Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/151

145

D’ATTRACTION.

${\begin{aligned}&={\frac {2\operatorname {Sin} .{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}c\left(\operatorname {Cos} .^{2}{\tfrac {1}{2}}b+\operatorname {Cos} .^{2}{\tfrac {1}{2}}c-\operatorname {Sin} .^{2}{\tfrac {1}{2}}a\right)}{\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c}}\\\\&={\frac {2\operatorname {Sin} .{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}a\left(\operatorname {Cos} .^{2}{\tfrac {1}{2}}c+\operatorname {Cos} .^{2}{\tfrac {1}{2}}a-\operatorname {Sin} .^{2}{\tfrac {1}{2}}b\right)}{\operatorname {Cos} .{\tfrac {1}{2}}c\operatorname {Cos} .{\tfrac {1}{2}}a}}\\\\&={\frac {2\operatorname {Sin} .{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}b\left(\operatorname {Cos} .^{2}{\tfrac {1}{2}}a+\operatorname {Cos} .^{2}{\tfrac {1}{2}}b-\operatorname {Sin} .^{2}{\tfrac {1}{2}}c\right)}{\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b}}\,;\end{aligned}}$

ou, en réduisant au même dénominateur,

$(YZ\operatorname {Cos} .\alpha +ZX\operatorname {Cos} .\beta +XY\operatorname {Cos} .\gamma )\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c$

${\begin{aligned}&=\operatorname {Sin} .{\tfrac {1}{2}}a\operatorname {Cos} .^{2}{\tfrac {1}{2}}a\left(\operatorname {Sin} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c+\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}c\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad -\operatorname {Sin} .^{2}{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}c\\\\&+\operatorname {Sin} .{\tfrac {1}{2}}b\operatorname {Cos} .^{2}{\tfrac {1}{2}}b\left(\operatorname {Sin} .{\tfrac {1}{2}}c\operatorname {Cos} .{\tfrac {1}{2}}a+\operatorname {Cos} .{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}a\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad -\operatorname {Sin} .^{2}{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}a\\\\&+\operatorname {Sin} .{\tfrac {1}{2}}c\operatorname {Cos} .^{2}{\tfrac {1}{2}}c\left(\operatorname {Sin} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b+\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}b\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad -\operatorname {Sin} .^{2}{\tfrac {1}{2}}c\operatorname {Cos} .{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}b\,;\end{aligned}}$

ou encore

$(YZ\operatorname {Cos} .\alpha +ZX\operatorname {Cos} .\beta +XY\operatorname {Cos} .\gamma )\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c$

${\begin{aligned}&=\operatorname {Sin} .{\tfrac {1}{2}}a\operatorname {Cos} .^{2}{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}(b+c)-\operatorname {Sin} .^{2}{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}c\\\\&+\operatorname {Sin} .{\tfrac {1}{2}}b\operatorname {Cos} .^{2}{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}(c+a)-\operatorname {Sin} .^{2}{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Sin} .{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}a\\\\&+\operatorname {Sin} .{\tfrac {1}{2}}c\operatorname {Cos} .^{2}{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}(a+b)-\operatorname {Sin} .^{2}{\tfrac {1}{2}}c\operatorname {Cos} .{\tfrac {1}{2}}c\operatorname {Sin} .{\tfrac {1}{2}}a\operatorname {Sin} .{\tfrac {1}{2}}b\,;\end{aligned}}$

ou enfin