Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/35

les coefficients $\mathrm {A,B} ,\ldots$ étant déterminés de la manière suivante

${\begin{alignedat}{5}\mathrm {A} &=\beta \\\mathrm {B} &=\gamma ,&\mathrm {B} _{1}&=\beta \mathrm {A} ,\\\mathrm {C} &=\delta ,&\mathrm {C} _{1}&=\mathrm {\gamma A+\beta B} ,&\mathrm {C} _{2}&=\beta \mathrm {B} _{1},&\\\mathrm {D} &=\varepsilon ,&\mathrm {D} _{1}&=\delta \mathrm {A+\gamma B+\beta C} ,&\mathrm {D} _{2}&=\gamma \mathrm {B_{1}+\beta C_{1}} ,\\\mathrm {E} &=\zeta ,\quad &\mathrm {E} _{1}&=\varepsilon \mathrm {A+\delta B+\gamma C+\beta D} ,\quad &\mathrm {E} _{2}&=\delta \mathrm {B_{1}+\gamma C_{1}+\beta D_{1}} ,\\\ldots &\ldots &\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots ,&\ldots &\ldots \ldots \ldots \ldots \ldots \ldots ,\end{alignedat}}$

${\begin{alignedat}{2}\mathrm {D} _{3}&=\beta \mathrm {C} _{2},\\\mathrm {E} _{3}&=\gamma \mathrm {C_{2}+\beta D_{2}} ,\quad &\mathrm {E} _{4}&=\beta \mathrm {D} _{3},\\\ldots &\ldots \ldots \ldots \ldots ,&\ldots &\ldots \ldots .\end{alignedat}}$

Ainsi, remettant à la place de $u$ sa valeur $\alpha ^{p-q-1}y^{p-q}z$ on aura

${\frac {\psi '(y)}{z\left[1-z{\cfrac {\varphi (\alpha y)}{\alpha }}\right]}}$

${\begin{aligned}={\frac {\psi '(y)}{z}}&+\mathrm {A} \alpha ^{p-1}y^{p}\psi '(y)\\&+\mathrm {B} \alpha ^{p+q-1}y^{p+q}\psi '(y)+\mathrm {B} _{1}\alpha ^{2p-2}zy^{2p}\psi '(y)\\&+\mathrm {C} \alpha ^{p+2q-1}y^{p+2q}\psi '(y)+\mathrm {C} _{1}\alpha ^{2p+q-2}zy^{2p+q}\psi '(y)+\mathrm {C} _{2}\alpha ^{3p+q-3}z^{2}y^{3p}\psi '(y)\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Donc, pratiquant les transformations enseignées dans le n^o 18, on aura la valeur de $\psi \left({\frac {x}{\alpha }}\right)$ exprimée par la série suivante, dans laquelle il faudra se souvenir de faire $y=1$ après toutes les différentiations,

${\begin{aligned}\psi \left({\frac {x}{\alpha }}\right)=&\psi (y)\\&+\mathrm {A} \alpha ^{p-1}y^{p}\psi '(y)\\&+\mathrm {B} \alpha ^{p+q-1}y^{p+q}\psi '(y)+\mathrm {B} _{1}\alpha ^{2p-2}.{\frac {1}{2}}{\frac {d\left[y^{2p}\psi '(y)\right]}{dy}}\\&+\mathrm {C} \alpha ^{p+2q-1}y^{p+2q}\psi '(y)+\mathrm {C} _{1}\alpha ^{2p+q-2}.{\frac {1}{2}}{\frac {d\left[y^{2p+q}\psi '(y)\right]}{dy}}\\&\quad +\mathrm {C} _{2}\alpha ^{3p+q-3}.{\frac {1}{2.3}}{\frac {d^{2}\left[y^{3p}\psi '(y)\right]}{dy^{2}}}\\&+\mathrm {D} \alpha ^{p+3q-1}y^{p+3q}\psi '(y)+\mathrm {D} _{1}\alpha ^{2p+2q-2}.{\frac {1}{2}}{\frac {d\left[y^{2p+2q}\psi '(y)\right]}{dy}}\\&\quad +\mathrm {D} _{2}\alpha ^{3p+q-3}.{\frac {1}{2.3}}{\frac {d^{2}\left[y^{3p+q}\psi '(y)\right]}{dy^{2}}}+\mathrm {D} _{3}\alpha ^{4p-4}.{\frac {1}{2.3.4}}{\frac {d^{3}\left[y^{4p}\psi '(y)\right]}{dy^{3}}}\end{aligned}}$