Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/42

qui ont en même temps l’avantage d’être toujours plus petites que celles-là.

27. Je ferai donc

$s=\operatorname {tang} {\frac {\omega }{2}}\sin \varphi ,\quad u=\operatorname {tang} {\frac {\omega }{2}}\cos \varphi ,\quad \theta =\varphi +\psi ,$

et, substituant d’abord dans les formules du n^o 23

${\frac {2\operatorname {tang} {\cfrac {\omega }{2}}}{1+\operatorname {tang} ^{2}{\cfrac {\omega }{2}}}},\quad 1-{\frac {2\operatorname {tang} ^{2}{\cfrac {\omega }{2}}}{1+\operatorname {tang} ^{2}{\cfrac {\omega }{2}}}},$

à la place de $\sin \omega ,\cos \omega ,$ ensuite $\theta -\varphi ,s,u$ à la place de $\psi ,\ \operatorname {tang} {\frac {\omega }{2}}\sin \varphi ,$$\operatorname {tang} {\frac {\omega }{2}}\cos \varphi$ et $s^{2}+u^{2}$ à la place de $\operatorname {tang} ^{2}{\frac {\omega }{2}},$ on aura

${\begin{aligned}\xi '\ \ =&{\frac {\left(1+u^{2}-s^{2}\right)\cos \theta +2su\sin \theta }{1+s^{2}+u^{2}}},\\\xi ''\ =&-{\frac {\left(1+s^{2}-u^{2}\right)\sin \theta +2su\cos \theta }{1+s^{2}+u^{2}}},\\\xi '''=&{\frac {2u\sin \theta -2s\cos \theta }{1+s^{2}+u^{2}}}\,;\\\eta '\ \ =&{\frac {\left(1+u^{2}-s^{2}\right)\sin \theta -2su\cos \theta }{1+s^{2}+u^{2}}},\\\eta ''\ =&{\frac {\left(1+s^{2}-u^{2}\right)\cos \theta -2su\sin \theta }{1+s^{2}+u^{2}}},\\\eta '''=&-{\frac {2u\cos \theta +2s\sin \theta }{1+s^{2}+u^{2}}}\,;\\\zeta '\ \ =&{\frac {2s}{1+s^{2}+u^{2}}},\\\zeta ''\ =&{\frac {2u}{1+s^{2}+u^{2}}},\\\zeta '''=&{\frac {1-s^{2}-u^{2}}{1+s^{2}+u^{2}}}.\end{aligned}}$