It is shown that the automorphism group of a relation algebra
$\mla P$ constructed from a projective geometry $P$ is isomorphic
to the collineation group of $P$. Also, the base automorphism
group of a representation of $\mla P$ over an affine geometry $D$
is isomorphic to the quotient of the collineation group of $D$ by
the dilatation subgroup. Consequently, the total number of
inequivalent representations of $\mla P$, for finite geometries
$P$, is the sum of the numbers
\[
\frac{|\aut P|}{|\aut {D}|\mathbin{/}|\dil{D}|},
\]
where $D$ ranges over a list of the non-isomorphic
affine geometries having $P$ as their geometry at infinity. This
formula is used to compute the number of inequivalent
representations of relation algebras constructed over projective
lines of order at most 10. For instance, the relation algebra
constructed over the projective line of order 9 has 56,700
mutually inequivalent representations.