Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/53

49

INDÉTERMINÉES.

${\begin{array}{rrr}&&p+i{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}},&s+i{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x\operatorname {d} y^{2}}},\ldots ,\\&&&t+i{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} y^{3}}},\ldots ,\\&&&\ldots \,;\end{array}}$

en conséquence, on trouvera, par l’application de la série de Taylor au développement des fontions des polynômes, que $V$ doit devenir (40)

$V+\left\{{\begin{array}{r}KZ+L{\frac {\operatorname {d} Z}{\operatorname {d} x}}+N\ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}+Q\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{3}}}+\ldots \\\\+M{\frac {\operatorname {d} Z}{\operatorname {d} y}}+O{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x\operatorname {d} y}}+R{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{2}\operatorname {d} y}}+\ldots \\\\+P\ \ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}+S{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x\operatorname {d} y^{2}}}+\ldots \\\\+T\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} y^{3}}}+\ldots \\\\+\ldots \end{array}}\right\}{\frac {i}{1}}+\ldots$

d’où il suit que $\iint V\operatorname {d} x\operatorname {d} y,$ deviendra

$\iint V\operatorname {d} x\operatorname {d} y+{\frac {i}{1}}\iint \left\{{\begin{array}{r}KZ+L{\frac {\operatorname {d} Z}{\operatorname {d} x}}+N\ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}+Q\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{3}}}+\ldots \\\\+M{\frac {\operatorname {d} Z}{\operatorname {d} y}}+O{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x\operatorname {d} y}}+R{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{2}\operatorname {d} y}}+\ldots \\\\+P\ \ \ {\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}+S{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x\operatorname {d} y^{2}}}+\ldots \\\\+T\quad {\frac {\operatorname {d} ^{3}Z}{\operatorname {d} y^{3}}}+\ldots \\\\+\ldots \end{array}}\right\}\operatorname {d} x\operatorname {d} y+\ldots$