Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/243

${\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}t+{\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}u+{\frac {\operatorname {d} \Omega '}{\operatorname {d} v'}}v={\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}t'+{\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}u'+{\frac {\operatorname {d} \Omega '}{\operatorname {d} v'}}v',\qquad$(54)

sous la condition

$0=\Omega '=$

${\begin{alignedat}{2}{\frac {\operatorname {d} S}{\operatorname {d} x}}t'&+{\frac {\operatorname {d} S}{\operatorname {d} y}}u'+&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}{\frac {t'^{2}}{1.2}}&+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}{\frac {t'u'}{1.2}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}{\frac {u'^{2}}{1.2}}+\ldots \mathrm {(55)} \\\\&+{\frac {\operatorname {d} S}{\operatorname {d} z}}v'&&+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}{\frac {v't'}{1.2}}+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {u'v'}{1.2}}\\\\&&&+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {v'^{2}}{1.2}},\end{alignedat}}$

en ne conservant, pour la facilité de l’écriture, que les accens de $t,u,v.$ Or, cela donne

$\left.{\begin{aligned}&{\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}={\frac {\operatorname {d} S}{\operatorname {d} x}}+\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v'\right)+\ldots \\\\&{\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}={\frac {\operatorname {d} S}{\operatorname {d} y}}+\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'\right)+\ldots \\\\&{\frac {\operatorname {d} \Omega '}{\operatorname {d} v'}}={\frac {\operatorname {d} S}{\operatorname {d} z}}+\left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}t'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'\right)+\ldots \end{aligned}}\right\}\quad$(56)

et, par suite,

${\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}t'+{\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}u'+{\frac {\operatorname {d} \Omega '}{\operatorname {d} v'}}v'=$

${\begin{alignedat}{2}{\frac {\operatorname {d} S}{\operatorname {d} x}}t'&+{\frac {\operatorname {d} S}{\operatorname {d} y}}u'+&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}t'^{2}&+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}t'u'+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}u'^{2}+\ldots \\\\&+{\frac {\operatorname {d} S}{\operatorname {d} z}}v'&&+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z\operatorname {d} x}}v't'+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}u'v'\\\\&&&+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}v'^{2}\end{alignedat}}$

ou bien, en ayant égard à la relation (55),