Page:Annales de mathématiques pures et appliquées, 1821-1822, Tome 12.djvu/182

176

QUESTIONS

$\operatorname {d} s=\left\{1-{\frac {\operatorname {f} '(x)}{\operatorname {f} '(t)}}\right\}\operatorname {d} x+\left\{1-{\frac {\operatorname {f} '(y)}{\operatorname {f} '(t)}}\right\}\operatorname {d} y+\left\{1-{\frac {\operatorname {f} '(z)}{\operatorname {f} '(t)}}\right\}\operatorname {d} z,$

$\operatorname {d} p=yz\left\{t-x{\frac {\operatorname {f} '(x)}{\operatorname {f} '(t)}}\right\}\operatorname {d} x+zx\left\{t-y{\frac {\operatorname {f} '(y)}{\operatorname {f} '(t)}}\right\}\operatorname {d} y+xy\left\{t-z{\frac {\operatorname {f} '(z)}{\operatorname {f} '(t)}}\right\}\operatorname {d} z\,;$

on voit d’abord que, si l’on a $x=y=z=t,$ il en résulte $\operatorname {d} s=0,\ \operatorname {d} p=0,$ conditions communes au maximum et au minimum.

Par une nouvelle différentiation, on a, en considérant $x,y,z,$ comme fonctions de $t,$ et mettant toujours pour $\operatorname {d} t$ sa valeur donnée par l’équation (1)

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