Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/31

Or, en développant ces valeurs suivant les dimensions de ${\frac {zy}{b}},$ elles se trouveront exprimées de cette manière

${\begin{aligned}\mathrm {P} &=\mathrm {A} {\frac {zy}{b}},\\\mathrm {Q} &=\mathrm {B} {\frac {zy}{b}}+\mathrm {B} _{1}\left({\frac {zy}{b}}\right)^{2},\\\mathrm {R} &=\mathrm {C} {\frac {zy}{b}}+\mathrm {C} _{1}\left({\frac {zy}{b}}\right)^{2}+\mathrm {C} _{2}\left({\frac {zy}{b}}\right)^{3},\\\mathrm {S} &=\mathrm {D} {\frac {zy}{b}}+\mathrm {D} _{1}\left({\frac {zy}{b}}\right)^{2}+\mathrm {D} _{2}\left({\frac {zy}{b}}\right)^{3}+\mathrm {D} _{3}\left({\frac {zy}{b}}\right)^{4},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

De sorte que l’on aura

${\begin{aligned}\mathrm {A} \ \ &=c,\\\mathrm {B} \ \ &=-d,\\\mathrm {C} \ \ &=e,\\\mathrm {D} \ \ &=-f,\\\ldots &\ldots \ldots ,\\\mathrm {B} _{1}&=c\mathrm {A} ,\\\mathrm {C} _{1}&=-d\mathrm {A} +c\mathrm {B} ,\\\mathrm {D} _{1}&=e\mathrm {A} -d\mathrm {B} +c\mathrm {C} ,\\\ldots &\ldots \ldots \ldots \ldots \ldots ,\\\mathrm {C} _{2}&=c\mathrm {B} _{1},\\\mathrm {D} _{2}&=-d\mathrm {B} _{1}+c\mathrm {C} _{1},\\\ldots &\ldots \ldots \ldots \ldots \ldots ,\\\mathrm {D} _{3}&=c\mathrm {C} _{2},\\\ldots &\ldots \ldots .\end{aligned}}$

Donc, on aura,

${\begin{aligned}{\frac {\psi '(y)}{z\left[1-{\cfrac {z\varphi (\alpha y)}{\alpha }}\right]}}=&{\frac {\psi '(y)}{z}}+a{\frac {\mathrm {A} }{b^{2}}}y^{2}\psi '(y)\\&+a^{2}\left({\frac {\mathrm {B} y^{3}}{b^{3}}}+{\frac {\mathrm {B} _{1}zy^{4}}{b^{4}}}\right)\psi '(y)\\&+a^{3}\left({\frac {\mathrm {C} y^{4}}{b^{4}}}+{\frac {\mathrm {C} _{1}zy^{5}}{b^{5}}}+{\frac {\mathrm {C} _{2}z^{2}y^{6}}{b^{6}}}\right)\psi '(y)\\&+a^{4}\left({\frac {\mathrm {D} y^{5}}{b^{5}}}+{\frac {\mathrm {D} _{1}zy^{6}}{b^{6}}}+{\frac {\mathrm {D} _{2}z^{2}y^{7}}{b^{7}}}+{\frac {\mathrm {D} _{3}z^{3}y^{8}}{b^{8}}}\right)\psi '(y)\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$