View source for Goursat congruence

A congruence of straight lines in which the first point invariant of the focal net of one focal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446401.png" /> is equal to the second point invariant of a second focal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446402.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446403.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446404.png" /> be the Laplace transforms (cf. [[Laplace transformation (in geometry)|Laplace transformation (in geometry)]]) of the focal surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446406.png" />. Then for each straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446407.png" /> of a Goursat congruence there exists a second-order surface passing through the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446408.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g0446409.png" />) having a third-order osculation with the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g04464010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g04464011.png" /> and with the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g04464012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044640/g04464013.png" /> [[#References|[1]]]. If two adjacent congruences in a [[Laplace sequence|Laplace sequence]] are Goursat congruences, the complete sequence consists of Goursat congruences.

Named after E. Goursat, who studied congruences of this type.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Tzitzeica,   "Sur certaines congruences de droites"  ''J. Math. Pures Appl. (9)'' , '''7'''  (1928)  pp. 189–208</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.P. Finikov,   "Projective differential geometry" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR></table>



====Comments====
Goursat congruences, which are rarely encountered anymore, can also be defined by the property that in their Laplace sequence both adjacent congruences have the same Laplace invariant [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,   "Projective Differentialgeometrie" , '''2''' , Vandenhoeck &amp; Ruprecht  (1954)</TD></TR></table>