equation from which Pearson starts. But it should bo remarked that the integral (33), as it stands, is not convergent. For when z is very great,
^V©008^-*)'.....................<34)
so that (r + 0)
i rx i p
H- xdx J0 (rx) J"0 (las) = . ,' .. das {sin (r +l)x + cos (r -1) x],
ATT J ATT y (ri) J
and this is not convergent when x = oo.
The criticism does not apply to (29) itself when n — 1, 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, arc finito for the values of r in question. But this condition is not satisfied in the differentiation under the integral sign of (29) when n — 1. For the next operation upon (32) then yields
(rx) </o (1%). o
When we substitute for J"0 (loo) from (34) and for Jl (rx) from
/2
r j (^ \ (^ i\
we get xdx cos -; — rx\ cos .- — ton ) ,
J \ 4 / V4 /
which becomes infinite with x, even for general values of r and I. e and G. Thus 1