L’équation (17) de l’article XI donne, en y substituant, au lieu de δ r , a δ μ ′ y , {\displaystyle \delta r,a\delta \mu 'y,} et en négligeant les quantités de l’ordre α , {\displaystyle \alpha ,}
Or, on a par l’article XI, aux quantités près de l’ordre α 2 , c a 2 = n , {\displaystyle \alpha ^{2},\ {\frac {c}{a^{2}}}=n,} ensuite, n 2 = S + P a 3 ; {\displaystyle n^{2}={\frac {\mathrm {S+P} }{a^{3}}}\,;} donc
d’où l’on tire, en intégrant,
et en négligeant les quantités de l’ordre 
![{\displaystyle {\begin{aligned}0={\frac {d^{2}y}{dt^{2}}}-{\frac {2c\delta c}{a^{4}\delta \mu '}}&+y\left[{\frac {3c^{2}}{a^{4}}}-{\frac {2(\mathrm {S+P} )}{a^{3}}}\right]\\+{\frac {2c}{a^{2}}}{\frac {n^{2}}{n'-n}}&\left\{{\begin{aligned}\quad \ \ \left(\mathrm {C} -{\frac {1}{i^{2}}}\right)&\cos \ \ (n't-nt+\mathrm {B} )\\+{\frac {1}{4}}\sideset {^{1}}{}{\mathrm {C} }&\cos 2(n't-nt+\mathrm {B} )\\+{\frac {1}{9}}\sideset {^{2}}{}{\mathrm {C} }&\cos 3(n't-nt+\mathrm {B} )\\+\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}\\\\+n^{2}&\left\{{\begin{aligned}\mathrm {F} +\left(\sideset {^{1}}{}{\mathrm {F} }+{\frac {1}{i^{2}}}\right)&\cos \ \ (n't-nt+\mathrm {B} )\\+{\frac {1}{2}}\sideset {^{2}}{}{\mathrm {F} }&\cos 2(n't-nt+\mathrm {B} )\\+{\frac {1}{3}}\sideset {^{3}}{}{\mathrm {F} }&\cos 3(n't-nt+\mathrm {B} )\\+\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}.\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7000fa0dc41f2d556a1aeddac6d39a764c81f630)
ensuite, 
donc
![{\displaystyle \left\{{\begin{aligned}\left[\sideset {^{1}}{}{\mathrm {F} }+{\frac {1}{i^{2}}}+{\frac {2n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)\right]&\cos \ \ (n't-nt+\mathrm {B} )\\+\left({\frac {1}{2}}\sideset {^{2}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{4}}\sideset {^{1}}{}{\mathrm {C} }\right)&\cos 2(n't-nt+\mathrm {B} )\\+\left({\frac {1}{3}}\sideset {^{3}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{9}}\sideset {^{2}}{}{\mathrm {C} }\right)&\cos 3(n't-nt+\mathrm {B} )\\+\ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/338f61228f84d36199a5a61bea87c8b38a87e40f)
![{\displaystyle {\begin{aligned}&+{\frac {n^{2}}{(n'-n)^{2}-n^{2}}}\left[\sideset {^{1}}{}{\mathrm {F} }+{\frac {1}{i^{2}}}+{\frac {2n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)\right]\cos(n't-nt+\mathrm {B} )\\&+{\frac {n^{2}}{(2n'-2n)^{2}-n^{2}}}\left[{\frac {1}{2}}\sideset {^{2}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{4}}\sideset {^{1}}{}{\mathrm {C} }\right]\cos 2(n't-nt+\mathrm {B} )\\&+{\frac {n^{2}}{(3n'-3n)^{2}-n^{2}}}\left[{\frac {1}{3}}\sideset {^{3}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{9}}\sideset {^{2}}{}{\mathrm {C} }\right]\cos 3(n't-nt+\mathrm {B} )\\&+\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/8c865ff33fe3f4e8c9a043cb8dd42f815a36992b)
