De même je change les expressions
![{\displaystyle \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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/710e4cee11ff6a4ba68689fb4fb8c28e1872ca1e)
en celles-ci
![{\displaystyle \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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11c2eb24e879fa467908c9c0fa40460e50a2e8d)
Par ce moyen, l’équation se trouve composée de deux parties, l’une finie et l’autre indéfinie, laquelle renferme les quantités intégrales
![{\displaystyle \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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c335b06c6c285eef6e2ddb2012f7d193a3cb600)
![{\displaystyle \int re^{{\text{V}}_{1}t}dt,\ \ \int \mathrm {R} e^{{\text{V}}_{1}t}dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51f4a696d79ee96948b155478f17f172a4011e9e)
Je fais les coefficients de ces quantités chacun égal à zéro, ce qui me donne sept équations entre les sept inconnues
et
savoir
(1o)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47a934218c3d77a4548e828de306b822f1edb782)
(2o)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f64395f2267bd6179de2397f0f27f1cf7b20a9aa)