Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/380

${\begin{array}{ll}\operatorname {Cos} .(Q-Q')=\operatorname {Cos} .(P+P'),\qquad &\operatorname {Sin} .(Q-Q')=\operatorname {Sin} .(P+P'),\\\operatorname {Cos} .(R-R')\,=\operatorname {Cos} .(Q+Q'),\qquad &\operatorname {Sin} .(R-R')\,=\operatorname {Sin} .(Q+Q'),\\\operatorname {Cos} .(S-S')\ \ =\operatorname {Cos} .(R+R'),\qquad &\operatorname {Sin} .(S-S')\ \ =\operatorname {Sin} .(R+R').\end{array}}$

Ajoutant les équations dans les deux séries et transposant, il viendra

$\left.{\begin{aligned}&\operatorname {Cos} .(A-A')+\operatorname {Cos} .(B-B')+\operatorname {Cos} .(C-C')+\ldots \\&\qquad \qquad +\operatorname {Cos} .(Q-Q')+\operatorname {Cos} .(R-R')+\operatorname {Cos} .(S-S')\\-&\operatorname {Cos} .(A+A')-\operatorname {Cos} .(B+B')-\operatorname {Cos} .(C+C')-\ldots \\&\qquad \qquad -\operatorname {Cos} .(Q+Q')-\operatorname {Cos} .(R+R')-\operatorname {Cos} .(S+S')\end{aligned}}\right\}=0.$ (9)

$\left.{\begin{aligned}&\operatorname {Sin} .(A+A')+\operatorname {Sin} .(B+B')+\operatorname {Sin} .(C+C')+\ldots \\&\qquad \qquad +\operatorname {Sin} .(Q+Q')+\operatorname {Sin} .(R+R')+\operatorname {Sin} .(S+S')\\-&\operatorname {Sin} .(A-A')-\operatorname {Sin} .(B-B')-\operatorname {Sin} .(C-C')-\ldots \\&\qquad \qquad -\operatorname {Sin} .(Q-Q')-\operatorname {Sin} .(R-R')-\operatorname {Sin} .(S-S')\end{aligned}}\right\}=0.$ (10)

En faisant toujours usage des deux théorèmes

${\begin{aligned}&2\operatorname {Sin} .x\operatorname {Sin} .y=\operatorname {Cos} .(x-y)-\operatorname {Cos} .(x+y),\\&2\operatorname {Cos} .x\operatorname {Sin} .y=\operatorname {Sin} .(x+y)-\operatorname {Sin} .(x-y),\end{aligned}}$

on trouvera

${\begin{aligned}&2\operatorname {Sin} .A\operatorname {Sin} .A'=\operatorname {Cos} .(A-A')-\operatorname {Cos} .(A+A'),\\&2\operatorname {Sin} .B\operatorname {Sin} .B'=\operatorname {Cos} .(B-B')-\operatorname {Cos} .(B+B'),\\&2\operatorname {Sin} .C\operatorname {Sin} .C'=\operatorname {Cos} .(C-C')-\operatorname {Cos} .(C+C'),\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&2\operatorname {Sin} .Q\operatorname {Sin} .Q'=\operatorname {Cos} .(Q-Q')-\operatorname {Cos} .(Q+Q'),\\&2\operatorname {Sin} .R\operatorname {Sin} .R'=\operatorname {Cos} .(R-R')-\operatorname {Cos} .(R+R'),\\&2\operatorname {Sin} .S\operatorname {Sin} .S'=\operatorname {Cos} .(S-S')-\operatorname {Cos} .(S+S')\,;\end{aligned}}$