190
MÉCANIQUE ANALYTIQUE.
19. On peut avoir de la même manière l’angle
correspondant ; pour cela, on fera
![{\displaystyle {\begin{aligned}{\frac {\cos \beta -\cos \alpha }{2+\cos \alpha +\cos \beta }}=&\sin 2\mu ,\\{\frac {\cos \beta -\cos \alpha }{2-\cos \alpha -\cos \beta }}=&\sin 2\nu ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf41cea79cc1b40659c0cdc81dabd8a26299b7d9)
et l’on aura
![{\displaystyle {\begin{aligned}{\frac {1}{1+\cos \psi }}=&{\frac {2}{(2+\cos \alpha +\cos \beta )\cos ^{2}\mu \left(1+\operatorname {tang} ^{2}\mu +2\operatorname {tang} \mu \cos 2\sigma \right)}},\\{\frac {1}{1-\cos \psi }}=&{\frac {2}{(2-\cos \alpha -\cos \beta )\cos ^{2}\nu \left(1+\operatorname {tang} ^{2}\nu -2\operatorname {tang} \nu \cos 2\sigma \right)}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed13a30379add1b59633a32e46d575c6216653e)
Si, dans les mêmes formules de l’article 98 (Sect. VII), on fait
on a
![{\displaystyle n'=1,\quad n''=1,\qquad \ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/030d53d1f537ecfc2a0edce15f0930d1d12d47ca)
donc
![{\displaystyle {\begin{aligned}&\left(1+\operatorname {tang} ^{2}\mu +2\operatorname {tang} \mu \cos 2\sigma \right)^{-1}\\&\quad =\mathrm {(A)+(B)\cos 2\sigma +(C)\cos 4\sigma +(D)\cos 6\sigma } +\ldots ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5be36e153e4c6f97e3e25a8ca0c44e864eda3c10)
où
![{\displaystyle {\begin{aligned}(\mathrm {A} )=&1+\operatorname {tang} ^{2}\mu +\operatorname {tang} ^{4}\mu +\operatorname {tang} ^{6}\mu +\ldots ={\frac {1}{1-\operatorname {tang} ^{2}\mu }},\\(\mathrm {B} )=&-2\operatorname {tang} \mu \left(1+\operatorname {tang} ^{2}\mu +\operatorname {tang} ^{4}\mu +\ldots \right)=-{\frac {2\operatorname {tang} \mu }{1-\operatorname {tang} ^{2}\mu }},\\(\mathrm {C} )=&2\operatorname {tang} ^{2}\mu \left(1+\operatorname {tang} ^{2}\mu +\operatorname {tang} ^{4}\mu +\ldots \right)={\frac {2\operatorname {tang} ^{2}\mu }{1-\operatorname {tang} ^{2}\mu }},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c2bb516211edc400a0b90faaad31333625a459)
Ainsi l’on aura
![{\displaystyle \left(1+\operatorname {tang} ^{2}\mu +2\operatorname {tang} \mu \cos 2\sigma \right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e33a6ebb0606e96fa04c2c6328a78fa343e0b2a)
![{\displaystyle ={\frac {1}{1-\operatorname {tang} ^{2}\mu }}\left(1-2\operatorname {tang} \mu \cos 2\sigma +2\operatorname {tang} ^{2}\mu \cos 4\sigma -2\operatorname {tang} ^{3}\mu \cos 6\sigma +\ldots \right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e89cf71ab8ed5eea63d21835924c687304d4097)
Si l’on multiplie cette série par la suivante
![{\displaystyle \mathrm {A+B\cos 2\sigma +C\cos 4\sigma } +\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2357b8424d9e1d758eaab77c4a9e98a44a3988)
le produit sera de nouveau de la forme
![{\displaystyle \mathrm {A'+B'\cos 2\sigma +C'\cos 4\sigma } +\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ef2a7077c75a1f5ebc088ae2bb783dd49ba2ac)