Page:Gauss - Théorie du mouvement des corps célestes, traduction Dubois, 1864.djvu/380

361

MÉTHODE D’OLBERS.

 Les corrections sont expliquées en page de discussion

${\begin{aligned}\mathrm {T} ''\!-\mathrm {T} &={\frac {r^{\frac {1}{2}}r''^{\frac {3}{2}}\mathrm {C} \sin {\dfrac {1}{2}}(v''\!-v)}{q^{\frac {1}{2}}}}+{\frac {r''^{\frac {1}{2}}r^{\frac {3}{2}}\mathrm {C} \sin {\dfrac {1}{2}}(v''\!-v)}{q^{\frac {1}{2}}}}\\&+{\frac {rr''\mathrm {C} \sin {\dfrac {1}{2}}(v''\!-v)\cos {\dfrac {1}{2}}(v''\!-v)}{q^{\frac {1}{2}}}}.\end{aligned}}$

Les relations (β) donnent aussi,

$q$	${}=\left(rr''\right)^{\frac {1}{2}}\cos {\frac {1}{2}}v\cos {\frac {1}{2}}v''$
	${}={\frac {1}{2}}\left(rr''\right)^{\frac {1}{2}}\left(\cos {\frac {1}{2}}(v''\!-v)+\cos {\frac {1}{2}}(v''\!+v)\right)$
	${}={\frac {1}{2}}\left(rr''\right)^{\frac {1}{2}}\left(\cos {\frac {1}{2}}(v''\!-v)+{\frac {\cos ^{2}\!{\dfrac {1}{4}}(v+v'')-\sin ^{2}\!{\dfrac {1}{4}}(v+v'')}{\cos ^{2}\!{\dfrac {1}{4}}(v+v'')+\sin ^{2}\!{\dfrac {1}{4}}(v+v'')}}\right)$
	${}={\frac {1}{2}}\left(rr''\right)^{\frac {1}{2}}\left(\cos {\frac {1}{2}}(v''\!-v)+{\frac {1-\operatorname {tang} ^{2}\!{\dfrac {1}{4}}(v+v'')}{1+\operatorname {tang} ^{2}\!{\dfrac {1}{4}}(v+v'')}}\right).$

Ces mêmes relations (β) donnent encore

${\frac {\cos {\dfrac {1}{2}}v}{\cos {\dfrac {1}{2}}v''}}={\frac {r''^{\frac {1}{2}}}{r^{\frac {1}{2}}}}$

d’où

${\begin{aligned}{\frac {r''^{\frac {1}{2}}+r^{\frac {1}{2}}}{r''^{\frac {1}{2}}-r^{\frac {1}{2}}}}&={\frac {\cos {\dfrac {1}{2}}v+\cos {\dfrac {1}{2}}v''}{\cos {\dfrac {1}{2}}v-\cos {\dfrac {1}{2}}v''}}={\frac {\cos {\dfrac {1}{4}}(v+v'')\cos {\dfrac {1}{4}}(v''\!-v)}{\sin {\dfrac {1}{4}}(v+v'')\sin {\dfrac {1}{4}}(v''\!-v)}}\\&=\operatorname {cotang} {\frac {1}{4}}(v''\!+v)\operatorname {cotang} {\frac {1}{4}}(v''\!-v)\,;\end{aligned}}$

de là, en posant

$\operatorname {tang} z=\left({\frac {r}{r''}}\right)^{\frac {1}{2}},$

on déduit l’équation

${\begin{aligned}\operatorname {tang} {\frac {1}{4}}(v''\!+v)=\operatorname {tang} (45^{\circ }\!-z)\operatorname {cotang} {\frac {1}{4}}(v''\!-v).\end{aligned}}$