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THÉORIE DE JUPITER ET DE SATURNE.
La formule (9) de l’article VII, transportée à Saturne, donnera ainsi
![{\displaystyle {\begin{aligned}&m\delta v'_{1}=\left[{\frac {1}{2}}-{\frac {2n'(3n'-n)}{(n-2n')(4n'-n)}}\right]\\&\qquad \qquad \qquad \times {\frac {e'}{a'}}0{,}0053605\cos \left(3n't-nt+3\varepsilon '-\varepsilon -\varpi '-77^{\circ }50'46''\right)\\&+\left[{\frac {5n'(3n'-n)}{2(n-2n')(4n'-n)}}-{\frac {1}{2}}\right]{\frac {e'^{2}}{a'}}0{,}00081435\sin(3n't-nt+3\varepsilon '-\varepsilon -2\varpi ')\\&+m\left\{a'\mathrm {Q} \left[{\frac {9n'^{2}}{(3n'-n)^{2}}}+{\frac {12n'^{2}}{(n-2n')(4n'-n)}}\right]\right.\\&+\left.2a'^{2}{\frac {\partial \mathrm {Q} }{\partial a'}}\left[{\frac {n'}{3n'-n}}+{\frac {n'(3n'-n)}{(n-2n')(4n'-n)}}\right]\right\}\sin(3n't-nt+3\varepsilon '-\varepsilon +\mathrm {A} ),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2a5d225b9c400aeaa96fecbdb4203f156521c7)
d’où l’on tire, en négligeant les termes insensibles,
![{\displaystyle {\begin{aligned}m\delta v'_{1}=&-5''{,}9\cos \left(3n't-nt+3\varepsilon '-\varepsilon -\varpi '-77^{\circ }50'46''\right)\\&+m(50{,}0811a'\mathrm {Q} +5{,}2805a'^{2}{\frac {\partial \mathrm {Q} }{\partial a'}})\sin \left(3n't-nt+3\varepsilon '-\varepsilon +\mathrm {A} \right).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbacea5a5b770db6f9f6f2dc730992ae0471778)
Il ne s’agit plus que de déterminer
et
Pour cela, j’observe que la partie de
qui dépend de l’angle
![{\displaystyle 3n't-nt+3\varepsilon '-\varepsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a39da36b46bdb909f8d6c59a4a05b17f9abb5090)
peut être mise sous cette forme
![{\displaystyle {\begin{aligned}\mathrm {R} =&\quad \ \mathrm {N} ^{(0)}e'^{2}\cos(3n't-nt+3\varepsilon '-\varepsilon -2\varpi ')\\&+\mathrm {N} ^{(1)}ee'\cos(3n't-nt+3\varepsilon '-\varepsilon -\ \ \varpi -\varpi ')\\&+\mathrm {N} ^{(2)}e^{2}\ \cos(3n't-nt+3\varepsilon '-\varepsilon -2\varpi )\\&+\mathrm {N} ^{(3)}\gamma ^{2}\ \cos(3n't-nt+3\varepsilon '-\varepsilon -2\Pi ),\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f0177b0988b9e4cb397f9dea410f8d6d47465c)
et l’on trouve
![{\displaystyle {\begin{aligned}a'\mathrm {N} ^{(0)}=&{\frac {3}{8\alpha ^{2}}}-{\frac {17}{8}}b_{\frac {1}{2}}^{(1)}-{\frac {5}{4}}\alpha {\frac {db_{\frac {1}{2}}^{(1)}}{d\alpha }}-{\frac {1}{8}}\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(1)}}{d\alpha ^{2}}},\\a'\mathrm {N} ^{(1)}=&\qquad -\ \ 5\,\ b_{\frac {1}{2}}^{(2)}+{\frac {5}{2}}\alpha {\frac {db_{\frac {1}{2}}^{(2)}}{d\alpha }}+{\frac {1}{4}}\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(2)}}{d\alpha ^{2}}},\\a'\mathrm {N} ^{(2)}=&\qquad -{\frac {21}{8}}b_{\frac {1}{2}}^{(3)}-{\frac {5}{4}}\alpha {\frac {db_{\frac {1}{2}}^{(3)}}{d\alpha }}-{\frac {1}{8}}\alpha ^{2}{\frac {d^{2}b_{\frac {1}{2}}^{(3)}}{d\alpha ^{2}}},\\a'\mathrm {N} ^{(3)}=&-{\frac {1}{8}}\alpha b_{\frac {1}{2}}^{(3)}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad67f76f97c1ddba7fd047fb0045a7f83988dbc)
On a ensuite généralement,
étant une fonction homogène de
et