Page:Annales de mathématiques pures et appliquées, 1810-1811, Tome 1.djvu/265

257

RÉSOLUES.

$(\mathrm {C'} )\left\{{\begin{array}{ll}{\tfrac {\mathrm {dM} }{\mathrm {d} x}}{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u^{2}}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x^{2}}}\left({\tfrac {\mathrm {d} x}{\mathrm {d} u}}\right)^{2}+2{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y\mathrm {d} z}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} z}{\mathrm {d} u}}\\+{\tfrac {\mathrm {dM} }{\mathrm {d} y}}{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u^{2}}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y^{2}}}\left({\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right)^{2}+2{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} z}}{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} z}{\mathrm {d} u}}\\+{\tfrac {\mathrm {dM} }{\mathrm {d} z}}{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u^{2}}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} z^{2}}}\left({\tfrac {\mathrm {d} z}{\mathrm {d} u}}\right)^{2}+2{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} y}}{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}\\\end{array}}\right\}=0,$

$(\mathrm {D'} )\left\{{\begin{array}{ll}{\tfrac {\mathrm {dM} }{\mathrm {d} x}}{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u\mathrm {d} v}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x^{2}}}{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y\mathrm {d} z}}\left\{{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} z}{\mathrm {d} v}}+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}{\tfrac {\mathrm {d} z}{\mathrm {d} u}}\right\}\\+{\tfrac {\mathrm {dM} }{\mathrm {d} y}}{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u\mathrm {d} v}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y^{2}}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} z}}\left\{{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} z}{\mathrm {d} v}}+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} z}{\mathrm {d} u}}\right\}\\+{\tfrac {\mathrm {dM} }{\mathrm {d} z}}{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u\mathrm {d} v}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} z^{2}}}{\tfrac {\mathrm {d} z}{\mathrm {d} u}}{\tfrac {\mathrm {d} z}{\mathrm {d} v}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} y}}\left\{{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right\}\\\end{array}}\right\}=0,$

$(\mathrm {E'} )\left\{{\begin{array}{ll}{\tfrac {\mathrm {dM} }{\mathrm {d} x}}{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} v^{2}}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x^{2}}}\left({\tfrac {\mathrm {d} x}{\mathrm {d} v}}\right)^{2}+2{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y\mathrm {d} z}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}{\tfrac {\mathrm {d} z}{\mathrm {d} v}}\\+{\tfrac {\mathrm {dM} }{\mathrm {d} y}}{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} v^{2}}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y^{2}}}\left({\tfrac {\mathrm {d} y}{\mathrm {d} v}}\right)^{2}+2{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} z}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} z}{\mathrm {d} v}}\\+{\tfrac {\mathrm {dM} }{\mathrm {d} z}}{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} v^{2}}}+{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} z^{2}}}\left({\tfrac {\mathrm {d} z}{\mathrm {d} v}}\right)^{2}+2{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} y}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}\\\end{array}}\right\}=0.$

Alors, si entre les dix équations

$\mathrm {(A),(B),(C),(D),(E),(A'),(B'),(C'),(D'),(E')}$, on élimine $p$ et $q,$ et en outre cinq des neuf fonctions

${\tfrac {\mathrm {dM} }{\mathrm {d} x}},\;{\tfrac {\mathrm {dM} }{\mathrm {d} y}},\;{\tfrac {\mathrm {dM} }{\mathrm {d} z}},\;{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x^{2}}},\;{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y^{2}}},\;{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} z^{2}}},\;{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} y\mathrm {d} z}},\;{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} z}},\;{\tfrac {\mathrm {d^{2}M} }{\mathrm {d} x\mathrm {d} y}},$

les quatre autres disparaîtront d’elles-mêmes, et les valeurs de $r,s,t,$ tirées des trois équations finales seront les mêmes que ci-dessus.