Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/340

Donc, si l’on fait pour plus de simplicité

${\begin{aligned}z=2\left[x'^{3}\left(x''x^{\scriptscriptstyle {\text{V}}}+x'''x^{\scriptscriptstyle {\text{IV}}}\right)\right.&+x''^{3}\left(x'x'''+x^{\scriptscriptstyle {\text{IV}}}x^{\scriptscriptstyle {\text{V}}}\right)+x'''^{3}\left(x''x^{\scriptscriptstyle {\text{IV}}}+x'x^{\scriptscriptstyle {\text{V}}}\right)\\&+\left.x^{{\scriptscriptstyle {\text{IV}}}3}\left(x'''x^{\scriptscriptstyle {\text{V}}}+x'x''\right)+x^{{\scriptscriptstyle {\text{V}}}3}\left(x'x^{\scriptscriptstyle {\text{IV}}}+x''x'''\right)\right]\\+3\left[x'\left(x''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)\right.&+x''\left(x'^{2}x'''^{2}+x^{{\scriptscriptstyle {\text{IV}}}2}x^{{\scriptscriptstyle {\text{V}}}2}\right)+x'''\left(x''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\\&+\left.x^{\scriptscriptstyle {\text{IV}}}\left(x'''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'^{2}x''^{2}\right)+x^{\scriptscriptstyle {\text{V}}}\left(x'^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x''^{2}x'''^{2}\right)\right],\end{aligned}}$

on aura

$\mathrm {T} =50z-4m^{5}+25\left[m^{3}n-m^{2}p+m\left(q-n^{2}\right)+np-25r\right],$

et l’on trouvera que la quantité $z$ ne sera susceptible que des six valeurs suivantes, que nous désignerons par $z',z'',z''',z^{\scriptscriptstyle {\text{IV}}},z^{\scriptscriptstyle {\text{V}}},z^{\scriptscriptstyle {\text{VI}}}\,:$

