Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/59

55

INDÉTERMINÉES.

dernières en deux autres ; de sorte que finalement l’équation (XXVI) donne naissance aux quatre suivantes :

$\left.{\begin{array}{r}0=K-{\frac {\operatorname {d} L}{\operatorname {d} x}}+\ \ {\frac {\operatorname {d} ^{2}N}{\operatorname {d} x^{2}}}-\quad {\frac {\operatorname {d} ^{3}Q}{\operatorname {d} x^{3}}}+\ldots \\\\-{\frac {\operatorname {d} M}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}O}{\operatorname {d} x\operatorname {d} y}}-{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{2}\operatorname {d} y}}+\ldots \\\\+\ \ \ {\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}-{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y^{2}}}+\ldots \\\\-\quad {\frac {\operatorname {d} ^{3}T}{\operatorname {d} y^{3}}}+\ldots \\\\+\ldots \end{array}}\right\}\quad$(XXIX)

$X=\left\{{\begin{array}{r}M-{\frac {\operatorname {d} O}{\operatorname {d} x}}+{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}-\ldots \\\\-{\frac {\operatorname {d} P}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}-\ldots \\\\+{\frac {\operatorname {d} ^{2}T}{\operatorname {d} y^{2}}}-\ldots \\\\-\ldots \end{array}}\right\}Z+\left\{{\begin{array}{r}P-{\frac {\operatorname {d} S}{\operatorname {d} x}}+\ldots \\\\-{\frac {\operatorname {d} T}{\operatorname {d} y}}+\ldots \\\\+\ldots \end{array}}\right\}{\frac {\operatorname {d} Z}{\operatorname {d} y}}+\left\{{\begin{array}{r}T-\ldots \\\\-\ldots \end{array}}\right\}{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}+\ldots ,\mathrm {(XXX)}$

$Y=\left\{{\begin{array}{r}L-{\frac {\operatorname {d} N}{\operatorname {d} x}}+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}-\ldots \\\\-{\frac {\operatorname {d} O}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x\operatorname {d} y}}-\ldots \\\\+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}-\ldots \\\\-\ldots \end{array}}\right\}Z+\left\{{\begin{array}{r}N-{\frac {\operatorname {d} Q}{\operatorname {d} x}}+\ldots \\\\-{\frac {\operatorname {d} R}{\operatorname {d} y}}+\ldots \\\\+\ldots \end{array}}\right\}{\frac {\operatorname {d} Z}{\operatorname {d} x}}+\left\{{\begin{array}{r}Q-\ldots \\\\-\ldots \end{array}}\right\}{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}+\ldots ,\mathrm {(XXXI)}$