Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/237

$\left.{\begin{alignedat}{3}A=&{\frac {\operatorname {d} S}{\operatorname {d} x}},&B=&{\frac {\operatorname {d} S}{\operatorname {d} y}},&C=&{\frac {\operatorname {d} S}{\operatorname {d} z}},\\\\D=&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}},&E=&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}},&F=&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}},\\\\G=&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}},&\quad H=&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}},&\quad K=&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}.\end{alignedat}}\right\}\quad$(41)

et l’on obtiendra ainsi les propositions suivantes :

1.^o Il existe, en général, sur toute surface courbe une courbe à double courbure le long de laquelle les points de contact des plans tangens sont tous paraboliques. Cette courbe partage la surface courbe en deux régions dans l’une desquelles les points de contact des plans tangens sont hyperboliques tandis, que, dans l’autre, ils sont elliptiques.

Si $S=\operatorname {f} (x,y,z)=0$ est l’équation de la surface courbe dont il s’agit, rapportée à trois axes rectangulaires quelconques, la courbe à double courbure, dont il vient d’être question, sera donnée sur cette surface, par son intersection avec une autre surface ayant (33) pour équation

$\left\{{\begin{aligned}&\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}\left[\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}\right]\\&\quad \qquad +2{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} S}{\operatorname {d} z}}\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)\\\\+&\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}\left[\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}\right]\\&\quad \qquad +2{\frac {\operatorname {d} S}{\operatorname {d} z}}{\frac {\operatorname {d} S}{\operatorname {d} x}}\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}\right)\\\\+&\left({\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}\left[\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\right]\\&\quad \qquad +2{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} S}{\operatorname {d} y}}\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}\right)\end{aligned}}\right\}=0,$ (42)

Un point $(x,y,z)$ de la surface $S=0$ appartient à la région des points de contact hyperboliques ou à la région des points de con-