Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/252

${\frac {t'}{B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}}}={\frac {u'}{C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}}}={\frac {v'}{A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}}}\,;\qquad$(87)

de sorte que, $\lambda$ étant un muiliplicateur convenable, on aura

$t'=\lambda \left(B{\frac {\operatorname {d} S'}{\operatorname {d} z'}}-C{\frac {\operatorname {d} S'}{\operatorname {d} y'}}\right),\ u'=\lambda \left(C{\frac {\operatorname {d} S'}{\operatorname {d} x'}}-A{\frac {\operatorname {d} S'}{\operatorname {d} z'}}\right),$

$v'=\lambda \left(A{\frac {\operatorname {d} S'}{\operatorname {d} y'}}-B{\frac {\operatorname {d} S'}{\operatorname {d} x'}}\right).$

En substituant ces valeurs dans la formule (85), $\lambda$ disparaîtra de lui-même, et nous aurons

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

d’où nous conclurons (81)

$\rho =-{\frac {\left\{\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}\right\}^{\frac {3}{2}}}{\left\{{\begin{aligned}&\left(B{\frac {\operatorname {d} S'}{\operatorname {d} z'}}-C{\frac {\operatorname {d} S'}{\operatorname {d} y'}}\right)^{2}{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x^{2}}}\\&\ \ +2\left(C{\frac {\operatorname {d} S'}{\operatorname {d} x'}}-A{\frac {\operatorname {d} S'}{\operatorname {d} z'}}\right)\left(A{\frac {\operatorname {d} S'}{\operatorname {d} y'}}-B{\frac {\operatorname {d} S'}{\operatorname {d} x'}}\right){\frac {\operatorname {d} ^{2}S'}{\operatorname {d} y'\operatorname {d} z'}}\\\\+&\left(C{\frac {\operatorname {d} S'}{\operatorname {d} x'}}-A{\frac {\operatorname {d} S'}{\operatorname {d} z'}}\right)^{2}{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} y^{2}}}\\&\ \ +2\left(A{\frac {\operatorname {d} S'}{\operatorname {d} y'}}-B{\frac {\operatorname {d} S'}{\operatorname {d} x'}}\right)\left(B{\frac {\operatorname {d} S'}{\operatorname {d} z'}}-C{\frac {\operatorname {d} S'}{\operatorname {d} y'}}\right){\frac {\operatorname {d} ^{2}S'}{\operatorname {d} z'\operatorname {d} x'}}\\\\+&\left(A{\frac {\operatorname {d} S'}{\operatorname {d} y'}}-B{\frac {\operatorname {d} S'}{\operatorname {d} x'}}\right)^{2}{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} z^{2}}}\\&\ \ +2\left(B{\frac {\operatorname {d} S'}{\operatorname {d} z'}}-C{\frac {\operatorname {d} S'}{\operatorname {d} y'}}\right)\left(C{\frac {\operatorname {d} S'}{\operatorname {d} x'}}-A{\frac {\operatorname {d} S'}{\operatorname {d} z'}}\right){\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x'\operatorname {d} y'}}\end{aligned}}\right\}}}.\mathrm {(88)}$

Comme $t',u',v'$ ont disparu, nous pouvons les supposer nuls ; de sorte que cette valeur de $\rho$ est rigoureusement celle du rayon de courbure, à l’origine des coordonnées, de la section faite dans la surface $S=0$ par le plan (69). En substituant cette valeur dans les formules (82), nous obtiendrons rigoureusement les coordonnées du centre de courbure de la section.

La formule générale (88), que nous n’avons encore rencontrée nulle part, est susceptible d’un grand nombre de conséquences qu’il nous reste présentement à développer.