= Solution
For an ordinary intrinsic density, inversion of the <Abel transform> gives
$$
\mathcal P(q)
=\frac{2\sqrt{1-q^2}}\pi
\frac d{dq}\int_0^q
\frac{\mathcal P(Q)}{\sqrt{q^2-Q^2}}\,dQ.
$$
If $\mathcal P(Q)=1$, the integral is $\pi/2$ for every $q>0$, so its derivative vanishes. The missing probability is an endpoint atom: all systems must be infinitely thin,
$$
\boxed{\mathcal P(q)=\delta(q)}.
$$
Indeed, putting $q=0$ directly into part c gives $\mathcal P(Q\mid0)=1$. Thus a uniform apparent-axis-ratio distribution corresponds to a <Dirac delta distribution> of ideal zero-thickness disks, rather than to a regular intrinsic density.
Back to article page
