Page:Annales de mathématiques pures et appliquées, 1812-1813, Tome 3.djvu/271

263

DES ÉCHELLES.

$(78)\qquad \Delta _{\xi ,\nu }^{m,n}=\Delta _{\xi ,}^{m,}\Delta _{,\nu }^{,n}=\left(\mathrm {E} ^{\xi ,}-1\right)^{m}\left(\mathrm {E} ^{,\nu }-1\right)^{n}$

$=\left\{\left(1+\Delta ^{1,}\right)^{\xi }-1\right\}^{m}\left\{\left(1+\Delta ^{,1}\right)^{\nu }-1\right\}^{n}=\left(e^{\xi \partial ^{1,}}-1\right)^{m}\left(e^{\nu \partial ^{,1}}-1\right)^{n}$

$(79)\qquad \Delta ^{m,n}=\Delta ^{m,}\Delta ^{,n}=\left(\mathrm {E} ^{1,}-1\right)^{m}\left(\mathrm {E} ^{,1}-1\right)^{n}$

$=\left\{\left(1+\Delta _{\xi ,}^{1,}\right)^{\frac {1}{\xi }}-1\right\}^{m}\left\{\left(1+\Delta _{,\nu }^{,1}\right)^{\frac {1}{\nu }}-1\right\}^{n}=\left(e^{\partial ^{1,}}-1\right)^{m}\left(e^{\partial ^{,1}}-1\right)^{n}$

$(80)\quad \Delta _{\xi ,\nu }^{m}=\left(\mathrm {E} ^{\xi ,\nu }-1\right)^{m}$

$=\left\{\left(1+\Delta _{\xi ,}^{1,}\right)\left(1+\Delta _{,\nu }^{,1}\right)-1\right\}^{m}\left\{\left(1+\Delta ^{,1}\right)^{\xi }\left(1+\Delta ^{,1}\right)^{\nu }-1\right\}^{m}$

$=\left(\Delta _{\xi ,}^{1,}+\Delta _{,\nu }^{,1}.\mathrm {E} ^{\xi ,}\right)^{m}=\left(\Delta _{,\nu }^{,1}+\Delta _{\xi ,}^{1,}.\mathrm {E} ^{,\nu }\right)^{m}=\left(e^{\xi \partial ^{1,}+\nu \partial ^{,1}}-1\right)^{m}.$

$(81)\qquad \Delta _{\xi ,\nu }^{-m}=\Sigma _{\xi ,\nu }^{m}=\left(\mathrm {E} ^{\xi ,\nu }-1\right)^{-m}$

$=\left(\Delta _{\xi ,}^{1,}+\Delta _{,\nu }^{,1}.\mathrm {E} ^{\xi ,}\right)^{-m}=\left(\Delta _{,\nu }^{,1}+\Delta _{\xi ,}^{1,}.\mathrm {E} ^{,\nu }\right)^{-m}$

$=\Sigma _{\xi ,}^{m,}\left(1+\Delta _{,\nu }^{,1}.\Sigma _{\xi ,}^{1,}.\mathrm {E} ^{\xi ,}\right)^{-m}=\Sigma _{,\nu }^{,m}\left(1+\Delta _{\xi ,}^{1,}.\Sigma _{,\nu }^{,1}.\mathrm {E} ^{,\nu }\right)^{-m}=\left(e^{\xi \partial ^{1,}+\nu \partial ^{,1}}-1\right)^{-m}.$

On déduit aussi des équations (72) et (73)

$(82)\left\{{\begin{aligned}\partial ^{1,}&=\operatorname {Log} .\mathrm {E} ^{1,}=\operatorname {Log} .\left(1+\Delta ^{1,}\right)\\&={\frac {1}{\xi }}\operatorname {Log} .\left(1+\Delta _{\xi ,}^{1,}\right)=\operatorname {Log} .\left(1+\Delta _{\xi ,}^{1,}\right)^{\frac {1}{\xi }},\\\partial ^{,1}&=\operatorname {Log} .\mathrm {E} ^{,1}=\operatorname {Log} .\left(1+\Delta ^{,1}\right)\\&={\frac {1}{\nu }}\operatorname {Log} .\left(1+\Delta _{,\nu }^{,1}\right)=\operatorname {Log} .\left(1+\Delta _{,\nu }^{,1}\right)^{\frac {1}{\nu }},\\\end{aligned}}\right.$

$(83)\quad \partial =\partial ^{1,}+\partial ^{,1}=\operatorname {Log} .\mathrm {E} ^{1,}+\operatorname {Log} .\mathrm {E} ^{,1}$

$=\operatorname {Log} .\left(\mathrm {E} ^{1,}.\mathrm {E} ^{,1}\right)=\operatorname {Log} .\mathrm {E} =\operatorname {Log} .\left(1+\Delta ^{1,}\right)+\operatorname {Log} .\left(1+\Delta ^{,1}\right)$

$=\operatorname {Log} .\left\{\left(1+\Delta ^{1,}\right)\left(1+\Delta ^{,1}\right)\right\}={\frac {1}{\xi }}\operatorname {Log} .\left(1+\Delta _{\xi ,}^{1,}\right)+{\frac {1}{\nu }}\operatorname {Log} .\left(1+\Delta _{,\nu }^{,1}\right)$

$=\left\{\operatorname {Log} .\left(1+\Delta _{\xi ,}^{1,}\right)^{\frac {1}{\xi }}\left(1+\Delta _{,\nu }^{,1}\right)^{\frac {1}{\nu }}\right\}.$

$(84)\quad \partial ^{m}=\left(\partial ^{1,}+\partial ^{,1}\right)^{m}=\partial ^{m,}\left(1+{\mathcal {f}}^{1,}.\partial ^{,1}\right)^{m}=\partial ^{,m}\left(1+{\mathcal {f}}^{,1}.\partial ^{1,}\right)^{m},$

$(85)\quad \partial ^{-m}{\mathcal {f}}^{m}=\left(\partial ^{1,}+\partial ^{,1}\right)^{-m}={\mathcal {f}}^{m,}\left(1+{\mathcal {f}}^{1,}.\partial ^{,1}\right)^{-m}={\mathcal {f}}^{,m}\left(1+{\mathcal {f}}^{,1}.\partial ^{1,}\right)^{-m},$

où ${\mathcal {f}}$ indique l’opération inverse de $\partial \,$ ; de sorte que ${\mathcal {f}}=\partial ^{-1}.$ Il a, avec le signe d’intégration ordinaire, le rapport suivant : ${\mathcal {f}}^{1,}=\int \mathrm {d} x,\ {\mathcal {f}}^{,1}=\int \mathrm {d} y.$

22. Toutes ces équations ne sont que des relations de définition, entre les différentes échelles de différenciation, ou des résultats qui en dérivent immédiatement. En les multipliant par $\phi (x,y),$ elles offrent autant de