Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/160

Cela posé, on trouvera (Article XXII), $q$ étant égal à ${\frac {a_{1}}{a_{2}}},$

${\begin{aligned}&\Pi \ \,(a_{1},a_{2})={\frac {q(\mathrm {B} )-2q^{2}(\mathrm {A} )}{2a_{2}^{3}}},\\&\Pi _{1}(a_{1},a_{2})={\frac {q(\mathrm {C} )-2q^{2}(\mathrm {B} )+2q(\mathrm {A} )}{2a_{2}^{3}}},\\&\Pi _{2}(a_{1},a_{2})={\frac {q(\mathrm {D} )-2q^{2}(\mathrm {C} )+q(\mathrm {B} )}{2a_{2}^{3}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

et de là, en faisant, pour abréger,

${\begin{aligned}&\mathrm {P} \ \,={\frac {q(\mathrm {B} )-q^{2}(\mathrm {A} )}{2}},\\&\mathrm {P} _{1}={\frac {q(\mathrm {C} )-2q^{2}(\mathrm {B} )+2q(\mathrm {A} )}{2}},\\&\mathrm {P} _{2}={\frac {q(\mathrm {D} )-q^{2}(\mathrm {C} )+q(\mathrm {B} )}{2}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

on aura par l’Article XXIV

${\begin{aligned}&{\breve {\Pi }}\ \,(a_{1},a_{2})=3{\frac {q^{2}\mathrm {P} _{1}-2q^{3}\mathrm {P} }{2}}-q^{3}\mathrm {A} ,\\&{\breve {\Pi }}_{1}(a_{1},a_{2})=3{\frac {q^{2}\mathrm {P} _{2}-2q^{3}\mathrm {P} _{1}+2q^{2}\mathrm {P} }{2}}-q^{3}\mathrm {B} ,\\&{\breve {\Pi }}_{2}(a_{1},a_{2})=3{\frac {q^{2}\mathrm {P} _{3}-2q^{3}\mathrm {P} _{2}+q^{2}\mathrm {P} _{1}}{2}}-q^{3}\mathrm {C} ,\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

On trouvera des expressions semblables pour les fonctions ${\breve {\Pi }}$ de $(a_{1},a_{3}),$$(a_{2},a_{3}),\ldots ,$ en faisant $q={\frac {a_{1}}{a_{3}}},\ ={\frac {a_{2}}{a_{3}}},\ldots .$