Page:Annales de mathématiques pures et appliquées, 1827-1828, Tome 18.djvu/221

 Les corrections sont expliquées en page de discussion

$\left.{\begin{aligned}12P^{2}T\ \ \ &=2P'P''P'''+{\sqrt[{3}]{36\Delta P^{2}P'^{2}P''^{2}P'''^{2}}}+6P{\sqrt[{3}]{36\Delta ^{2}PP'P''P'''}},\\\\12P'^{2}T'\ \,&=2P''P'''P+{\sqrt[{3}]{36\Delta P^{2}P'^{2}P''^{2}P'''^{2}}}+6P'{\sqrt[{3}]{36\Delta ^{2}PP'P''P'''}},\\\\12P''^{2}T''\,&=2P'''PP'+{\sqrt[{3}]{36\Delta P^{2}P'^{2}P''^{2}P'''^{2}}}+6P'{\sqrt[{3}]{36\Delta ^{2}PP'P''P'''}},\\\\12P'''^{2}T'''&=2PP'P''+{\sqrt[{3}]{36\Delta P^{2}P'^{2}P''^{2}P'''^{2}}}+6P'{\sqrt[{3}]{36\Delta ^{2}PP'P''P'''}}\,;\end{aligned}}\right\}$ (III)

équations qui donneront les valeurs de $T,T',T'',T'''.$

Au moyen des équations (11) et (12), les équations (II) deviennent facilement

$\left.{\begin{aligned}2P''P'''(pp')&=PP'(P''+P''')+2{\sqrt[{3}]{6\Delta P^{2}P'^{2}P''^{2}P'''^{2}}},\\\\2P'''P'(pp'')&=PP''(P'''+P')+2{\sqrt[{3}]{6\Delta P^{2}P'^{2}P''^{2}P'''^{2}}},\\\\2P'P''(pp''')&=PP'''(P'+P'')+2{\sqrt[{3}]{6\Delta P^{2}P'^{2}P''^{2}P'''^{2}}},\\\\2PP'(p''p''')&=P''P'''(P+P')+2{\sqrt[{3}]{6\Delta P^{2}P'^{2}P''^{2}P'''^{2}}},\\\\2PP''(p'''p')&=P'''P'(P+P'')+2{\sqrt[{3}]{6\Delta P^{2}P'^{2}P''^{2}P'''^{2}}},\\\\2PP'''(p'p'')&=P'P''(P+P''')+2{\sqrt[{3}]{6\Delta P^{2}P'^{2}P''^{2}P'''^{2}}},\\\\\end{aligned}}\right\}$ (IV)

équations délivrées de $T,T',T'',T'''.$

En prenant la racine cubique de $36$ fois le produit des équations (1) on obtient