Page:Gauss - Théorie du mouvement des corps célestes, traduction Dubois, 1864.djvu/379

360

NOTES DU TRADUCTEUR.

comète, on aura donc, en appelant $\theta$ l’époque du passage du périhélie, et en représentant par $\mathrm {C}$ la quantité 27^j,403895,

${\begin{aligned}\mathrm {T} -\theta &=q^{\frac {3}{2}}\mathrm {C} \left(\operatorname {tang} ^{3}{\frac {1}{2}}v+3\operatorname {tang} {\frac {1}{2}}v\right)\\\mathrm {T} ''\!-\theta &=q^{\frac {3}{2}}\mathrm {C} \left(\operatorname {tang} ^{3}\!{\frac {1}{2}}v''+3\operatorname {tang} {\frac {1}{2}}v''\right),\end{aligned}}$

d’où

${\begin{aligned}\mathrm {T} ''\!-\mathrm {T} &=q^{\frac {3}{2}}\mathrm {C} \left[\operatorname {tang} ^{3}{\frac {1}{2}}v''\!-\operatorname {tang} ^{3}{\frac {1}{2}}v+3\left(\operatorname {tang} {\frac {1}{2}}v''-\operatorname {tang} {\frac {1}{2}}v\right)\right]\\&=q^{\frac {3}{2}}\mathrm {C} \left(\operatorname {tang} {\frac {1}{2}}v''\!-\operatorname {tang} {\frac {1}{2}}v\right)\left(3+{\frac {\operatorname {tang} ^{3}\!{\dfrac {1}{2}}v''\!-\operatorname {tang} ^{3}\!{\dfrac {1}{2}}v}{\operatorname {tang} {\dfrac {1}{2}}v''-\operatorname {tang} {\dfrac {1}{2}}v}}\right)\end{aligned}}$

ou

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

ou enfin,

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

Mais on sait qu’on a les relations

[β″]

${\begin{aligned}r&={\frac {q}{\cos ^{2}\!{\dfrac {1}{2}}v}},&r''&={\frac {q}{\cos ^{2}\!{\dfrac {1}{2}}v''}}\,;\end{aligned}}$

il vient alors, en ayant égard à ces expressions,