Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/271

261

DES ORBITES.

${\frac {X-X_{_{l}}}{t-t_{_{l}}}}={\frac {X'-X}{t'-t}},\quad {\frac {Y-Y_{_{l}}}{t-t_{_{l}}}}={\frac {Y'-Y}{t'-t}},\quad {\frac {Z-Z_{_{l}}}{t-t_{_{l}}}}={\frac {Z'-Z}{t'-t}}\,;$

c’est-à-dire,

$\left.{\begin{aligned}(t'-t)X_{l}-(t'-t_{_{l}})X+(t-t_{_{l}})X'&=0,\\(t'-t)Y_{l}-(t'-t_{_{l}})Y+(t-t_{_{l}})Y'&=0,\\(t'-t)Z_{_{l}}-(t'-t_{_{l}})Z+(t-t_{_{l}})Z'&=0.\\\end{aligned}}\right\}$(3)

Mettant dans ces dernières équations les valeurs données (2 et 3), elles deviendront

$\left.{\begin{aligned}&\left.{\begin{aligned}&(t'-t)\operatorname {Cos} .\beta _{l}\operatorname {Cos} .\gamma _{l}.D_{l}+(t'-t)r_{l}\operatorname {Cos} .\alpha _{l}\\-&(t'-t_{l})\operatorname {Cos} .\beta \operatorname {Cos} .\gamma .D-(t'-t_{l})r\operatorname {Cos} .\alpha \\+&(t-t_{l})\operatorname {Cos} .\beta '\operatorname {Cos} .\gamma '.D'+(t-t_{l})r'\operatorname {Cos} .\alpha '\\\end{aligned}}\right\}=0\\&\left.{\begin{aligned}&(t'-t)\operatorname {Sin} .\beta _{l}\operatorname {Cos} .\gamma _{l}.D_{l}+(t'-t)r_{l}\operatorname {Sin} .\alpha _{l}\\-&(t'-t_{l})\operatorname {Sin} .\beta \operatorname {Cos} .\gamma .D-(t'-t_{l})r\operatorname {Sin} .\alpha \\+&(t-t_{l})\operatorname {Sin} .\beta '\operatorname {Cos} .\gamma '.D'+(t-t_{l})r'\operatorname {Sin} .\alpha '\\\end{aligned}}\right\}=0\\(t'-t)&\operatorname {Sin} .\gamma _{l}.D_{l}-(t'-t_{l})\operatorname {Sin} .\gamma .D+(t-t_{l})\operatorname {Sin} .\gamma '.D'=0\,;\\\end{aligned}}\right\}(4)$

desquelles il s’agira de tirer les valeurs de $D_{l},D,D'.$

On simplifie un peu ces équations en posant

$\left.{\begin{array}{llll}&(t'-t)D_{l}\operatorname {Cos} .\gamma _{l}&=\Delta _{l},&(t'-t)r_{l}&=\rho _{l},\\&(t'-t_{l})D\operatorname {Cos} .\gamma &=\Delta ,&(t'-t_{l})r&=\rho ,\\&(t-t_{l})D'\operatorname {Cos} .\gamma '&=\Delta ',&(t-t_{l})r'&=\rho '\,;\\\end{array}}\right\}$(5)

elles deviennent ainsi

$\left.{\begin{array}{lll}&\Delta _{l}\operatorname {Cos} .\beta _{l}-\Delta \operatorname {Cos} .\beta +\Delta '\operatorname {Cos} .\beta '+\left(\rho _{l}\operatorname {Cos} .\alpha _{l}-\rho \operatorname {Cos} .\alpha +\rho '\operatorname {Cos} .\alpha '\right)&=0,\\&\Delta _{l}\operatorname {Sin} .\beta _{l}-\Delta \operatorname {Sin} .\beta +\Delta '\operatorname {Sin} .\beta '+\left(\rho _{l}\operatorname {Sin} .\alpha _{l}-\rho \operatorname {Sin} .\alpha +\rho '\operatorname {Sin} .\alpha '\right)&=0,\\&\Delta _{l}\operatorname {Tang} .\gamma _{l}-\Delta \operatorname {Tang} .\gamma +\Delta '\operatorname {Tang} .\gamma '&=0.\\\end{array}}\right\}$(6)

On en tire la valeur de $\Delta _{l}$ en prenant la somme de leurs produits respectifs par

$\operatorname {Sin} .\beta \operatorname {Tang} .\gamma '-\operatorname {Sin} .\beta '\operatorname {Tang} .\gamma ,\ \operatorname {Cos} .\beta '\operatorname {Tang} .\gamma -\operatorname {Cos} .\beta \operatorname {Tang} .\gamma ',\ \operatorname {Sin} .(\beta '-\beta )\,;$