Laplace sequence

A sequence of congruences in the three-dimensional projective (affine, Euclidean) space in which every two adjacent congruences are formed by the tangents to two families of curves of the conjugate net of a surface (the focal surface of the congruence). Each of the two adjacent congruences of a Laplace sequence is called the Laplace transform of the other (cf. Laplace transformation (in geometry)). Analytic transformations of the Laplace equation are connected with the geometrical transition from one focal surface of a congruence to another focal surface of it (see [1]). With every Laplace sequence of congruences there is associated a Laplace sequence of focal surfaces (see [2]). A Laplace sequence of $p$-dimensional Cartan manifolds of singular projective type in the projective $n$-space $P_n$ (see [3]) has been generalized to the case of arbitrary $p$-conjugate systems in $P_n$ (see [4]).

References

[1]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 2 , Chelsea, reprint (1972)
[2]	S.P. [S.P. Finikov] Finikow, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)
[3]	S.S. Chern, "Laplace transforms of a class of higher-dimensional varieties in a projective space of $n$ dimensions" Proc. Nat. Acad. Sci. USA , 30 (1944) pp. 95–97
[4]	R.V. Smirnov, "Laplace transforms of $p$-conjugate systems" Dokl. Akad. Nauk SSSR , 71 : 3 (1950) pp. 437–439 (In Russian)