Page:Joseph Louis de Lagrange - Œuvres, Tome 4.djvu/479

Maintenant, puisque

$\operatorname {tang} {\frac {\varphi '}{2}}=\operatorname {tang} {\frac {\psi -\omega }{2}}={\frac {\operatorname {tang} {\cfrac {\psi }{2}}-\operatorname {tang} {\cfrac {\omega }{2}}}{1+\operatorname {tang} {\cfrac {\psi }{2}}\operatorname {tang} {\cfrac {\omega }{2}}}},$

si l’on substitue pour $\operatorname {tang} {\frac {\psi }{2}}$ sa valeur trouvée ci-dessus, on aura

$\operatorname {tang} {\frac {\varphi '}{2}}={\frac {{\sqrt {r''}}\cos \omega -{\sqrt {r'}}}{{\sqrt {r''}}\sin \omega }},$

et l’on trouvera de la même manière, à cause de $\varphi ''=\psi +\omega ,$

$\operatorname {tang} {\frac {\varphi ''}{2}}={\frac {{\sqrt {r''}}-{\sqrt {r'}}\cos \omega }{{\sqrt {r'}}\sin \omega }}\,;$

donc

$\operatorname {tang} {\frac {\varphi ''}{2}}-\operatorname {tang} {\frac {\varphi '}{2}}={\frac {r'+r''-2{\sqrt {r'r''}}\cos \omega }{{\sqrt {r'r''}}\sin \omega }},$

et

$\operatorname {tang} ^{3}{\frac {\varphi ''}{2}}-\operatorname {tang} ^{3}{\frac {\varphi '}{2}}={\frac {\left(r''-{\sqrt {r'r''}}\cos \omega \right)^{3}+\left(r'-{\sqrt {r'r''}}\cos \omega \right)^{3}}{r'r''{\sqrt {r'r''}}\sin ^{3}\omega }}$

$={\frac {\begin{aligned}r'^{3}+r''^{3}-3\left(r'^{2}+r''^{2}\right){\sqrt {r'r''}}\cos \omega +3(r'+r'')r'r''\cos ^{2}\omega &\\-2r'r''{\sqrt {r'r''}}\cos ^{3}\omega &\end{aligned}}{r'r''{\sqrt {r'r''}}\sin ^{3}\omega }}.$

Donc

$3\left(\operatorname {tang} {\frac {\varphi ''}{2}}-\operatorname {tang} {\frac {\varphi '}{2}}\right)+\operatorname {tang} ^{3}{\frac {\varphi ''}{2}}-\operatorname {tang} ^{3}{\frac {\varphi '}{2}}$

égale

$3r'r''\left(r'+r''-2{\sqrt {r'r''}}\cos \omega \right)\left(1-\cos ^{2}\omega \right)+r'^{3}+r''^{3}$

$-3\left(r'^{2}+r''^{2}\right){\sqrt {r'r''}}\cos \omega +3(r'+r'')r'r''\cos ^{2}\omega -2r'r''{\sqrt {r'r''}}\cos ^{3}\omega ,$

divisé par $r'r''{\sqrt {r'r''}}\sin ^{3}\omega \,;$ ce qui se réduit à cette quantité

${\frac {(r'+r'')^{3}-3(r'+r'')^{2}{\sqrt {r'r''}}\cos \omega +4r'r''{\sqrt {r'r''}}\cos ^{3}\omega }{r'r''{\sqrt {r'r''}}\sin ^{3}\omega }}$

$={\frac {\left(r'+r''-2{\sqrt {r'r''}}\cos \omega \right)^{2}\left(r'+r''+{\sqrt {r'r''}}\cos \omega \right)}{r'r''{\sqrt {r'r''}}\sin ^{3}\omega }}.$