Page:Mémoires de l’Académie des sciences, Tome 10.djvu/737

${\begin{aligned}\omega &=\varphi +\left(\sigma ''_{0}-\sigma '_{0}\right)\left[{\frac {1}{2}}\varepsilon \cos .\lambda _{0}-{\frac {1}{16}}\varepsilon ^{2}\cos .\lambda _{0}\left(6+\sin .^{2}\lambda _{0}\right)\right]\\&+{\frac {1}{32}}\varepsilon ^{2}\sin .^{2}\lambda _{0}\cos .\lambda _{0}\left(\sin .2\sigma ''_{0}-\sin .2\sigma '_{0}\right)\\&-{\frac {1}{4}}\varepsilon ^{2}\mathrm {M} \left(\sigma ''_{0}-\sigma '_{0}\right)^{2}\cos .^{2}\lambda _{0}\operatorname {tang} .\mathrm {L} ''\\&-{\frac {1}{4}}\varepsilon ^{2}\mathrm {M} \left(\sigma ''_{0}-\sigma '_{0}\right)\cot .^{2}\lambda _{0}\operatorname {tang} .\mathrm {L} ''\left(\operatorname {tang} .\sigma ''_{0}+\cos .^{2}\lambda _{0}\cot .\sigma '_{0}\right).\end{aligned}}$

Arrivé ainsi à cette valeur de $\omega$ on tirera celles de $\lambda ''$ de la relation

$\cot .\mathrm {V} ''\sin .\omega =\operatorname {tang} .\lambda '\cos .\lambda ''-\cos .\omega \sin .\lambda ''.$

Au surplus on a en série

$\lambda ''=\mathrm {L} ''+\left({\frac {d\lambda ''}{d\mu }}\right)\mu +{\frac {1}{2}}\left({\frac {d^{2}\lambda ''}{d\mu ^{2}}}\right)\mu ^{2}+\ldots$

et à cause de

${\begin{aligned}\left({\frac {d\lambda ''}{d\mu }}\right)&={\frac {\operatorname {tang} .\mathrm {L} ''-\cos .\varphi \operatorname {tang} .\lambda '}{\sin .\varphi \left(\operatorname {tang} .\mathrm {L} ''\operatorname {tang} .\lambda '+\cos .\varphi \right)}}=\mathrm {M} ,\\\left({\frac {d^{2}\lambda ''}{d\mu ^{2}}}\right)&={\frac {\operatorname {tang} .\lambda '+2\mathrm {M} \sin .\varphi -\mathrm {M} ^{2}\left(\operatorname {tang} .\lambda '+\operatorname {tang} .\mathrm {L} ''\cos .\varphi \right)}{\operatorname {tang} .\mathrm {L} ''\operatorname {tang} .\lambda '+\cos .\varphi }}=\mathrm {N} ,\end{aligned}}$

il vient définitivement

${\begin{aligned}\lambda ''=&\mathrm {L''+M} \left(\sigma ''_{0}-\sigma '_{0}\right)\left[{\frac {1}{2}}\varepsilon \cos .\lambda _{0}-{\frac {1}{16}}\varepsilon ^{2}\cos .\lambda _{0}\left(6+\sin .^{2}\lambda _{0}\right)\right]\\&+{\frac {1}{32}}\mathrm {M} \varepsilon ^{2}\sin .^{2}\lambda _{1}\cos .\lambda _{0}\left(\sin .2\sigma ''_{0}-\sin .2\sigma '_{0}\right)\\&-{\frac {1}{4}}\mathrm {M} \varepsilon ^{2}\left(\sigma ''_{0}-\sigma '_{0}\right)\cot .^{2}\lambda _{0}\operatorname {tang} .\mathrm {L} ''\left[\operatorname {tang} .\sigma ''_{0}+\cos .^{2}\lambda _{0}\cot .\sigma '_{0}\right]\\&+{\frac {1}{8}}\varepsilon ^{2}\left(\sigma ''_{0}-\sigma '_{0}\right)^{2}\cos .^{2}\lambda _{0}\mathrm {\left[N-2M^{2}\operatorname {tang} .L''\right]} .\end{aligned}}$

Il reste à trouver $\mathrm {V} '$ et $s\,:$ or on a rigoureusement

$\sin .\mathrm {V} '=\sin .\mathrm {V} ''{\frac {\cos .\lambda ''}{\cos .\lambda '}},$