![{\displaystyle {\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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a36d4cc2704867b2f4097f50f021f4428dd7d1)
aient même signe, contraire à celui de la fonction
![{\displaystyle {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48d6bbb9a6c10f2f310530e0fb354449e4b8a58)
![{\displaystyle -{\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}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a580d28eb7d4adfd8dc657c9a0c8bf16e1ce74d5)
et qu’en outre on ait une quelconque des trois conditions
![{\displaystyle \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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49239844aa04ec3456fa770a56c7cbe594ecc9d5)
![{\displaystyle \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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f05e08032d1ff6066f27184345d382ac0989c3d)
Il y aura d’ailleurs maximum ou minimum suivant que les coefficiens différentiels
![{\displaystyle {\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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a36d4cc2704867b2f4097f50f021f4428dd7d1)
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
![{\displaystyle \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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57fc2c0d0a8d0720b5d05f8d7ffd0097c124bdd1)
![{\displaystyle \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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/221b38ffb9313ce10e725b76cbd1e8c84a97de4c)
la fonction
renfermera le quarré d’une fonction du premier degré ; or, si la racine de ce quarré se trouve facteur dans
, sans l’être dans
, et qu’en outre cette dernière fonction, pour toutes les valeurs des variations
conserve constamment le même signe que
, il y aura, pour ce système de valeurs,