Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/139

Car :

1^o En ajoutant ensemble les carrés des équations (B), on a

${\frac {(xdy-ydx)^{2}}{dt^{2}}}+{\frac {(xdz-zdx)^{2}}{dt^{2}}}+{\frac {(ydz-zdy)^{2}}{dt^{2}}}$

$={\frac {\left(x^{2}+y^{2}+z^{2}\right)(dx^{2}+dy^{2}+dz^{2})}{dt^{2}}}-{\frac {(xdx+ydy+zdz)^{2}}{dt^{2}}}$

$=\mathrm {P^{2}+Q^{2}+R^{2}} =\Pi ^{2},$

c’est-à-dire

${\frac {\rho ^{2}\left(dx^{2}+dy^{2}+dz^{2}\right)}{dt^{2}}}-{\frac {\rho ^{2}d\rho ^{2}}{dt^{2}}}=\Pi ^{2}\,;$

d’où l’on tire

$dx^{2}+dy^{2}+dz^{2}={\frac {\Pi ^{2}}{\rho ^{2}}}dt^{2}+d\rho ^{2}.$

2^o En ajoutant aussi ensemble les carrés des équations (E), mais après y avoir transposé les termes $\mathrm {N,M,L} ,$ on a

$\left({\frac {gx}{\rho }}-\mathrm {N} \right)^{2}+\left({\frac {gy}{\rho }}-\mathrm {M} \right)^{2}+\left({\frac {gz}{\rho }}-\mathrm {L} \right)^{2}=g^{2}-{\frac {2g}{\rho }}(\mathrm {N} x+\mathrm {M} y+\mathrm {L} z)+\lambda ^{2}$

${\begin{aligned}=&{\frac {(\mathrm {R} dy+\mathrm {Q} dz)^{2}}{dt^{2}}}+{\frac {(\mathrm {P} dz-\mathrm {R} dx)^{2}}{dt^{2}}}+{\frac {(\mathrm {Q} dx+\mathrm {P} dy)^{2}}{dt^{2}}}\\=&{\frac {\Pi ^{2}\left(dx^{2}+dy^{2}+dz^{2}\right)}{dt^{2}}}-{\frac {(\mathrm {P} dx-\mathrm {Q} dy+\mathrm {R} dz)^{2}}{dt^{2}}}\,;\end{aligned}}$

mais on a (4)

$\mathrm {P} dx-\mathrm {Q} dy+\mathrm {R} dz=0,$

et (7)

$\mathrm {N} x+\mathrm {M} y+\mathrm {L} z=g\rho -\Pi ^{2}\,;$

donc

$dx^{2}+dy^{2}+dz^{2}=\left({\frac {2g}{\rho }}-\Delta \right)dt^{2}.$

Égalant ces deux valeurs de $dx^{2}+dy^{2}+dz^{2},$ on en tire comme ci-dessus l’équation

$dt=-{\frac {\rho d\rho }{\sqrt {-\Delta \rho ^{2}+2g\rho -\Pi ^{2}}}}.$