Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/375

363

DES SUITES.

${\begin{aligned}{\text{Pour trois,}}\;4\operatorname {Cos} .t\operatorname {Cos} .u\operatorname {Cos} .v=\operatorname {Cos} .s&+\operatorname {Cos} .(s-2t),\\&+\operatorname {Cos} .(s-2u)\\&+\operatorname {Cos} .(s-2v)\end{aligned}}$

${\begin{aligned}{\text{Pour quatre,}}\;8\operatorname {Cos} .t\operatorname {Cos} .u\operatorname {Cos} .v\operatorname {Cos} .x&=\\\operatorname {Cos} .s+\operatorname {Cos} .(s-2t)&+{\frac {1}{2}}\operatorname {Cos} .\left[s-2(t+u)\right]\\+\operatorname {Cos} .(s-2u)&+{\frac {1}{2}}\operatorname {Cos} .\left[s-2(t+v)\right]\\+\operatorname {Cos} .(s-2v)&+{\frac {1}{2}}\operatorname {Cos} .\left[s-2(u+v)\right]\\+\operatorname {Cos} .(s-2x)&+{\frac {1}{2}}\operatorname {Cos} .\left[s-2(t+x)\right]\\&+{\frac {1}{2}}\operatorname {Cos} .\left[s-2(u+x)\right]\\&+{\frac {1}{2}}\operatorname {Cos} .\left[s-2(v+x)\right]\end{aligned}}$

et ainsi de suite.

3. Si, dans les résultats auxquels nous venons de parvenir, on change respectivement $t,u,v,x$ en ${\frac {1}{2}}\varpi -t,\ {\frac {1}{2}}\varpi -u,\ {\frac {1}{2}}\varpi -v,\ {\frac {1}{2}}\varpi -x\,;$ il viendra

${\begin{aligned}{\text{Pour un arc}},&\operatorname {Sin} .t=+\operatorname {Sin} .s,\\{\text{Pour deux,}}\;2\operatorname {Sin} .t\operatorname {Sin} .u=-\operatorname {Cos} .s&+{\frac {1}{2}}\operatorname {Cos} .(s-2t),\\&+{\frac {1}{2}}\operatorname {Cos} .(s-2u)\\{\text{Pour trois,}}\;4\operatorname {Sin} .t\operatorname {Sin} .u\operatorname {Sin} .v=-\operatorname {Sin} .s&+\operatorname {Sin} .(s-2t),\\&+\operatorname {Sin} .(s-2u)\\&+\operatorname {Sin} .(s-2v)\end{aligned}}$