Discussion Page:Henri Poincaré - Les méthodes nouvelles de la mécanique céleste, Tome 1, 1892.djvu/130

sauf erreur de ma part, il manque un facteur $(1+2x_{1}^{2}x_{2})$ dans le hessien de la fonction

\varphi =\left({\frac {1}{2x_{1}^{2}}}+x_{2}\right)^{2}={\frac {1}{4x_{1}^{4}}}+{\frac {x_{2}}{x_{1}^{2}}}+x_{2}^{2}

ce qui donne

{\begin{aligned}{\frac {\partial \varphi }{\partial x_{1}}}&=-{\frac {1}{x_{1}^{5}}}-{\frac {2x_{2}}{x_{1}^{3}}}\\{\frac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}&={\frac {5}{x_{1}^{6}}}+{\frac {6x_{2}}{x_{1}^{4}}}\\{\frac {\partial \varphi }{\partial x_{2}}}&={\frac {1}{x_{1}^{2}}}+2x_{2}\\{\frac {\partial ^{2}\varphi }{\partial x_{2}^{2}}}&=2\\{\frac {\partial ^{2}\varphi }{\partial x_{1}\,\partial x_{2}}}&=-{\frac {2}{x_{1}^{3}}}\\\end{aligned}}

Le hessien de la fonction des 2 variables $\varphi (x_{1},x_{2})$ étant

{\frac {\partial ^{2}\varphi }{\partial x_{1}^{2}}}{\frac {\partial ^{2}\varphi }{dx_{2}^{2}}}-\left({\frac {\partial \varphi }{\partial x_{1}\,\partial x_{2}}}\right)^{2}

,

ça nous donne

{\frac {6}{x_{1}^{6}}}+{\frac {12x_{2}}{x_{1}^{4}}}