Line congruences associated to Appell's hypergeometric functions of rank-4

Line congruences are the genesis of important examples of transformations of projective surfaces, such as the Laplace transform. We survey and review results related to this historical subject, then derive original formulae for the Laplace transform of the entire rank-4 linear system associated to such an immersed projective surface. We apply our results to study the geometry of surfaces defined by Appell's hypergeometric functions of rank-4: namely, $F_2$ and $F_4$. We show that the sequence of Laplace invariants for each is determined respectively by the Euler-Poisson-Darboux equation for $F_2$, and Darboux's Harmonic equation for $F_4$. Further, we show the natural line congruences generated by the Laplace transforms of each constitute a $W$-congruence, an important example of line congruence in which a surface and its Laplace transform are simultaneously locally conformally equivalent.