Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/346

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}},$

aient même signe, contraire à celui de la fonction

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}\right)^{2}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}\right)^{2}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)^{2}$

$-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}-2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}},$

et qu’en outre on ait une quelconque des trois conditions

$\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}<0,\quad \left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}<0,$

$\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)^{2}-{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}<0.$

Il y aura d’ailleurs maximum ou minimum suivant que les coefficiens différentiels

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}},$

seront tous trois négatifs ou tous trois positifs.

Ce cas est sujet, au surplus, à une exception analogue à celle que nous avons signalée à l’égard des fonctions de deux variables. Si l’on a, à la fois, pour un système de valeur de $x,y,z,$

$\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}\right)^{2}={\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}},\quad \left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}\right)^{2}={\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}},$

$\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\right)^{2}={\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}},$

la fonction ${\frac {\operatorname {d} ^{2}S}{\operatorname {d} s^{2}}}$ renfermera le quarré d’une fonction du premier degré ; or, si la racine de ce quarré se trouve facteur dans ${\frac {\operatorname {d} ^{3}S}{\operatorname {d} s^{3}}}$, sans l’être dans ${\frac {\operatorname {d} ^{4}S}{\operatorname {d} s^{4}}}$, et qu’en outre cette dernière fonction, pour toutes les valeurs des variations $\delta x,\delta y,\delta z,$ conserve constamment le même signe que ${\frac {\operatorname {d} ^{2}S}{\operatorname {d} s^{2}}}$, il y aura, pour ce système de valeurs,