Page:Lagrange - Œuvres (1867) vol. 1.djvu/715

et l’on remarquera que les quantités ${\mathfrak {A_{1},B_{1},C_{1},\ldots ,P_{1},Q_{1},R_{1}}},\ldots$ restent nécessairement les mêmes, en changeant $a$ en $a'$ et $a'$ en $a,$ de sorte qu’on aura aussi

${\mathfrak {A}}'_{1}={\frac {\mathfrak {A}}{a'^{3}}},\quad {\mathfrak {B}}'_{1}={\frac {\mathfrak {B}}{a'^{3}}},\ldots ,\quad {\mathfrak {P}}_{1}={\frac {\mathfrak {P}}{a'^{5}}},\quad {\mathfrak {Q}}'_{1}={\frac {\mathfrak {Q}}{a'^{5}}},\ldots .$

Faisant donc ces substitutions dans les formules du n^o 75, et mettant partout $\mathrm {a}$ au lieu de ${\frac {a}{a'}},$ on trouvera d’abord

${\begin{aligned}{\mathfrak {A}}_{2}&=a^{3}{\mathfrak {A}}-a^{2}{\frac {\mathfrak {B}}{2}},\\{\mathfrak {A}}_{3}&=a^{2}{\frac {2{\mathfrak {A-C}}}{2}}-a^{2},\\{\mathfrak {A}}_{4}&=a^{3}{\mathfrak {A-A_{2}}}=a^{2}{\frac {\mathfrak {B}}{2}},\\{\mathfrak {B}}_{2}&=a^{3}{\mathfrak {B}}-a^{2}{\frac {2{\mathfrak {A+C}}}{2}}+a^{2},\\{\mathfrak {B}}_{5}&=-a^{2}{\mathfrak {B}},\\\end{aligned}}$

ensuite on aura (n^o 74)

${\begin{aligned}{\mathfrak {P}}_{2}&={\frac {3}{a'^{3}}}\left(\mathrm {a} {\frac {\mathfrak {Q}}{2}}-\mathrm {a} ^{2}{\mathfrak {P}}\right),\\{\mathfrak {Q}}_{2}&={\frac {3}{a'^{3}}}\left(\mathrm {a} {\frac {2{\mathfrak {P+R}}}{2}}-\mathrm {a} ^{2}{\mathfrak {Q}}\right),\\{\mathfrak {R}}_{2}&={\frac {3}{a'^{3}}}\left(\mathrm {a} {\frac {\mathfrak {Q+S}}{2}}-\mathrm {a} ^{2}{\mathfrak {R}}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\{\mathfrak {P}}_{3}&={\frac {3}{a'^{3}}}\left(\mathrm {a} {\frac {\mathfrak {Q}}{2}}-{\mathfrak {P}}\right),\\{\mathfrak {Q}}_{3}&={\frac {3}{a'^{3}}}\left(\mathrm {a} {\frac {\mathfrak {2P+R}}{2}}-{\mathfrak {Q}}\right),\\{\mathfrak {R}}_{3}&={\frac {3}{a'^{3}}}\left(\mathrm {a} {\frac {\mathfrak {Q+S}}{2}}-{\mathfrak {R}}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

ou bien, en faisant pour plus de simplicité

$\mathrm {p} =3\left(\mathrm {a} {\frac {\mathfrak {Q}}{2}}-\mathrm {a} ^{2}{\mathfrak {P}}\right),$