Page:Annales de mathématiques pures et appliquées, 1816-1817, Tome 7.djvu/187

181

DE RÉVOLUTION.

${\begin{array}{rr}A'^{2}-BC=a,&B+C-2A'\operatorname {Cos} .\alpha =a,\\B'^{2}-CA=b,&C+A-2B'\operatorname {Cos} .\beta =b,\\C'^{2}-AB=c,&A+B-2C'\operatorname {Cos} .\gamma =c,\\\end{array}}$

on trouvera d’abord

${\begin{alignedat}{2}&k&={\frac {a'\operatorname {Sin} .^{2}\beta -b'\operatorname {Sin} .^{2}\alpha }{a\operatorname {Sin} .^{2}\beta -b\operatorname {Sin} .^{2}\alpha }}&=-{\frac {a\operatorname {Sin} .^{2}\beta -b\operatorname {Sin} .^{2}\alpha }{ab'-ba'}},\\\\&k&={\frac {b'\operatorname {Sin} .^{2}\gamma -c'\operatorname {Sin} .^{2}\beta }{b\operatorname {Sin} .^{2}\gamma -c\operatorname {Sin} .^{2}\beta }}&=-{\frac {b\operatorname {Sin} .^{2}\gamma -c\operatorname {Sin} .^{2}\beta }{bc'-cb'}},\\\\&k&={\frac {c'\operatorname {Sin} .^{2}\alpha -a'\operatorname {Sin} .^{2}\gamma }{c\operatorname {Sin} .^{2}\alpha -a\operatorname {Sin} .^{2}\gamma }}&=-{\frac {c\operatorname {Sin} .^{2}\alpha -a\operatorname {Sin} .^{2}\gamma }{ca'-ac'}}.\end{alignedat}}$

Voilà donc la valeur de $k$ sous six formes différentes ; et nous avons en outre les trois équations de condition

${\begin{alignedat}{2}&(a\operatorname {Sin} .^{2}\beta -b\operatorname {Sin} .^{2}\alpha )^{2}&+(ab'-ba')(a'\operatorname {Sin} .^{2}\beta -b'\operatorname {Sin} .^{2}\alpha )&=0,\\&(b\operatorname {Sin} .^{2}\gamma -c\operatorname {Sin} .^{2}\beta )^{2}&+\,(bc'-cb')\ (b'\operatorname {Sin} .^{2}\gamma -c'\operatorname {Sin} .^{2}\beta )&=0,\\&(c\operatorname {Sin} .^{2}\alpha -a\operatorname {Sin} .^{2}\gamma )^{2}&+(ca'-ac')(c'\operatorname {Sin} .^{2}\alpha -a'\operatorname {Sin} .^{2}\gamma )&=0.\end{alignedat}}$

Nous aurons encore

$d={\sqrt {A.{\frac {b'\operatorname {Sin} .^{2}\gamma -c'\operatorname {Sin} .^{2}\beta }{b\operatorname {Sin} .^{2}\gamma -c\operatorname {Sin} .^{2}\beta }}}},\quad e={\sqrt {B.{\frac {c'\operatorname {Sin} .^{2}\alpha -a'\operatorname {Sin} .^{2}\gamma }{c\operatorname {Sin} .^{2}\alpha -a\operatorname {Sin} .^{2}\gamma }}}},$

$f={\sqrt {C.{\frac {a'\operatorname {Sin} .^{2}\beta -b'\operatorname {Sin} .^{2}\alpha }{a\operatorname {Sin} .^{2}\beta -b\operatorname {Sin} .^{2}\alpha }}}}\,;$

ce qui donnera pour la double équation de l’axe de révolution

$(x+y\operatorname {Cos} .\gamma +z\operatorname {Cos} .\beta ){\sqrt {\frac {b\operatorname {Sin} .^{2}\gamma -c\operatorname {Sin} .^{2}\beta }{A\left(b'\operatorname {Sin} .^{2}\gamma -c'\operatorname {Sin} .^{2}\beta \right)}}}=$