Page:Annales de mathématiques pures et appliquées, 1810-1811, Tome 1.djvu/260

252

QUESTIONS

mais, parce que $x$ et $y$ sont, l’un et l’autre, des fonctions de $u$ et $v,$ on doit avoir aussi

$\mathrm {d} x={\tfrac {\mathrm {d} x}{\mathrm {d} u}}\mathrm {d} u+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}\mathrm {d} v,\quad \mathrm {d} y={\tfrac {\mathrm {d} y}{\mathrm {d} u}}\mathrm {d} u+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}\mathrm {d} v$ ;

substituant donc dans l’équation précédente, elle deviendra

$\mathrm {(\mathrm {I} )} \quad \left\{{\tfrac {\mathrm {d} x}{\mathrm {d} u}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}q-{\tfrac {\mathrm {d} z}{\mathrm {d} u}}\right\}\mathrm {d} u+\left\{{\tfrac {\mathrm {d} x}{\mathrm {d} v}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}q-{\tfrac {\mathrm {d} z}{\mathrm {d} v}}\right\}\mathrm {d} v=0$.

La différentielle complète de cette équation, par rapport à $u$ et $v,$ sera

$(\mathrm {II} )\left\{{\begin{array}{ll}\left\{{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} u}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right)^{2}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u^{2}}}\right\}\mathrm {d} u^{2}\\+2\left\{{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u\mathrm {d} v}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u\mathrm {d} v}}q+{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}r+\left[{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right]s+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u\mathrm {d} v}}\right\}\mathrm {d} u\mathrm {d} v\\+\left\{{\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} v^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} v^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} v}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} v}}\right)^{2}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} v^{2}}}\right\}\mathrm {d} v^{2}=0;\\\end{array}}\right.$

or, à cause de l’indépendance des différentielles $\mathrm {d} u{\text{ et }}\mathrm {d} v$, les équations $\mathrm {(I)}$ et $\mathrm {(II)}$ se partagent dans les cinq suivantes :

$(1)\quad {\tfrac {\mathrm {d} x}{\mathrm {d} u}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}q={\tfrac {\mathrm {d} z}{\mathrm {d} u}},\quad (2)\quad {\tfrac {\mathrm {d} x}{\mathrm {d} v}}p+{\tfrac {\mathrm {d} y}{\mathrm {d} v}}q={\tfrac {\mathrm {d} z}{\mathrm {d} v}}$,

$(3)\quad {\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} u}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right)^{2}t={\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u^{2}}}$,

$(4)\quad {\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} u\mathrm {d} v}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} u\mathrm {d} v}}q+{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} x}{\mathrm {d} v}}r+\left[{\tfrac {\mathrm {d} x}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}+{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} u}}\right]s+{\tfrac {\mathrm {d} y}{\mathrm {d} u}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}t={\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} u\mathrm {d} v}}{\text{,}}$

$(5)\quad {\tfrac {\mathrm {d} ^{2}x}{\mathrm {d} v^{2}}}p+{\tfrac {\mathrm {d} ^{2}y}{\mathrm {d} v^{2}}}q+\left({\tfrac {\mathrm {d} x}{\mathrm {d} v}}\right)^{2}r+2{\tfrac {\mathrm {d} x}{\mathrm {d} v}}{\tfrac {\mathrm {d} y}{\mathrm {d} v}}s+\left({\tfrac {\mathrm {d} y}{\mathrm {d} v}}\right)^{2}t-{\tfrac {\mathrm {d} ^{2}z}{\mathrm {d} v^{2}}}$^[1].

1. Ces équations, en y changeant x et y en u et v, et vice versa, rentrent dans celles qu’a données M. Lacroix, pour une transformation analogue à celle-ci ; mais