Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/184

De même je change les expressions

$\int {\frac {d\mathrm {x} _{1}}{dt}}e^{{\text{V}}_{1}t}dt,\quad \int {\frac {dp}{dt}}e^{{\text{V}}_{1}t}dt,\ldots ,$

en celles-ci

$\mathrm {x} _{1}e^{{\text{V}}_{1}t}-\mathrm {V} _{1}\int \mathrm {x} _{1}e^{{\text{V}}_{1}t}dt,\quad pe^{{\text{V}}_{1}t}-\mathrm {V} _{1}\int pe^{{\text{V}}_{1}t}dt,\ldots .$

Par ce moyen, l’équation se trouve composée de deux parties, l’une finie et l’autre indéfinie, laquelle renferme les quantités intégrales

$\int \mathrm {x} _{1}e^{{\text{V}}_{1}t}dt,\ \ \int pe^{{\text{V}}_{1}t}dt,\ \ \int \mathrm {P} e^{{\text{V}}_{1}t}dt,\ \ \int qe^{{\text{V}}_{1}t}dt,\ \ \int \mathrm {Q} e^{{\text{V}}_{1}t}dt,$

$\int re^{{\text{V}}_{1}t}dt,\ \ \int \mathrm {R} e^{{\text{V}}_{1}t}dt.$

Je fais les coefficients de ces quantités chacun égal à zéro, ce qui me donne sept équations entre les sept inconnues $\alpha _{1},\beta _{1},\gamma _{1},\mathrm {A_{1},B_{1},C_{1}}$ et $\mathrm {V} _{1},$ savoir

(1^o)

${\begin{aligned}\mathrm {V} _{1}^{2}&-{\frac {n}{2}}\left[\alpha _{1}f_{2}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{2},a_{1})+\beta _{1}f_{3}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{1})+\gamma _{1}f_{4}\chi _{1}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{1})\right]\mathrm {V} _{1}\\&\quad +\mathrm {M} _{1}^{2}-{\frac {n}{2}}\mathrm {A} _{1}f_{2}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{2},a_{1})-{\frac {n}{2}}\mathrm {B} _{1}f_{3}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{3},a_{1})\\&\ \ \quad \qquad \qquad \qquad \qquad \qquad \qquad -{\frac {n}{2}}\mathrm {C} _{1}f_{4}\chi _{1}{\overset {\circ }{\Psi }}_{1}(a_{4},a_{1})=0.\end{aligned}}$

(2^o)

${\begin{aligned}\alpha _{1}\mathrm {V} _{1}^{2}&+\left[nf_{1}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{1},a_{2})-2\mathrm {A} _{1}(\mu _{2}-\mu _{1})+{\frac {n}{2}}\mathrm {B} _{1}f_{3}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{3},a_{2})\right.\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.+{\frac {n}{2}}\mathrm {C} _{1}f_{4}\chi _{2}{\overset {\circ }{\Pi }}_{1}(a_{4},a_{2})\right]\mathrm {V} _{1}\\&+\alpha _{1}\left[\mathrm {M} _{2}^{2}-(\mu _{2}-\mu _{1})^{2}\right]-{\frac {n}{2}}\beta _{1}f_{3}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{3},a_{2})-(\mu _{2}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{3},a_{2})\right]\\&\ \ \qquad \qquad \qquad \qquad \qquad -{\frac {n}{2}}\gamma _{1}f_{4}\chi _{2}\left[{\overset {\circ }{\Psi }}_{1}(a_{4},a_{2})-(\mu _{2}-\mu _{1}){\overset {\circ }{\Pi }}_{1}(a_{4},a_{2})\right]=0.\end{aligned}}$