Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/152

146

CAS DIVERS

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

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

mais, de la valeur de $X^{2}+Y^{2}+Z^{2}$ trouvée ci-dessus, on peut conclure

$\left(X^{2}+Y^{2}+Z^{2}\right)\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c$

$=2\left(\operatorname {Sin} .^{2}{\tfrac {1}{2}}a+\operatorname {Sin} .^{2}{\tfrac {1}{2}}b+\operatorname {Sin} .^{2}{\tfrac {1}{2}}c\right)\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c\,;$

ajoutant à cette expression le double de la précédente, et ayant égard à l’équation (2), il viendra

$R^{2}\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c$

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

ou encore

$R^{2}\operatorname {Cos} .{\tfrac {1}{2}}a\operatorname {Cos} .{\tfrac {1}{2}}b\operatorname {Cos} .{\tfrac {1}{2}}c$

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