263
SUR L’ÉQUATION SÉCULAIRE DE LA LUNE.
quent le produit de
par ces forces,
![{\displaystyle 2a\int \operatorname {d} \mathrm {R} ={\frac {2n'^{2}}{3n^{2}}}-{\frac {n'^{2}u}{n^{2}}}\left(1+{\frac {3}{2}}e'^{2}\right)+{\frac {3n'^{2}}{2n^{2}}}\int ue'de'\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/878b149f1e69bc3c32922923f6546210894f01aa)
mais on a, à fort peu près,
![{\displaystyle u=-{\frac {d^{2}u}{n^{2}dt^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b5f7729550d14d562d170e71b1005ac01d59dc)
ce qui donne, à cause de l’extrême lenteur avec laquelle
varie,
![{\displaystyle \int ue'de'=-{\frac {du}{ndt}}{\frac {e'de'}{ndt}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2498250a9a14565ee9737c89aef920e54b69446d)
on aura donc
![{\displaystyle 2a\int \operatorname {d} \mathrm {R} ={\frac {2n'^{2}}{3n^{2}}}-{\frac {n'^{2}u}{n^{2}}}\left(1+{\frac {3}{2}}e'^{2}\right)-{\frac {3n'^{2}}{2n^{2}}}{\frac {du}{ndt}}{\frac {e'de'}{ndt}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb48b1ddbc4b888bc983a286bb0ca834dddad925)
On trouvera pareillement
![{\displaystyle a\left(x{\frac {\partial \mathrm {R} }{\partial x}}+y{\frac {\partial \mathrm {R} }{\partial y}}+z{\frac {\partial \mathrm {R} }{\partial z}}\right)=-{\frac {n'^{2}}{2n^{2}}}-{\frac {3n'^{2}e'^{2}}{4n^{2}}}-{\frac {n'^{2}u}{n^{2}}}\left(1+{\frac {3}{2}}e'^{2}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d2b5a4dc8208a1701328afd507bf62463538531)
L’équation différentielle précédente deviendra ainsi, en y faisant
![{\displaystyle {\begin{aligned}0=&{\frac {d^{2}u'}{dt^{2}}}+n^{2}u'\left(1-{\frac {3\delta a}{a}}-{\frac {2n'^{2}}{n^{2}}}-{\frac {3n'^{2}e'^{2}}{n^{2}}}\right)\\&-{\frac {3n'^{2}}{2n^{2}}}{\frac {du'}{dt}}{\frac {e'de'}{dt}}+n^{2}{\frac {\delta a}{a}}+{\frac {1}{6}}n'^{2}-{\frac {3n'^{2}e'^{2}}{4}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f981f653585f7899c7087fc6e79db1fc8dfb096)
d’où l’on tire les deux équations suivantes ;
![{\displaystyle {\frac {\delta a}{a}}=-{\frac {n'^{2}}{6n^{2}}}+{\frac {3n'^{2}e'^{2}}{4n^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af60d3291aaa2128e3146a72fc0dc7b368527b3d)
![{\displaystyle 0={\frac {d^{2}u'}{dt^{2}}}+n^{2}u'\left(1-{\frac {3n'^{2}}{2n^{2}}}-{\frac {21n'^{2}e'^{2}}{4n^{2}}}\right)-{\frac {3n'^{2}}{2n^{2}}}{\frac {du'}{dt}}{\frac {e'de'}{dt}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e004ae3efe08d08fe6559455f2b8bbfc48a4bd)