Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/817

Or, en regardant $\Omega$ comme fonction de $a,b,c,f,g,h,$ et ces quantités comme fonctions de $\alpha ,\beta ,\gamma ,\lambda ,\mu ,\nu ,$ on a par les formules connues

${\begin{alignedat}{6}{\frac {d\Omega }{d\alpha }}=&{\frac {da}{d\alpha }}{\frac {d\Omega }{da}}&&+{\frac {db}{d\alpha }}{\frac {d\Omega }{db}}&&+{\frac {dc}{d\alpha }}{\frac {d\Omega }{dc}}&&+{\frac {df}{d\alpha }}{\frac {d\Omega }{df}}&&+{\frac {dg}{d\alpha }}{\frac {d\Omega }{dg}}&&+{\frac {dh}{d\alpha }}{\frac {d\Omega }{dh}},\\{\frac {d\Omega }{d\beta }}=&{\frac {da}{d\beta }}{\frac {d\Omega }{da}}&&+{\frac {db}{d\beta }}{\frac {d\Omega }{db}}&&+{\frac {dc}{d\beta }}{\frac {d\Omega }{dc}}&&+{\frac {df}{d\beta }}{\frac {d\Omega }{df}}&&+{\frac {dg}{d\beta }}{\frac {d\Omega }{dg}}&&+{\frac {dh}{d\beta }}{\frac {d\Omega }{dh}},\\\end{alignedat}}$

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .$

4. Faisant toutes ces substitutions dans les expressions précédentes de ${\frac {da}{dt}},{\frac {db}{dt}},\ldots ,$ et ordonnant les termes suivant les différences partielles de $\Omega ,$ on voit d’abord que le coefficient de ${\frac {d\Omega }{da}}$ est nul dans la valeur de ${\frac {da}{dt}},$ que celui de ${\frac {d\Omega }{db}}$ est nul dans la valeur de ${\frac {db}{dt}},$ et ainsi des autres ; qu’ensuite, en employant des symboles $[a,b],[a,c],[b,c],\ldots$ analogues à ceux du premier Mémoire, tels que l’on ait

${\begin{alignedat}{6}&[a,b]=-{\frac {da}{d\alpha }}{\frac {db}{d\lambda }}&&-{\frac {da}{d\beta }}{\frac {db}{d\mu }}&&-{\frac {da}{d\gamma }}{\frac {db}{d\nu }}&&+{\frac {da}{d\lambda }}{\frac {db}{d\alpha }}&&+{\frac {da}{d\mu }}{\frac {db}{d\beta }}&&+{\frac {da}{d\nu }}{\frac {db}{d\gamma }},\\&[a,c]=-{\frac {da}{d\alpha }}{\frac {dc}{d\lambda }}&&-{\frac {da}{d\beta }}{\frac {dc}{d\mu }}&&-{\frac {da}{d\gamma }}{\frac {dc}{d\nu }}&&+{\frac {da}{d\lambda }}{\frac {dc}{d\alpha }}&&+{\frac {da}{d\mu }}{\frac {dc}{d\beta }}&&+{\frac {da}{d\nu }}{\frac {dc}{d\gamma }},\\&[b,c]=-{\frac {db}{d\alpha }}{\frac {dc}{d\lambda }}&&-{\frac {db}{d\beta }}{\frac {dc}{d\mu }}&&-{\frac {db}{d\gamma }}{\frac {dc}{d\nu }}&&+{\frac {db}{d\lambda }}{\frac {dc}{d\alpha }}&&+{\frac {db}{d\mu }}{\frac {dc}{d\beta }}&&+{\frac {db}{d\nu }}{\frac {dc}{d\gamma }},\\\end{alignedat}}$

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,$

on aura ces formules

${\begin{alignedat}{5}{\frac {da}{dt}}=&\quad [a,b]{\frac {d\Omega }{db}}&&+[a,c]{\frac {d\Omega }{dc}}&&+[a,f]{\frac {d\Omega }{df}}&&+[a,g]{\frac {d\Omega }{dg}}&&+[a,h]{\frac {d\Omega }{dh}},\\{\frac {db}{dt}}=&-[a,b]{\frac {d\Omega }{da}}&&+[b,c]{\frac {d\Omega }{dc}}&&+[b,f]{\frac {d\Omega }{df}}&&+[b,g]{\frac {d\Omega }{dg}}&&+[b,h]{\frac {d\Omega }{dh}},\\{\frac {dc}{dt}}=&-[a,c]{\frac {d\Omega }{da}}&&-[b,c]{\frac {d\Omega }{db}}&&+[c,f]{\frac {d\Omega }{df}}&&+[c,g]{\frac {d\Omega }{dg}}&&+[c,h]{\frac {d\Omega }{dh}},\end{alignedat}}$

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .$