Page:Gauss - Théorie du mouvement des corps célestes, traduction Dubois, 1864.djvu/272

253

DÉTERMINATION D’UNE ORBITE D’APRÈS QUATRE OBSERVATIONS.

${\begin{aligned}0&=(n12)\mathrm {R} \cos \mathrm {B} \sin(l-\alpha )\\&-(n02)\left[\rho '\cos \beta '\sin(\alpha '-\alpha )+\mathrm {R} '\cos \mathrm {B} '\sin(l'-\alpha )\right]\\&+(n01)\left[\rho ''\cos \beta ''\sin(\alpha ''-\alpha )+\mathrm {R} ''\cos \mathrm {B} ''\sin(l''-\alpha )\right],\\[0.75ex]0&=(n23)\left[\rho '\cos \beta '\sin(\alpha '''-\alpha ')+\mathrm {R} '\cos \mathrm {B} '\sin(\alpha '''-l')\right]\\&-(n13)\left[\rho ''\cos \beta ''\sin(\alpha '''-\alpha '')+\mathrm {R} ''\cos \mathrm {B} ''\sin(\alpha '''-l'')\right]\\&+(n12)\mathrm {R} '''\cos \mathrm {B} '''\sin(\alpha '''-l''').\end{aligned}}$

Ces équations, en posant,

${\frac {\mathrm {R} '\cos \mathrm {B} '\sin(l'-\alpha )}{\cos \beta '\sin(\alpha '-\alpha )}}-\;\mathrm {R} '\cos \delta '={}$	$b',$
${\frac {\mathrm {R} ''\cos \mathrm {B} ''\sin(\alpha '''-l'')}{\cos \beta ''\sin(\alpha '''-\alpha '')}}-\mathrm {R} ''\cos \delta ''={}$	$b'',$
${\frac {\mathrm {R} '\cos \mathrm {B} '\sin(\alpha '''-l')}{\cos \beta '\sin(\alpha '''-\alpha ')}}-\;\mathrm {R} '\cos \delta '={}$	$\varkappa ',$
${\frac {\mathrm {R} ''\cos \mathrm {B} ''\sin(l''-\alpha )}{\cos \beta ''\sin(\alpha ''-\alpha )}}-\mathrm {R} ''\cos \delta ''={}$	$\varkappa '',$

${\frac {\mathrm {R} \cos \mathrm {B} \sin(l-\alpha )}{\cos \beta ''\sin(\alpha ''-\alpha )}}={}$	$\lambda ,$
${\frac {\mathrm {R} '''\cos \mathrm {B} '''\sin(\alpha '''-l''')}{\cos \beta '\sin(\alpha '''-\alpha ')}}={}$	$\lambda ''',$
${\frac {\cos \beta '\sin(\alpha '-\alpha )}{\cos \beta ''\sin(\alpha ''-\alpha )}}={}$	$\mu ',$
${\frac {\cos \beta ''\sin(\alpha '''-\alpha '')}{\cos \beta '\sin(\alpha '''-\alpha ')}}={}$	$\mu '',$

et toutes les réductions étant convenablement faites, se changent en les équations suivantes :

${\begin{aligned}{\frac {\mu '(1+\mathrm {P} ')(x'+b')}{1+{\dfrac {\mathrm {Q} '}{\left(x'^{2}+a'^{2}\right)^{\frac {3}{2}}}}}}&=x''+\varkappa ''+\lambda \mathrm {P} ',\\[1ex]{\frac {\mu ''(1+\mathrm {P} '')(x''+b'')}{1+{\dfrac {\mathrm {Q} ''}{\left(x''^{2}+a''^{2}\right)^{\frac {3}{2}}}}}}&=x'+\varkappa '+\lambda '''\mathrm {P} ''\,;\end{aligned}}$

ou, en posant en outre,

${\begin{aligned}-\varkappa ''-\;\lambda \;\mathrm {P} '&=c',&\mu '(1+\mathrm {P} ')&=d'\\-\varkappa '-\lambda '''\mathrm {P} ''&=c'',&\mu ''(1+\mathrm {P} '')&=d'',\end{aligned}}$