  We extend finding geometrically-significant preserved quantities by solving
specific PDEs to the affine transformations and subgroups. This can be viewed
not only as a purely geometrical problem but also as a subcase of finding
physical observables, and furthermore as part of the comparative study of
Background Independence level-by-level in mathematical structure. While cross
and scalar-triple products (combined with differences and ratios) suffice to
formulate these preserved quantities in 2- and 3-$d$ respectively, the
arbitrary-dimensional generalization evokes the theory of forms. The affine
preserved quantities are ratios of $d$-volume forms of differences, $d$-volume
forms being the `top forms' supported by dimension $d$, and referring moreover
to $d$-volumes of relationally-defined subsystems.
