The name is absent

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 ≈ f_g^sτf(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

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