Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/341

${\begin{aligned}z^{\text{vı}}=2\left[x'^{3}\left(x''x^{\text{ıv}}+x'''x^{\mathrm {v} }\right)\right.&+x''^{3}\left(x'x^{\mathrm {v} }+x'''x^{\text{ıv}}\right)+x^{\mathrm {v3} }\left(x''x'''+x'x^{\text{ıv}}\right)\\&+\left.x'''^{3}\left(x^{\text{ıv}}x^{\mathrm {v} }+x'x''\right)+x^{\mathrm {iv3} }\left(x'x'''+x''x^{\mathrm {v} }\right)\right]\\+3\left[x'\left(x''^{2}x^{\mathrm {iv2} }+x'''^{2}x^{\mathrm {v2} }\right)\right.&+x''\left(x'^{2}x^{\mathrm {v2} }+x'''^{2}x^{\mathrm {iv2} }\right)+x^{\mathrm {v} }\left(x''^{2}x'''^{2}+x'^{2}x^{\mathrm {iv2} }\right)\\&+\left.x'''\left(x^{\mathrm {iv2} }x^{\mathrm {v2} }+x'^{2}x''^{2}\right)+x^{\text{ıv}}\left(x'^{2}x'''^{2}+x''^{2}x^{\mathrm {v} }\right)\right].\end{aligned}}$

En effet, si l’on fait dans ces formules telles permutations que l’on voudra entre les racines $x',x'',x''',\ldots ,$ on verra toujours renaître les mêmes f’ormules ; d’où il s’ensuit que les six quantités $z',z'',z''',\ldots ,$ seront nécessairement les racines d’une équation du sixième degré, telle que

$z^{6}-\mathrm {A} z^{5}+\mathrm {B} z^{4}-\mathrm {C} z^{3}+\mathrm {D} z^{2}-\mathrm {E} z+\mathrm {F} =0,$

dont les coefficients $\mathrm {A,B} ,\ldots$ pourront par conséquent se déterminer par les règles connues.

On aura, par exemple,

$\mathrm {A} =z'+z''+z'''+z^{\text{ıv}}+z^{\mathrm {v} }+z^{\text{vı}}\,;$

c’est-à-dire

${\begin{aligned}\mathrm {A} =&4x'^{3}\,\ \left(x''x'''+x''x^{\text{ıv}}+x''x^{\mathrm {v} }+x'''x^{\text{ıv}}+x'''x^{\mathrm {v} }+x^{\text{ıv}}x^{\mathrm {v} }\right)\\+&4x''^{3}\ \left(x'x'''\,+x'x^{\text{ıv}}\,+x'x^{\mathrm {v} }\,+x'''x^{\text{ıv}}+x'''x^{\mathrm {v} }+x^{\text{ıv}}x^{\mathrm {v} }\right)\\+&4x'''^{3}\left(x'x''\ \,+x'x^{\text{ıv}}\,+x'x^{\mathrm {v} }\,+x''x^{\text{ıv}}\,+x''x^{\mathrm {v} }\ +x^{\text{ıv}}x^{\mathrm {v} }\right)\\+&4x^{\mathrm {iv3} }\left(x'x''\ \,+x'x'''\,+x'x^{\mathrm {v} }\ +x''x'''\,+x''x^{\mathrm {v} }\ +x'''x^{\mathrm {v} }\right)\\+&4x^{\mathrm {v3} }\,\left(x'x''\ \,+x'x'''\,+x'x^{\text{ıv}}+x''x'''\,+x''x^{\text{ıv}}+x'''x^{\text{ıv}}\right)\\+&6x'\ \,\left[(x''x''')^{2}+(x''x^{\text{ıv}})^{2}+(x''x^{\mathrm {v} })^{2}+(x'''x^{\text{ıv}})^{2}+(x'''x^{\mathrm {v} })^{2}+(x^{\text{ıv}}x^{\mathrm {v} })^{2}\right]\\+&6x''\ \left[(x'x''')^{2}\,+(x'x^{\text{ıv}})^{2}\,+(x'x^{\mathrm {v} })^{2}\,+(x'''x^{\text{ıv}})^{2}+(x'''x^{\mathrm {v} })^{2}+(x^{\text{ıv}}x^{\mathrm {v} })^{2}\right]\\+&6x'''\left[(x'x'')^{2}\,\ +(x'x^{\text{ıv}})^{2}\,+(x'x^{\mathrm {v} })^{2}\,+(x''x^{\text{ıv}})^{2}\,+(x''x^{\mathrm {v} })^{2}\ +(x^{\text{ıv}}x^{\mathrm {v} })^{2}\right]\\+&6x^{\text{ıv}}\left[(x'x'')^{2}\,\ +(x'x''')^{2}\,+(x'x^{\mathrm {v} })^{2}\,+(x''x''')^{2}\,+(x''x^{\mathrm {v} })^{2}\ +(x'''x^{\mathrm {v} })^{2}\right]\\+&6x^{\mathrm {v} }\,\left[(x'x'')^{2}\,\ +(x'x''')^{2}\,+(x'x^{\text{ıv}})^{2}+(x''x''')^{2}\,+(x''x^{\text{ıv}})^{2}+(x'''x^{\text{ıv}})^{2}\right]\\\end{aligned}}$

Or on a dans l’équation proposée

${\begin{aligned}-m=&x'+x''+x'''+x^{\text{ıv}}+x^{\mathrm {v} },\\n=&x'x''+x'x'''+x'x^{\text{ıv}}+x'x^{\mathrm {v} }+x''x'''+x''x^{\text{ıv}}+x''x^{\mathrm {v} }\end{aligned}}$

$+x'''x^{\text{ıv}}+x'''x^{\mathrm {v} }+x^{\text{ıv}}x^{\mathrm {v} }\,;$