Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/273

263

DES ORBITES.

$\left.{\begin{aligned}x=X,&\qquad {\frac {\operatorname {d} x}{\operatorname {d} t}}={\frac {X'-X_{l}}{t'-t_{l}}},\\y=Y,&\qquad {\frac {\operatorname {d} y}{\operatorname {d} t}}={\frac {Y'-Y_{l}}{t'-t_{l}}},\\z=Z,&\qquad {\frac {\operatorname {d} z}{\operatorname {d} t}}={\frac {Z'-Z_{l}}{t'-t_{l}}},\\\end{aligned}}\right\}$(11)

et avec ces six quantités on déterminera complètement les élémens de l’astre suivant ce qui a été dit (Annales, tome II, page 8).

III. Voyons présentement ce que deviennent ces résultats à la limite, c’est-à-dire, lorsque l’intervalle de temps qui sépare les observations extrêmes est infiniment petit ou nul. Posons d’abord, pour abréger,

$\left.{\begin{array}{llll}r_{l}\operatorname {Cos} .\alpha _{l}=g_{l},&r_{l}\operatorname {Sin} .\alpha _{l}=h_{l},&\operatorname {Cos} .\beta _{l}\operatorname {Cot} .\gamma _{l}=m_{l},&\operatorname {Sin} .\beta _{l}\operatorname {Cot} .\gamma _{l}=n_{l},\\r\operatorname {Cos} .\alpha =g,&r\operatorname {Sin} .\alpha =h,&\operatorname {Cos} .\beta \operatorname {Cot} .\gamma =m,&\operatorname {Sin} .\beta \operatorname {Cot} .\gamma =n,\\r'\operatorname {Cos} .\alpha '=g'\,;&r'\operatorname {Sin} .\alpha '=h'\,;&\operatorname {Cos} .\beta '\operatorname {Cot} .\gamma '=m'\,;&\operatorname {Sin} .\beta '\operatorname {Cot} .\gamma '=n'\,;\\\end{array}}\right\}$(12)

Nous aurons conséquemment

$\left.{\begin{array}{ll}r_{l}\operatorname {Sin} .(\alpha _{l}-\beta _{l})&=(m_{l}h_{l}-n_{l}g_{l})\operatorname {Tang} .\gamma _{l},\\r\ \operatorname {Sin} .(\alpha \ -\beta _{l})&=(m_{l}h\ -n_{l}g\ )\operatorname {Tang} .\gamma _{l},\\r'\operatorname {Sin} .(\alpha '-\beta _{l})&=(m_{l}h'-n_{l}g')\operatorname {Tang} .\gamma _{l}\,;\\r_{l}\operatorname {Sin} .(\alpha _{l}-\beta _{l})&=(m\ h_{l}-n\ g_{l})\operatorname {Tang} .\gamma ,\\r\ \operatorname {Sin} .(\alpha \ -\beta )&=(m\ h\ -n\ g)\operatorname {Tang} .\gamma ,\\r'\operatorname {Sin} .(\alpha '-\beta )&=(m\ h'-n\ g')\operatorname {Tang} .\gamma \,;\\r_{l}\operatorname {Sin} .(\alpha _{l}-\beta ')&=(m'h_{l}-n'g_{l})\operatorname {Tang} .\gamma ',\\r\ \operatorname {Sin} .(\alpha \ -\beta ')&=(m'h\ -n'g)\operatorname {Tang} .\gamma ',\\r'\operatorname {Sin} .(\alpha '-\beta ')&=(m'h'-n'g')\operatorname {Tang} .\gamma '\,;\\\end{array}}\right\}$(13)

$\left.{\begin{array}{ll}\operatorname {Sin} .(\beta '-\beta )&=(m\ n'-m'n\ )\operatorname {Tang} .\gamma '\operatorname {Tang} .\gamma ,\\\operatorname {Sin} .(\beta '-\beta _{l})&=(m_{l}n'-m'n_{l})\operatorname {Tang} .\gamma '\operatorname {Tang} .\gamma _{l},\\\operatorname {Sin} .(\beta \ -\beta _{l})&=(m_{l}n\ -m\ n_{l})\operatorname {Tang} .\gamma \ \operatorname {Tang} .\gamma _{l}.\\\end{array}}\right\}$(14)