Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/56

52

INTÉGRALES

Ces développemens, que l’on peut d’ailleurs vérifier en effectuant les différentiations indiquées, étant substitués dans l’équation (XXV), elle deviendra

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

$+{\frac {\operatorname {d} .}{\operatorname {d} x}}\left\{{\begin{aligned}\left\{{\begin{array}{r}L-{\frac {\operatorname {d} N}{\operatorname {d} x}}+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}-\ldots \\\\-{\frac {\operatorname {d} O}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x\operatorname {d} y}}-\ldots \\\\+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}-\ldots \\\\-\ldots \end{array}}\right\}Z+\left\{{\begin{array}{r}N-{\frac {\operatorname {d} Q}{\operatorname {d} x}}+\ldots \\\\-{\frac {\operatorname {d} R}{\operatorname {d} y}}+\ldots \\\\+\ldots \end{array}}\right\}{\frac {\operatorname {d} Z}{\operatorname {d} x}}+\left\{{\begin{array}{r}Q-\ldots \\\\-\ldots \end{array}}\right\}{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}+\ldots \end{aligned}}\right\}$

$+{\frac {\operatorname {d} .}{\operatorname {d} y}}\left\{{\begin{aligned}\left\{{\begin{array}{r}M-{\frac {\operatorname {d} O}{\operatorname {d} x}}+{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}-\ldots \\\\-{\frac {\operatorname {d} P}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}-\ldots \\\\+{\frac {\operatorname {d} ^{2}T}{\operatorname {d} y^{2}}}-\ldots \\\\-\ldots \end{array}}\right\}Z+\left\{{\begin{array}{r}P-{\frac {\operatorname {d} S}{\operatorname {d} x}}+\ldots \\\\-{\frac {\operatorname {d} T}{\operatorname {d} y}}+\ldots \\\\+\ldots \end{array}}\right\}{\frac {\operatorname {d} Z}{\operatorname {d} y}}+\left\{{\begin{array}{r}T-\ldots \\\\-\ldots \end{array}}\right\}{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}+\ldots \end{aligned}}\right\}$