Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/26

qu’on y change respectivement $t$ et $u$ en $t'$ et $u'$ ; si $(t',u')$ est un point de la courbe (4), de sorte qu’on ait $\Omega '=0,$ l’équation de la normale menée, par ce même point, à la courbe $\Omega =0$ sera (27)

${\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}(t-t')-{\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}(u-u')=0\,;\qquad$(52)

les deux constantes $t'$ et $u'$ étant liées entre elles par l’équation $\Omega '=0,$ c’est-à-dire, par l’équation

${\begin{aligned}&{\begin{array}{r|}0={\frac {\operatorname {d} S'}{\operatorname {d} x'}}t'^{2}+{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x'^{2}}}t'^{2}\\\\+{\frac {\operatorname {d} S'}{\operatorname {d} y'}}u'+2{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x'\operatorname {d} y'}}t'u'\\\\+{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} y''^{2}}}u'^{2}\end{array}}\\&{\begin{aligned}\\\\\\\\\end{aligned}}\\\end{aligned}}{\begin{aligned}&{\begin{array}{r|}{\frac {1}{1.2}}+{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'^{3}}}t'^{3}\\\\+3{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'^{2}\operatorname {d} y'}}t'^{2}u'\\\\+3{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'\operatorname {d} y'^{2}}}t'u'^{2}\\\\+{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} y'^{3}}}u'^{3}\end{array}}\\&{\begin{aligned}\\\\\end{aligned}}\\\end{aligned}}\left.{\begin{array}{r}{\frac {1}{1.2.3}}+\ldots \\\\\\+\ldots \\\\+\ldots \\\\+\ldots \\\\+\ldots \end{array}}\right\}\,;$ (53)

ce qui donner

${\frac {\operatorname {d} \Omega '}{\operatorname {d} t'}}={\frac {\operatorname {d} S'}{\operatorname {d} x'}}+\left({\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x'^{2}}}t'+{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x'\operatorname {d} y'}}u'\right)$

$+{\frac {1}{1.2}}\left({\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'^{3}}}t'^{3}+2{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'^{2}\operatorname {d} y'}}t'u'+{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'\operatorname {d} y'^{2}}}u'^{2}\right)+\ldots ,$

${\frac {\operatorname {d} \Omega '}{\operatorname {d} u'}}={\frac {\operatorname {d} S'}{\operatorname {d} y'}}+\left({\frac {\operatorname {d} ^{2}S'}{\operatorname {d} y'^{2}}}u'+{\frac {\operatorname {d} ^{2}S'}{\operatorname {d} x'\operatorname {d} y'}}t'\right)$

$+{\frac {1}{1.2}}\left({\frac {\operatorname {d} ^{3}S'}{\operatorname {d} y'^{3}}}u'^{3}+2{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'\operatorname {d} y'^{2}}}t'u'+{\frac {\operatorname {d} ^{3}S'}{\operatorname {d} x'^{2}\operatorname {d} y'}}t'^{2}\right)+\ldots ,$

au moyen de quoi l’équation (51) de la normale au point $(t',u')$ deviendra