Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/251

en plus à devenir, pour cette origine, le centre de courbure de la courbe, et $\rho$ à être son rayon de courbure pour le même point.

Si présentement on substitue dans $M$ et $N$ (77) et (78), pour ${\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}},$${\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}},{\frac {\operatorname {d} \Omega '}{\operatorname {d} v'}},$ leurs valeurs (56), on trouvera

$\left\{{\begin{aligned}&A\left({\frac {\operatorname {d} S}{\operatorname {d} z}}u'-{\frac {\operatorname {d} S}{\operatorname {d} y}}v'\right)\\&\quad +(Bv'-Cu')\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v'\right)+\ldots \\\\+&B\left({\frac {\operatorname {d} S}{\operatorname {d} x}}v'-{\frac {\operatorname {d} S}{\operatorname {d} z}}t'\right)\\&\quad +(Ct'-Av')\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'\right)+\ldots \\\\+&C\left({\frac {\operatorname {d} S}{\operatorname {d} y}}t'-{\frac {\operatorname {d} S}{\operatorname {d} x}}u'\right)\\&\quad +(Au'-Bt')\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'\right)+\ldots \end{aligned}}\right\},\ (83)$

$N=\left\{{\begin{aligned}&\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v'\right)+\ldots \\\\+&\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'\right)+\ldots \\\\+&\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'\right)+\ldots \end{aligned}}\right\},\ (84)$

valeurs qui devront être substituées dans la formule (81). Mais, si l’on suppose que le point $(t',u',v')$ est très-voisin de l’origine des $t,u,v,$ on pourra, sans erreur sensible, négliger, dans $M$ et $N,$ les termes de plus d’une dimension, par rapport à ces coordonnées, ce qui donnera simplement

${\frac {M}{N}}=-{\frac {\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)t'+\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)u'+\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)v'}{\left\{{\begin{aligned}&\left(B{\frac {\operatorname {d} S}{\operatorname {d} z}}-C{\frac {\operatorname {d} S}{\operatorname {d} y}}\right)\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v'\right)+\ldots \\\\+&\left(C{\frac {\operatorname {d} S}{\operatorname {d} x}}-A{\frac {\operatorname {d} S}{\operatorname {d} z}}\right)\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'\right)+\ldots \\\\+&\left(A{\frac {\operatorname {d} S}{\operatorname {d} y}}-B{\frac {\operatorname {d} S}{\operatorname {d} x}}\right)\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'\right)+\ldots \end{aligned}}\right\}}}\,;\mathrm {(85)}$

mais, dans la même hypothèse, l’équation $\Omega '=0$ sa réduit à

${\frac {\operatorname {d} S}{\operatorname {d} x}}t'+{\frac {\operatorname {d} S}{\operatorname {d} x}}u'+{\frac {\operatorname {d} S}{\operatorname {d} x}}v'=0,\qquad$(86)

qui, combinée avec (71), donne