Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/250

on obtiendra, pour les équations du point de concours des deux normales,

$\left.{\begin{aligned}&t={\frac {M}{N}}\left\{{\frac {\operatorname {d} S}{\operatorname {d} x}}-A\left(A{\frac {\operatorname {d} S}{\operatorname {d} x}}+B{\frac {\operatorname {d} S}{\operatorname {d} y}}+C{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\right\},\\\\&u={\frac {M}{N}}\left\{{\frac {\operatorname {d} S}{\operatorname {d} y}}-B\left(A{\frac {\operatorname {d} S}{\operatorname {d} x}}+B{\frac {\operatorname {d} S}{\operatorname {d} y}}+C{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\right\},\\\\&v={\frac {M}{N}}\left\{{\frac {\operatorname {d} S}{\operatorname {d} z}}-C\left(A{\frac {\operatorname {d} S}{\operatorname {d} x}}+B{\frac {\operatorname {d} S}{\operatorname {d} y}}+C{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\right\}\,;\end{aligned}}\right\}\quad$(79)

de sorte qu’en désignant par $\rho$ la distance de ce point de concours à l’origine des $t,u,v$, ce qui donnera

$\rho ^{2}=t^{2}+u^{2}+v^{2},\qquad$(80)

on aura, en faisant toujours usage de la relation (70),

$\rho ^{2}={\frac {M^{2}}{N^{2}}}\left\{{\begin{aligned}&\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}\\\\+&\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}\\\\+&\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}\end{aligned}}\right\}\,;\quad$(81)

employant donc cette relation pour éliminer ${\frac {M}{N}}$ des formules (79), on trouvera

$\left.{\begin{aligned}&t={\frac {\rho \left\{{\frac {\operatorname {d} S}{\operatorname {d} x}}-A\left(A{\frac {\operatorname {d} S}{\operatorname {d} x}}+B{\frac {\operatorname {d} S}{\operatorname {d} y}}+C{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\right\}}{\sqrt {\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}+\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}+\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}}}}\\\\&u={\frac {\rho \left\{{\frac {\operatorname {d} S}{\operatorname {d} y}}-B\left(A{\frac {\operatorname {d} S}{\operatorname {d} x}}+B{\frac {\operatorname {d} S}{\operatorname {d} y}}+C{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\right\}}{\sqrt {\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}+\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}+\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}}}}\\\\&v={\frac {\rho \left\{{\frac {\operatorname {d} S}{\operatorname {d} z}}-C\left(A{\frac {\operatorname {d} S}{\operatorname {d} x}}+B{\frac {\operatorname {d} S}{\operatorname {d} y}}+C{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\right\}}{\sqrt {\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)^{2}+\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)^{2}+\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)^{2}}}}\end{aligned}}\right\}\mathrm {(82)}$

À mesure que le point $(x',y',z')$ se rapprochera de l’origine, en suivant le périmètre de la section, le point (82) tendra de plus