${\begin{aligned}z'\ \ =2\left[x'^{3}\left(x''x^{\scriptscriptstyle {\text{V}}}+x'''x^{\scriptscriptstyle {\text{IV}}}\right)\right.&+x''^{3}\left(x'x'''+x^{\scriptscriptstyle {\text{IV}}}x^{\scriptscriptstyle {\text{V}}}\right)+x'''^{3}\left(x''x^{\scriptscriptstyle {\text{IV}}}+x'x^{\scriptscriptstyle {\text{V}}}\right)\\&+\left.x^{{\scriptscriptstyle {\text{IV}}}3}\left(x'''x^{\scriptscriptstyle {\text{V}}}+x'x''\right)+x^{{\scriptscriptstyle {\text{V}}}3}\left(x'x^{\scriptscriptstyle {\text{IV}}}+x''x'''\right)\right]\\+3\left[x'\left(x''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)\right.&+x''\left(x'^{2}x'''^{2}+x^{{\scriptscriptstyle {\text{IV}}}2}x^{{\scriptscriptstyle {\text{V}}}2}\right)+x'''\left(x''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\\&+\left.x^{\scriptscriptstyle {\text{IV}}}\left(x'''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'^{2}x''^{2}\right)+x^{\scriptscriptstyle {\text{V}}}\left(x'^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x''^{2}x'''^{2}\right)\right],\\\\z''\ =2\left[x'^{3}\left(x''x^{\scriptscriptstyle {\text{IV}}}+x'''x^{\scriptscriptstyle {\text{V}}}\right)\right.&+x''^{3}\left(x'x'''+x^{\scriptscriptstyle {\text{IV}}}x^{\scriptscriptstyle {\text{V}}}\right)+x'''^{3}\left(x''x^{\scriptscriptstyle {\text{V}}}+x'x^{\scriptscriptstyle {\text{IV}}}\right)\\&+\left.x^{{\scriptscriptstyle {\text{V}}}3}\left(x'''x^{\scriptscriptstyle {\text{IV}}}+x'x''\right)+x^{{\scriptscriptstyle {\text{IV}}}3}\left(x'x^{\scriptscriptstyle {\text{V}}}+x''x'''\right)\right]\\+3\left[x'\left(x''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\right.&+x''\left(x'^{2}x'''^{2}+x^{{\scriptscriptstyle {\text{IV}}}2}x^{{\scriptscriptstyle {\text{V}}}2}\right)+x'''\left(x''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)\\&+\left.x^{\scriptscriptstyle {\text{V}}}\left(x'''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'^{2}x''^{2}\right)+x^{\scriptscriptstyle {\text{IV}}}\left(x'^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x''^{2}x'''^{2}\right)\right],\\\\z'''=2\left[x'^{3}\left(x''x^{\scriptscriptstyle {\text{V}}}+x'''x^{\scriptscriptstyle {\text{IV}}}\right)\right.&+x''^{3}\left(x'x^{\scriptscriptstyle {\text{IV}}}+x'''x^{\scriptscriptstyle {\text{V}}}\right)+x^{{\scriptscriptstyle {\text{IV}}}3}\left(x''x'''+x'x^{\scriptscriptstyle {\text{V}}}\right)\\&+\left.x'''^{3}\left(x^{\scriptscriptstyle {\text{IV}}}x^{\scriptscriptstyle {\text{V}}}+x'x''\right)+x^{{\scriptscriptstyle {\text{V}}}3}\left(x'x'''+x''x^{\scriptscriptstyle {\text{IV}}}\right)\right]\\+3\left[x'\left(x''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)\right.&+x''\left(x'^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)+x^{\scriptscriptstyle {\text{IV}}}\left(x''^{2}x'''^{2}+x'^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\\&+\left.x'''\left(x^{{\scriptscriptstyle {\text{IV}}}2}x^{{\scriptscriptstyle {\text{V}}}2}+x'^{2}x''^{2}\right)+x^{\scriptscriptstyle {\text{V}}}\left(x'^{2}x'''^{2}+x''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)\right],\\\\\end{aligned}}$ ${\begin{aligned}z^{\scriptscriptstyle {\text{IV}}}=2\left[x'^{3}\left(x''x'''+x^{\scriptscriptstyle {\text{IV}}}x^{\scriptscriptstyle {\text{V}}}\right)\right.&+x''^{3}\left(x'x^{\scriptscriptstyle {\text{IV}}}+x'''x^{\scriptscriptstyle {\text{V}}}\right)+x^{{\scriptscriptstyle {\text{IV}}}3}\left(x''x^{\scriptscriptstyle {\text{V}}}+x'x'''\right)\\&+\left.x^{{\scriptscriptstyle {\text{V}}}3}\left(x'''x^{\scriptscriptstyle {\text{IV}}}+x'x''\right)+x'''^{3}\left(x'x^{\scriptscriptstyle {\text{V}}}+x''x^{\scriptscriptstyle {\text{IV}}}\right)\right]\\+3\left[x'\left(x''^{2}x'''^{2}+x^{{\scriptscriptstyle {\text{IV}}}2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\right.&+x''\left(x'^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)+x^{\scriptscriptstyle {\text{IV}}}\left(x''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'^{2}x'''^{2}\right)\\&+\left.x^{\scriptscriptstyle {\text{V}}}\left(x'''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'^{2}x''^{2}\right)+x'''\left(x'^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)\right],\\\\z^{\scriptscriptstyle {\text{V}}}=2\left[x'^{3}\left(x''x'''+x^{\scriptscriptstyle {\text{IV}}}x^{\scriptscriptstyle {\text{V}}}\right)\right.&+x''^{3}\left(x'x^{\scriptscriptstyle {\text{V}}}+x'''x^{\scriptscriptstyle {\text{IV}}}\right)+x^{{\scriptscriptstyle {\text{V}}}3}\left(x''x^{\scriptscriptstyle {\text{IV}}}+x'x'''\right)\\&+\left.x^{{\scriptscriptstyle {\text{IV}}}3}\left(x'''x^{\scriptscriptstyle {\text{V}}}+x'x''\right)+x'''^{3}\left(x'x^{\scriptscriptstyle {\text{IV}}}+x''x^{\scriptscriptstyle {\text{V}}}\right)\right]\\+3\left[x'\left(x''^{2}x'''^{2}+x^{{\scriptscriptstyle {\text{IV}}}2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\right.&+x''\left(x'^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}\right)+x^{\scriptscriptstyle {\text{V}}}\left(x''^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x'^{2}x'''^{2}\right)\\&+\left.x^{\scriptscriptstyle {\text{IV}}}\left(x'''^{2}x^{{\scriptscriptstyle {\text{V}}}2}+x'^{2}x''^{2}\right)+x'''\left(x'^{2}x^{{\scriptscriptstyle {\text{IV}}}2}+x''^{2}x^{{\scriptscriptstyle {\text{V}}}2}\right)\right],\end{aligned}}$