171
D’ASTRONOMIE.
porter uniquement sur les deux excentricités
et
et que le temps, doit être regardé comme exempt de différentiation. On aura ainsi
![{\displaystyle {\begin{aligned}{\frac {\operatorname {d} \phi }{\operatorname {d} \lambda }}=-{\frac {\operatorname {Sin} .\phi (2-\operatorname {Sin} .\lambda \operatorname {Cos} .\phi )}{\operatorname {Cos} .\lambda }},&\quad {\frac {\operatorname {d} r}{\operatorname {d} \lambda }}=a\operatorname {Cos} .\lambda \operatorname {Cos} .\phi ,\\{\frac {\operatorname {d} \psi }{\operatorname {d} \mu }}=-{\frac {\operatorname {Sin} .\psi (2-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}{\operatorname {Cos} .\mu }}\,;&\quad {\frac {\operatorname {d} s}{\operatorname {d} \mu }}=b\operatorname {Cos} .\mu \operatorname {Cos} .\psi .\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d169e5d9f7a6f2c198794eae79196659f46a6524)
22. Enfin, de l’expression de
donnée ci-dessus, on tire l’expression générale de
ainsi qu’il suit
![{\displaystyle \operatorname {d} \nu \!\nu ={\frac {\begin{aligned}-r^{2}\operatorname {d} \phi +rs\operatorname {d} \phi &\operatorname {Cos} .(\alpha -\beta -\phi +\psi )-s\operatorname {d} r\operatorname {Sin} .(\alpha -\beta -\phi +\psi )\\-s^{2}\operatorname {d} \psi +rs\operatorname {d} \psi &\operatorname {Cos} .(\alpha -\beta -\phi +\psi )+r\operatorname {d} r\operatorname {Sin} .(\alpha -\beta -\phi +\psi )\\\end{aligned}}{r^{2}-2rs\operatorname {Cos} .(\alpha -\beta -\phi +\psi )+s^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8f9554ef67ed4f2d1c43dd7a87fcc49158df49)
ce qui donnera, pour les deux coefficiens partiels ![{\displaystyle {\frac {\operatorname {d} \nu \!\nu }{\operatorname {d} \lambda }},\ {\frac {\operatorname {d} \nu \!\nu }{\operatorname {d} \mu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8869428dc9f4b899bf9ba57d75a5adc4a1b2df9d)
![{\displaystyle {\frac {\operatorname {d} \nu \!\nu }{\operatorname {d} \lambda }}={\frac {\operatorname {Cos} .\lambda }{r^{2}-2rs\operatorname {Cos} .(\alpha -\beta -\phi +\psi )+s^{2}}}\left\{{\tfrac {a^{2}\operatorname {Cos} .^{2}\lambda \operatorname {Sin} .\phi (2-\operatorname {Sin} .\lambda \operatorname {Cos} .\phi )}{(1-\operatorname {Sin} .\lambda \operatorname {Cos} .\phi )^{2}}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dea6ff2ff3b4512efef2c8274d3810fede11df6)
![{\displaystyle -{\frac {ab\operatorname {Cos} .^{2}\mu \operatorname {Sin} .\phi (2-\operatorname {Sin} .\lambda \operatorname {Cos} .\phi )}{(1-\operatorname {Sin} .\lambda \operatorname {Cos} .\phi )(1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}}\operatorname {Cos} .(\alpha -\beta -\phi +\psi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c597553c75485d3def1d15ca75744f2c568be2)
![{\displaystyle \left.-{\frac {ab\operatorname {Cos} .^{2}\mu \operatorname {Cos} .\phi }{(1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}}\operatorname {Sin} .(\alpha -\beta -\phi +\psi )\right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/424c89d31e7203572afd5c2c20ea36d81724ef09)
![{\displaystyle {\frac {\operatorname {d} \nu \!\nu }{\operatorname {d} \mu }}={\frac {\operatorname {Cos} .\mu }{r^{2}-2rs\operatorname {Cos} .(\alpha -\beta -\phi +\psi )+s^{2}}}\left\{{\tfrac {a^{2}\operatorname {Cos} .^{2}\mu \operatorname {Sin} .\psi (2-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}{(1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )^{2}}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd252eb949d9bb26005ec7a4470f8475dc336a6b)
![{\displaystyle -{\frac {ab\operatorname {Cos} .^{2}\mu \operatorname {Sin} .\psi (2-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}{(1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )(1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}}\operatorname {Cos} .(\alpha -\beta -\phi +\psi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ef084293f74febb4f2284f917e7c5e1b9f422f)
![{\displaystyle \left.+{\frac {ab\operatorname {Cos} .^{2}\mu \operatorname {Cos} .\psi }{(1-\operatorname {Sin} .\mu \operatorname {Cos} .\psi )}}\operatorname {Sin} .(\alpha -\beta -\phi +\psi )\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/833dbf48be8c59ad29744070366570fd53954f43)
24. Pour en tirer les deux coefficiens
il faudra faire, dans les deux expressions,
on aura ainsi