Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/757

et qu’on observe que les trois premières équations de condition donnent

${\begin{aligned}&\xi '\ d\xi '\ \,+\eta '\ \,d\eta '\ \,+\zeta '\ \,d\zeta '\ \,=0,\\&\xi ''\,d\xi ''\,+\eta ''\ d\eta ''\,+\zeta ''\ d\zeta ''\,=0,\\&\xi '''d\xi '''+\eta '''d\eta '''+\zeta '''d\zeta '''=0,\\\end{aligned}}$

on aura la transformée

${\begin{aligned}&{\frac {\xi '\ d\xi ''\ +\eta '\ d\eta ''\ +\zeta '\ d\zeta ''\ }{df}}{\frac {d\mathrm {X} }{da}}{\frac {d\mathrm {Y} }{dt}}\\+&{\frac {\xi '\ d\xi '''+\eta '\ \,d\eta '''+\zeta '\ \,d\zeta '''}{df}}{\frac {d\mathrm {X} }{da}}{\frac {d\mathrm {Z} }{dt}}\\+&{\frac {\xi ''\,d\xi '\ +\eta ''\,d\eta '\,\ +\zeta ''\ d\zeta '\,\ }{df}}{\frac {d\mathrm {Y} }{da}}{\frac {d\mathrm {X} }{dt}}\\+&{\frac {\xi ''\,d\xi '''+\eta ''\,d\eta '''+\zeta ''\ d\zeta '''}{df}}{\frac {d\mathrm {Y} }{da}}{\frac {d\mathrm {Z} }{dt}}\\+&{\frac {\xi '''d\xi '\ \,+\eta '''d\eta '\ \,+\zeta '''d\zeta '\ \ }{df}}{\frac {d\mathrm {Z} }{da}}{\frac {d\mathrm {X} }{dt}}\\+&{\frac {\xi '''d\xi ''\ +\eta '''d\eta ''\ +\zeta '''d\zeta ''\ }{df}}{\frac {d\mathrm {Z} }{da}}{\frac {d\mathrm {Y} }{dt}}.\end{aligned}}$

Mais les trois dernières équations de condition entre $\xi ',\xi '',\ldots ,$ étant différentiées, donnent

${\begin{alignedat}{6}\xi '\ d\xi ''\,&+&\eta '\ d\eta ''\,&+&\zeta '\ d\zeta ''\,=&-\xi ''d\xi '&-&\eta ''d\eta '&-&\zeta ''d\zeta ',\\\xi '\ d\xi '''&+&\eta '\ d\eta '''&+&\zeta '\ d\zeta '''=&-\xi '''d\xi '&-&\eta '''d\eta '&-&\zeta '''d\zeta ',\\\xi ''d\xi '''&+&\eta ''d\eta '''&+&\zeta ''d\zeta '''=&-\xi '''d\xi ''&-&\eta '''d\eta ''&-&\zeta '''d\zeta ''.\end{alignedat}}$

Donc, si l’on fait, pour abréger, comme dans le n^o 23 de la Section citée de la Mécanigue analytique^[1],

${\begin{aligned}d\mathrm {P} =&\xi '''d\xi ''\ +\eta '''\,d\eta ''\ +\zeta '''d\zeta '',\\d\mathrm {Q} =&\xi '\ \ d\xi '''+\eta '\ \ d\eta '''+\zeta '\ \ d\zeta ''',\\d\mathrm {R} =&\xi ''\ d\xi '\ \ +\eta ''\ d\eta '\ \ +\zeta ''\ d\zeta ',\end{aligned}}$

1. Dans la deuxième édition, seconde Partie, Section IX, n^o 11.$\qquad$(Note de l’Éditeur.)