Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/210

On aura donc (Article XXIV)

${\breve {\Psi }}_{1}(a_{1},a_{2})=3{\frac {q^{2}\mathrm {Q} _{2}-2q^{3}\mathrm {Q} _{1}+2q^{2}\mathrm {Q} }{2}}+{\frac {q^{2}\mathrm {C} +2q^{2}\mathrm {A} }{2}}+2q^{2},$

expression qui servira aussi pour les quantités analogues ${\breve {\Psi }}_{1}(a_{1},a_{3}),$ ${\breve {\Psi }}_{1}(a_{2},a_{3}),\ldots ,$ en faisant successivement

$q={\frac {a_{1}}{a_{3}}},\quad q={\frac {a_{2}}{a_{3}}},\ldots .$

Ensuite on aura, pour les quantités réciproques,

${\breve {\Psi }}_{1}(a_{2},a_{1})=3{\frac {q\mathrm {P} _{2}-2\mathrm {P} _{1}+2q\mathrm {P} }{2}}+{\frac {q\mathrm {C} +2q\mathrm {A} }{2}}+{\frac {2}{q^{2}}}\,;$

et ainsi des autres.

Pareillement on trouvera (Article XXV)

${\begin{aligned}{\widehat {\Psi }}_{1}(a_{1},a_{2})=&3{\frac {q^{2}\mathrm {Q} _{2}-2q^{2}\mathrm {Q} }{2}}+{\widehat {\Gamma }}_{1}(a_{1},a_{2})-3q^{2},\\{\widehat {\Psi }}_{1}(a_{2},a_{1})=&3{\frac {q\mathrm {Q} _{2}-2q\mathrm {Q} }{2}}+{\widehat {\Gamma }}_{1}(a_{2},a_{1})-{\frac {3}{q^{2}}}\,;\end{aligned}}$

et ainsi des autres.

De là on tire, en employant les valeurs de la Table de l’Article LXXXI

${\begin{array}{|c|r|r|r|r|r|r|}\hline &(a_{1},a_{2})&(a_{1},a_{3})&(a_{1},a_{4})&(a_{2},a_{3})&(a_{2},a_{4})&(a_{3},a_{4})\\\hline \ {\breve {\Psi }}_{1}&-1{,}708&-0{,}181&-0{,}008&-1{,}733&-0{,}132&-0{,}887\\\hline \ {\widehat {\Psi }}_{1}&0{,}366&0{,}068&0{,}003&0{,}153&0{,}036&0{,}214\\\hline &(a_{2},a_{1})&(a_{3},a_{1})&(a_{4},a_{1})&(a_{3},a_{2})&(a_{4},a_{2})&(a_{4},a_{3})\\\hline \ {\breve {\Psi }}_{1}&1{,}203&11{,}699&29{,}330&1{,}580&14{,}871&3{,}489\\\hline \ {\widehat {\Psi }}_{1}&-6{,}181&-13{,}371&-40{,}052&-6{,}500&-16{,}188&-7{,}089\\\hline \end{array}}$