Page:Annales de mathématiques pures et appliquées, 1827-1828, Tome 18.djvu/168

$m-2,\ldots m-n,$ puis changeant respectivement $x$ et $y$ en $x-x'$ et $y-y'$, il viendra

$mu'_{n}+(m-1)\left\{{\frac {\operatorname {d} u'_{n}}{\operatorname {d} x}}(x-x')+{\frac {\operatorname {d} u'_{n}}{\operatorname {d} y}}(y-y')\right\}$

$+{\frac {m-2}{1.2}}\left\{{\frac {\operatorname {d} ^{2}u'_{n}}{\operatorname {d} x'^{2}}}(x-x')^{2}+2{\frac {\operatorname {d} ^{2}u'_{n}}{\operatorname {d} x'\operatorname {d} y'}}(x-x')(y-y')+{\frac {d^{2}u'_{n}}{\operatorname {d} y'^{2}}}(y-y')^{2}\right\}$

$+{\frac {m-3}{1.2.3}}\left\{{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'^{3}}}(x-x')^{3}+3{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'^{2}\operatorname {d} y'}}(x-x')^{2}(y-y')\right.$

$\left.+3{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'\operatorname {d} y'^{2}}}(x-x')(y-y')^{2}+{\frac {d^{3}u'_{n}}{\operatorname {d} y'^{3}}}(y-y')^{3}\right\}$

$+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

$+{\frac {m-n}{1.2\ldots n}}\left\{{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n}}}(x-x')^{n}+{\frac {n}{1}}{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n-1}\operatorname {d} y'}}(x-x')^{n-1}(y-y')+\right.$

$\left.\ldots +{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} y'^{n}}}(y-y')^{n}\right\}$

Or, le terme $t_{n}$ représentant la quantité indépendante de $x$ et de $y$ dans cette expression, il faudra, pour l’obtenir, y supposer $x=0$ et $y=0,$ ce qui donnera

$t_{n}=mu'_{n}-{\frac {(m-1)}{1}}\left({\frac {\operatorname {d} u'_{n}}{\operatorname {d} x'}}x'+{\frac {\operatorname {d} u'_{n}}{\operatorname {d} y'}}y'\right)$

$+{\frac {m-2}{1.2}}\left({\frac {\operatorname {d} ^{2}u'_{n}}{\operatorname {d} x'^{2}}}x'^{2}+2{\frac {\operatorname {d} ^{2}u'_{n}}{\operatorname {d} x'\operatorname {d} y'}}x'y'+{\frac {d^{2}u'_{n}}{\operatorname {d} y'^{2}}}y'^{2}\right)$

$-{\frac {m-3}{1.2.3}}\left({\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'^{3}}}x'^{3}+3{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'^{2}\operatorname {d} y'}}x'^{2}y'+3{\frac {\operatorname {d} ^{3}u'_{n}}{\operatorname {d} x'\operatorname {d} y'^{2}}}x'y'^{2}+{\frac {d^{3}u'_{n}}{\operatorname {d} y'^{3}}}y'^{3}\right)$

$+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

$\pm {\frac {m-n}{1.2\ldots n}}\left({\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n}}}x'^{n}+{\frac {n}{1}}{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n-1}\operatorname {d} y'}}x'^{n-1}y'\right.$

$\left.+{\frac {n}{1}}.{\frac {n-1}{2}}{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} x'^{n-2}\operatorname {d} y'^{2}}}x'^{n-2}y'^{2}+\ldots +{\frac {\operatorname {d} ^{n}u'_{n}}{\operatorname {d} y'^{n}}}y'^{n}\right)$