les coefficients
étant déterminés de la manière suivante
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e86aa5174a9c4c88b1aec1c4f313604c5984206)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c76f1b2cb2dad8561c6702b59bac9dcc09ad8a)
Ainsi, remettant à la place de
sa valeur
on aura
![{\displaystyle {\frac {\psi '(y)}{z\left[1-z{\cfrac {\varphi (\alpha y)}{\alpha }}\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d52117a45ffc8aef9bf2ab3b87190590171b4b1)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57e98e653a4450d64d53bff2684a315dccc8606e)
Donc, pratiquant les transformations enseignées dans le no 18, on aura la valeur de
exprimée par la série suivante, dans laquelle il faudra se souvenir de faire
après toutes les différentiations,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ece0f90899dea89d6664c9f2fe18e3685a050b89)