Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/58

54

INTÉGRALES

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

Cette partie étant ainsi supprimée, dans l’équation (XXVI), elle deviendra, en passant aux fonctions primitives,

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

$+{\frac {\operatorname {d} .}{\operatorname {d} x}}\left\{{\begin{aligned}\left\{{\begin{array}{r}O-{\frac {\operatorname {d} R}{\operatorname {d} x}}+\ldots \\\\-{\frac {\operatorname {d} S}{\operatorname {d} y}}+\ldots \\\\+\ldots \\\\\end{array}}\right\}Z+\left\{{\begin{array}{r}+R-\ldots \\\\-\ldots \\\\\end{array}}\right\}{\frac {\operatorname {d} Z}{\operatorname {d} x}}+\left\{{\begin{array}{r}S-\ldots \\\\-\ldots \end{array}}\right\}{\frac {\operatorname {d} Z}{\operatorname {d} y}}+\ldots \end{aligned}}\right\}.(\mathrm {XXVIII} )$

47. Or, présentement, les mêmes considérations qui nous ont conduits à partager l’équation (XXrVYI) dans les équations (XXVII) et (XVIII) conduisent également à décomposer chacune, de ces