  The differential equation governing distance-redshift for partially
filled-beam optics in pressure-freeFriedmann-Lema\^\i tre-Robertson-Walker
Cosmology (FLRW) is shown to be the Lame$^{\prime}$ equation. The
distance-redshift, $D(z)$, discussed is appropriate for observations in
inhomogeneous cosmologies for which lensing by masses external to the observing
beam are negligible and for which lensing by transparent matter within the beam
can be approximated by a homogeneous mass density expanding with the FLRW
background. Some solutions of the derived Lame$^{\prime}$ equation are given in
terms of Weierstrass elliptic integrals. A new simplified and useful expression
for filled-beam $D(z)$ in standard flat FLRW is also given.
