View source for Darboux net invariants

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The expressions $h$ and $k$,

$$h=c+ab-\frac{\partial a}{\partial u},\quad k=c+ab-\frac{\partial b}{\partial v},  $$
derived from the coefficients of the Laplace equation (in differential line geometry)

$$\frac{\partial^2\theta}{\partial u\partial v} = a\frac{\partial\theta}{\partial u}+b\frac{\partial\theta}{\partial v}+c\theta. \tag*{(*)}$$
Equation (*) is satisfied by the homogeneous coordinates of a point $x$ describing a [[Conjugate net|conjugate net]] of lines $u$ and $v$ on a two-dimensional surface in an $n$-dimensional projective space, where $n\geq 3$. It was shown by G. Darboux [[#References|[1]]] that the Darboux invariants $h$ and $k$ do not change their value when the normalization of the coordinates of the point $x$ is changed. Special forms of conjugate nets are obtained by imposing some condition on the Darboux invariants.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Darboux,   "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''2''' , Gauthier-Villars  (1889)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Tzitzeica,   "Géométrie différentielle projective des réseaux" , Gauthier-Villars &amp; Acad. Roumaine  (1924)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.P. Finikov,   "Theorie der Kongruenzen" , Akademie Verlag  (1959)  (Translated from Russian)</TD></TR></table>



====Comments====
The expressions $h$ and $k$ are more commonly referred to as the Darboux invariants of a net.