62 Lectures on Scientific Subjects
But the inner definite integral has the value τr∕2. Hence we
deduce
∫⅛ω√r=0;
and by differentiating as to p we conclude for all p and χ
that g(ρ) is zero. Thus g vanishes identically, contrary to
assumption.
We see therefore that in the case of a continuous symmetrical
density function F(r) there is at most one corresponding con-
tinuous distribution function, f, which, if it exists, must
likewise be symmetrical.
In consequence in the symmetric case we may take (1)
in the more special form
ʃ(r) =$‘T f(r sin (φ - θ))d<ρ ≈ fgsτf(r sin φ)dφ,
where F( — r) = Ffr), f( — ʃ) =f(s). Thus we need only to
consider the equation
F(r) =4jj∕(r sin <p)dφ,
or, making the change of variables r sin φ=s,
iJ0Vri-Si'
But this is precisely the integral equation solved by Abel
(1828). In order to solve it we have only to multiply through
by r∣Vpi-ri and integrate in r from 0 to ρ. This yields, as
in the special case treated above,
. /ʌz ч , [t>rFfr)dr
VV>dl-L
If there is a continuous solution, the integral on the right
must have a continuous derivative in p, namely f(pi).
Thus there exists such a continuous (symmetric) distribution
function f (ʃ) if and only if the integral