  We study the general properties of certain rank four rigid local systems
considered by Goursat. We analyze when they are irreducible, give an explicit
integral description as well as the invariant Hermitian form when it exists. By
a computer search we find what we expect are all irreducible such systems all
whose solutions are algebraic functions and give several explicit examples
defined over the rationals. We also exhibit one example with infinite monodromy
as arising from a family of genus two curves.
