Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/261

 Les corrections sont expliquées en page de discussion

sitives ou négatives, entières ou fractionnaires, commensurables ou incommensurables, réelles ou imaginaires, on aura successivement

$\operatorname {d} X=\alpha Ax^{\alpha -1}+\beta Bx^{\beta -1}+\gamma Cx^{\gamma -1}+\ldots ,$

$\operatorname {d} ^{2}X=\alpha (\alpha -1)Ax^{\alpha -2}+\beta (\beta -1)Bx^{\beta -2}+\gamma (\gamma -1)Cx^{\gamma -2}+\ldots ,$

$\operatorname {d} ^{3}X=\alpha (\alpha -1)(\alpha -2)Ax^{\alpha -3}+\beta (\beta -1)(\beta -2)Bx^{\beta -3}$

$+\gamma (\gamma -1)(\gamma -2)Cx^{\gamma -3}+\ldots ,$

. . . . . . . . . . . . . . . . . . . . . . . . .

et, en général,

$\operatorname {d} ^{n}X=\left\{{\begin{aligned}&\alpha (\alpha -1)(\alpha -2)(\alpha -3)\ldots (\alpha -n+1)Ax^{\alpha -n}\\\\+&\beta (\beta -1)(\beta -2)(\beta -3)\ldots (\beta -n+1)Bx^{\beta -n}\\\\+&\gamma (\gamma -1)(\gamma -2)(\gamma -3)\ldots (\gamma -n+1)Cx^{\gamma -n}\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}\,;$

de là on conclura, quelle que soit la constante g,

$X+\operatorname {d} X{\frac {g}{1}}+\operatorname {d} ^{2}X{\frac {g^{2}}{1.2}}+\operatorname {d} ^{3}X{\frac {g^{3}}{1.2.3}}+\ldots \operatorname {d} ^{n}X{\frac {g^{n}}{1.2.3\ldots n}}+\ldots$

$=\left\{{\begin{aligned}&A\left(x^{\alpha }+{\frac {\alpha }{1}}x^{\alpha -1}g+{\frac {\alpha }{1}}{\frac {\alpha -1}{2}}x^{\alpha -2}g^{2}+\ldots \right.\\&\qquad \qquad \qquad \qquad \left.{\frac {\alpha }{1}}{\frac {\alpha -1}{2}}\ldots {\frac {\alpha -n+1}{n}}x^{\alpha -n}g^{n}+\ldots \right)\\\\+&B\left(x^{\beta }+{\frac {\beta }{1}}x^{\beta -1}g+{\frac {\beta }{1}}{\frac {\beta -1}{2}}x^{\beta -2}g^{2}+\ldots \right.\\&\qquad \qquad \qquad \qquad \left.{\frac {\beta }{1}}{\frac {\beta -1}{2}}\ldots {\frac {\beta -n+1}{n}}x^{\beta -n}g^{n}+\ldots \right)\\\\+&C\left(x^{\gamma }+{\frac {\gamma }{1}}x^{\gamma -1}g+{\frac {\gamma }{1}}{\frac {\gamma -1}{2}}x^{\gamma -2}g^{2}+\ldots \right.\\&\qquad \qquad \qquad \qquad \left.{\frac {\gamma }{1}}{\frac {\gamma -1}{2}}\ldots {\frac {\gamma -n+1}{n}}x^{\gamma -n}g^{n}+\ldots \right)\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}\,;$

ou, plus brièvement,