Page:Mémoires de l’Académie des sciences, Tome 22.djvu/223

ces racines, et en vertu de l’équation (24) [§ 2],

(24)

${\frac {u}{2\pi }}={\frac {\varphi (\theta _{0})}{\operatorname {F} '(\theta _{0})}}e^{\theta _{0}t}+{\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}e^{\theta _{1}t}+\ldots +{\frac {\varphi (\theta _{n-1})}{\operatorname {F} '(\theta _{n-1})}}e^{\theta _{n-1}t},$

la variable $t$ étant considérée comme positive. La fonction $\varphi$ étant arbitraire et du degré $n-1,$ il est clair que les fractions

(25)

${\frac {\varphi (\theta _{0})}{\operatorname {F} '(\theta _{0})}},\qquad {\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}},\qquad {\frac {\varphi (\theta _{n-1})}{\operatorname {F} '(\theta _{n-1})}},$

représentent $n$ constantes arbitraires. Donc l’équation (24) fournit la valeur générale de $u.$ Il reste à substituer aux quantités (25) les constantes arbitraires $u_{o},u_{1},\ldots u_{n-1}.$ Or, les formules (17) donneront

(26)

$\left\{{\begin{alignedat}{2}{\frac {\varphi (\theta _{0})}{\operatorname {F} '(\theta _{0})}}+&&\quad {\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}+\ldots +&\qquad {\frac {\varphi (\theta _{n-1})}{\operatorname {F} '(\theta _{n-1})}}=&{\frac {u_{0}}{2\pi }},\\\theta _{0}{\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}+&&\theta _{1}{\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}+\ldots +&\theta _{n-1}{\frac {\varphi (\theta _{n-1})}{\operatorname {F} '(\theta _{n-1})}}=&{\frac {u_{1}}{2\pi }},\\{\text{etc}}\ldots \qquad &\\\theta _{0}^{n-1}{\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}+&&\theta _{1}^{n-1}{\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}+\ldots +&\theta _{n-1}^{n-1}{\frac {\varphi (\theta _{n-1})}{\operatorname {F} '(\theta _{n-1})}}=&{\frac {u_{n-1}}{2\pi }}.\end{alignedat}}\right.$

On aura d’ailleurs, d’après la formule de Lagrange,

(27)

${\begin{aligned}\varphi (\theta )=&{\frac {(\theta -\theta _{1})(\theta -\theta _{2})\ldots (\theta -\theta _{n-1})}{(\theta _{0}-\theta _{1})(\theta _{0}-\theta _{2})\ldots (\theta _{0}-\theta _{n-1})}}\varphi (\theta _{0})+{\text{etc}}\ldots \\=&{\frac {\operatorname {F} (\theta )-\operatorname {F} (\theta _{0})}{(\theta -\theta _{0})\operatorname {F} '(\theta _{0})}}\varphi (\theta _{0})+{\frac {\operatorname {F} (\theta )-\operatorname {F} (\theta _{1})}{(\theta -\theta _{1})\operatorname {F} '(\theta _{1})}}\varphi (\theta _{1})+{\text{etc}}\ldots \\=&{\frac {\operatorname {F} (\theta )-\operatorname {F} (\theta _{0})}{\theta -\theta _{0}}}{\frac {\varphi (\theta _{0})}{\operatorname {F} '(\theta _{0})}}+{\frac {\operatorname {F} (\theta )-\operatorname {F} (\theta _{1})}{\theta -\theta _{1}}}{\frac {\varphi (\theta _{1})}{\operatorname {F} '(\theta _{1})}}+{\text{etc}}\ldots \end{aligned}}$

En développant le second membre de l’équation (27) suivant les puissances de $\theta ,$ on obtiendra précisément le même développement qui résulterait de la fraction

(28)

${\frac {1}{2\pi }}{\frac {\operatorname {F} (\theta )-\operatorname {F} (\mathrm {U} )}{\theta -\mathrm {U} }},$