Page:Annales de mathématiques pures et appliquées, 1821-1822, Tome 12.djvu/183

177

RÉSOLUES.

 Les corrections sont expliquées en page de discussion

${\begin{aligned}&-2y\left\{zx{\frac {\operatorname {f} '(z)\operatorname {f} '(x)\operatorname {f} ''(t)}{\operatorname {f} '^{3}(t)}}+{\frac {z\operatorname {f} '(z)+x\operatorname {f} '(x)}{\operatorname {f} '(t)}}-t\right\}\operatorname {d} z\operatorname {d} x\\\\&-2z\left\{xy{\frac {\operatorname {f} '(x)\operatorname {f} '(y)\operatorname {f} ''(t)}{\operatorname {f} '^{3}(t)}}+{\frac {x\operatorname {f} '(x)+y\operatorname {f} '(y)}{\operatorname {f} '(t)}}-t\right\}\operatorname {d} x\operatorname {d} y.\end{aligned}}$

Si nous faisons $x=y=z=t,$ ces valeurs deviennent

$\operatorname {d} ^{2}s=-2{\frac {\operatorname {f} ''(t)}{\operatorname {f} '(t)}}\left(\operatorname {d} x^{2}+\operatorname {d} y^{2}+\operatorname {d} z^{2}+\operatorname {d} y\operatorname {d} z+\operatorname {d} z\operatorname {d} x+\operatorname {d} x\operatorname {d} y\right),$

$\operatorname {d} ^{2}p=2-t^{2}\left[1+{\frac {t\operatorname {f} ''(t)}{\operatorname {f} '(t)}}\right]\left(\operatorname {d} x^{2}+\operatorname {d} y^{2}+\operatorname {d} z^{2}+\operatorname {d} y\operatorname {d} z+\operatorname {d} z\operatorname {d} x+\operatorname {d} x\operatorname {d} y\right).$

ou bien encore

$\operatorname {d} ^{2}s=-{\frac {\operatorname {f} ''(t)}{\operatorname {f} '(t)}}\left\{\left(\operatorname {d} x^{2}+\operatorname {d} y^{2}+\operatorname {d} z^{2}\right)+(\operatorname {d} x+\operatorname {d} y+\operatorname {d} z)^{2}\right\},$

$\operatorname {d} ^{2}p=-t^{2}\left[1+t{\frac {\operatorname {f} ''(t)}{\operatorname {f} '(t)}}\right]\left\{\left(\operatorname {d} x^{2}+\operatorname {d} y^{2}+\operatorname {d} z^{2}\right)+(\operatorname {d} x+\operatorname {d} y+\operatorname {d} z)^{2}\right\}\,;$

d’où l’on voit que $s$ et $p$ seront maximums on minimums, suivant que ${\frac {\operatorname {f} ''(t)}{\operatorname {f} '(t)}}$ et $1+t{\frac {\operatorname {f} ''(t)}{\operatorname {f} '(t)}}$ seront positives ou négatives.

Genève, le 26 août 1821.