Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/239

$\left\{{\begin{aligned}&{\frac {\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}-2{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} S}{\operatorname {d} z}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}+\left({\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}}{\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}+\left({\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}}}\\\\=&{\frac {\left({\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}-2{\frac {\operatorname {d} S}{\operatorname {d} z}}{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}+\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}}{\left({\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}+\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}}}\\\\=&{\frac {\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}-2{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} S}{\operatorname {d} y}}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}+\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}}{\left({\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}+\left({\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}}}\end{aligned}}\right\}.\quad$(44)

On peut ajouter à tout ceci que, généralement pour que deux droites menées par un quelconque $(x,y,z)$, des points de la surface $S=0,$ et formant avec les axes des coordonnées des angles $\alpha$ et $\alpha ',\ \beta$ et $\beta '\ \gamma$ et $\gamma ',$ soient des diamètres conjugués du point de contact de cette surface avec son plan tangent en ce point, il faut qu’on ait (14), (15), (16), (17), (22),

$\operatorname {Cos} .^{2}\alpha +\operatorname {Cos} .^{2}\beta +\operatorname {Cos} .^{2}\gamma =1,\quad$(45)

$\operatorname {Cos} .^{2}\alpha '+\operatorname {Cos} .^{2}\beta '+\operatorname {Cos} .^{2}\gamma '=1,\quad$(46)

${\frac {\operatorname {d} S}{\operatorname {d} x}}\operatorname {Cos} .\alpha +{\frac {\operatorname {d} S}{\operatorname {d} y}}\operatorname {Cos} .\beta +{\frac {\operatorname {d} S}{\operatorname {d} z}}\operatorname {Cos} .\gamma =0,\quad$(47)

${\frac {\operatorname {d} S}{\operatorname {d} x}}\operatorname {Cos} .\alpha '+{\frac {\operatorname {d} S}{\operatorname {d} y}}\operatorname {Cos} .\beta '+{\frac {\operatorname {d} S}{\operatorname {d} z}}\operatorname {Cos} .\gamma '=0,\quad$(48)

$\left\{{\begin{aligned}&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}\operatorname {Cos} .\alpha \operatorname {Cos} .\alpha '+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}\left(\operatorname {Cos} .\beta \operatorname {Cos} .\gamma '+\operatorname {Cos} .\gamma \operatorname {Cos} .\beta '\right)\\+&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\operatorname {Cos} .\beta \operatorname {Cos} .\beta '+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}\left(\operatorname {Cos} .\gamma \operatorname {Cos} .\alpha '+\operatorname {Cos} .\alpha \operatorname {Cos} .\gamma '\right)\\+&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}\operatorname {Cos} .\gamma \operatorname {Cos} .\gamma '+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\left(\operatorname {Cos} .\alpha \operatorname {Cos} .\beta '+\operatorname {Cos} .\beta \operatorname {Cos} .\alpha '\right)\end{aligned}}\right\}=0\,;\mathrm {(49)}$