Page:Annales de mathématiques pures et appliquées, 1814-1815, Tome 5.djvu/92

86

DISCUSSION DES LIGNES.

$\left.{\begin{aligned}(Ar^{2}-\Delta )m^{2}+(Br^{2}-\Delta )n^{2}&+2(C'r^{2}-\Delta \operatorname {Cos} .\gamma )mn\\+2(B'r^{2}-\Delta \operatorname {Cos} .\beta )m&+2(A'r^{2}-\Delta \operatorname {Cos} .\alpha )n\\&+(Cr^{2}-\Delta )=0\\\end{aligned}}\right\}$(25)

éliminant enfin $m$ et $n$ entre cette équation et les équations (20) on aura d’abord

$\left(ABC-AA'^{2}-BB'^{2}-CC'^{2}+2A'B'C'\right)r^{6}$

$-\Delta \left\{{\begin{aligned}&(BC-A'^{2})+2(B'C'-AA')\operatorname {Cos} .\alpha \\+&(CA-B'^{2})+2(C'A'-BB')\operatorname {Cos} .\beta \\+&(AB-C'^{2})+2(A'B'-CC')\operatorname {Cos} .\gamma \\\end{aligned}}\right\}r^{4}$

(26)

$+\Delta ^{2}\left\{{\begin{aligned}&A\operatorname {Sin} .^{2}\alpha -2A'(\operatorname {Cos} .\alpha -\operatorname {Cos} .\beta \operatorname {Cos} .\gamma )\\+&B\operatorname {Sin} .^{2}\beta -2B'(\operatorname {Cos} .\beta -\operatorname {Cos} .\gamma \operatorname {Cos} .\alpha )\\+&C\operatorname {Sin} .^{2}\gamma -2C'(\operatorname {Cos} .\gamma -\operatorname {Cos} .\alpha \operatorname {Cos} .\beta )\\\end{aligned}}\right\}r^{2}$

$-\delta ^{3}(1-\operatorname {Cos} .^{2}\alpha -\operatorname {Cos} .^{2}\beta -\operatorname {Cos} .^{2}\gamma +\operatorname {Cos} .\alpha \operatorname {Cos} .\beta \operatorname {Cos} .\gamma =0,$

et ensuite

$\left.{\begin{aligned}m&={\tfrac {(A'r^{2}-\Delta \operatorname {Cos} .\alpha )(C'r^{2}-\Delta \operatorname {Cos} .\gamma )-(Br^{2}-\Delta )(B'r^{2}-\Delta \operatorname {Cos} .\beta )}{(Ar^{2}-\Delta )(Br^{2}-\Delta )-(C'r^{2}-\Delta \operatorname {Cos} .\gamma )^{2}}},\\n&={\tfrac {(B'r^{2}-\Delta \operatorname {Cos} .\beta )(C'r^{2}-\Delta \operatorname {Cos} .\gamma )-(Ar^{2}-\Delta )(A'r^{2}-\Delta \operatorname {Cos} .\alpha )}{(Ar^{2}-\Delta )(Br^{2}-\Delta )-(C'r^{2}-\Delta \operatorname {Cos} .\gamma )^{2}}}.\\\end{aligned}}\right\}$(27)

L’équation (26) donnera les longueurs des demi-diamètres principaux ; les formules (27) en détermineront la direction, et ensuite