sitives ou négatives, entières ou fractionnaires, commensurables ou incommensurables, réelles ou imaginaires, on aura successivement
![{\displaystyle \operatorname {d} X=\alpha Ax^{\alpha -1}+\beta Bx^{\beta -1}+\gamma Cx^{\gamma -1}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48cf1b61d78bb410f5f02c9c20d07935972fbdc4)
![{\displaystyle \operatorname {d} ^{2}X=\alpha (\alpha -1)Ax^{\alpha -2}+\beta (\beta -1)Bx^{\beta -2}+\gamma (\gamma -1)Cx^{\gamma -2}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15c057f95bb6d86d9d81973dc7cd62bfe65546d)
![{\displaystyle \operatorname {d} ^{3}X=\alpha (\alpha -1)(\alpha -2)Ax^{\alpha -3}+\beta (\beta -1)(\beta -2)Bx^{\beta -3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/618a99073da76fc60f60e3ec4657d7630a274965)
![{\displaystyle +\gamma (\gamma -1)(\gamma -2)Cx^{\gamma -3}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b66cb400ebc51806a4dfc4030438d6ab9571bf8)
. . . . . . . . . . . . . . . . . . . . . . . . .
et, en général,
![{\displaystyle \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\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b916f8d163c0d98dc8abb5c7242c54922b2223c4)
de là on conclura, quelle que soit la constante g,
![{\displaystyle 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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2471f3cf4ff181059bc419c37e65c6cfb52d0e)
![{\displaystyle =\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\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/159d6e810bda4c55eb46345d3e2f1e552d3af3a0)
ou, plus brièvement,