Page:Mémoires de l’Académie des sciences, Tome 8.djvu/356

(5)

$u=\mathrm {U} +{\frac {\mathrm {U} '}{1}}\,{\frac {t}{\sqrt {n}}}+{\frac {\mathrm {U} ''}{1.2}}\,{\frac {t^{2}}{n}}+{\text{etc}}.,$

(6)

${\begin{aligned}\operatorname {l} (u)&=\operatorname {l} (\mathrm {U} )+\operatorname {l} \left(1+\mathrm {\frac {U'}{U}} \,{\frac {t}{\sqrt {n}}}+\mathrm {\frac {U''}{2U}} \,{\frac {t^{2}}{n}}+\ldots \right)\\&=\operatorname {l} (\mathrm {U} )+\mathrm {\frac {U'}{U}} \,{\frac {t}{\sqrt {n}}}-\mathrm {\frac {U'^{2}-UU''}{2U^{2}}} \,{\frac {t^{2}}{n}}+{\text{etc}}.\end{aligned}}$

On aura donc à très-peu près, lorsque $n$ sera très-grand,

(7)

$u=\mathrm {U} e^{\mathrm {\frac {U'}{U}} \,{\frac {t}{\sqrt {n}}}-\mathrm {\frac {U'^{2}-UU''}{2U^{2}}} \,{\frac {t^{2}}{n}}},$

et par suite

(8)

$u^{n}=\mathrm {U} ^{n}e^{\mathrm {\frac {U'}{U}} \,t{\sqrt {n}}-\mathrm {\frac {U'^{2}-UU''}{2U^{2}}} \,t^{2}},$

ou, ce qui revient au même,

(9)

$u^{n}=\mathrm {U} ^{n}e^{{\frac {n}{2}}\mathrm {\frac {U'^{2}}{U'^{2}-UU''}} }e^{-\mathrm {\frac {U'^{2}-UU''}{2U^{2}}} \left(t-\mathrm {\frac {UU'}{U'^{2}-UU''}} {\sqrt {n}}\right)^{2}}.$

On en conclura, à très-peu près,

(10)

$\int _{\mathrm {X} -{\frac {a}{\sqrt {n}}}}^{\mathrm {X} +{\frac {a}{\sqrt {n}}}}u^{n}vdx={\frac {1}{\sqrt {n}}}\int _{-a}^{+a}\mathrm {U} ^{n}e^{{\frac {n}{2}}\mathrm {\frac {U'^{2}}{U'^{2}-UU''}} }e^{-\mathrm {\frac {U'^{2}-UU''}{2U^{2}}} (t-c)^{2}}\mathrm {V} dt,$

$c$ désignant pour abréger la constante $\mathrm {\frac {UU'}{U'^{2}-UU''}} {\sqrt {n}}.$

Si, pour plus de commodité, on posait

(11)

$u=e^{w},\quad {\text{et}}\quad \mathrm {U} =e^{\mathrm {W} }\quad {\text{ou}}\quad \mathrm {W} =\operatorname {l} (\mathrm {U} ),$

on trouverait sensiblement

(12)

$\mathrm {{\frac {U'}{U}}=W',\quad {\frac {-U'^{2}+UU''}{U^{2}}}=W''} ,$