Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/395

383

DES SUITES.

$=+{\frac {1}{4}}\left\{{\begin{aligned}&\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(t+u+v)+a^{2}}}\\+&\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(t+u-v)+a^{2}}}\\+&\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(u+v-t)+a^{2}}}\\+&\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(v+t-u)+a^{2}}}\end{aligned}}\right\}.$

2.^o ${\frac {a\operatorname {Sin} .t\operatorname {Sin} .u\operatorname {Sin} .v}{1}}-{\frac {a^{2}\operatorname {Sin} .2t\operatorname {Sin} .2u\operatorname {Sin} .2v}{2}}+{\frac {a^{3}\operatorname {Sin} .3t\operatorname {Sin} .3u\operatorname {Sin} .3v}{3}}-\ldots$

$=-{\frac {1}{4}}\left\{{\begin{aligned}&\operatorname {Arc} .\left[\operatorname {Tang} .={\frac {a\operatorname {Sin} .(t+u+v)}{1+a\operatorname {Cos} .(t+u+v)}}\right]\\-&\operatorname {Arc} .\left[\operatorname {Tang} .={\frac {a\operatorname {Sin} .(t+u-v)}{1+a\operatorname {Cos} .(t+u-v)}}\right]\\-&\operatorname {Arc} .\left[\operatorname {Tang} .={\frac {a\operatorname {Sin} .(u+v-t)}{1+a\operatorname {Cos} .(u+v-t)}}\right]\\-&\operatorname {Arc} .\left[\operatorname {Tang} .={\frac {a\operatorname {Sin} .(v+t-u)}{1+a\operatorname {Cos} .(v+t-u)}}\right]\end{aligned}}\right\}.$

3.^o ${\frac {a\operatorname {Cos} .^{3}t}{1}}-{\frac {a^{2}\operatorname {Cos} .^{3}2t}{2}}+{\frac {a^{3}\operatorname {Cos} .^{3}3t}{3}}-{\frac {a^{4}\operatorname {Cos} .^{4}4t}{4}}+\ldots$

$=+{\frac {1}{4}}\left\{\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .3t+a^{2}}}+3\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .t+a^{2}}}\right\}.$

4.^o ${\frac {a\operatorname {Sin} .^{3}t}{1}}-{\frac {a^{2}\operatorname {Sin} .^{3}2t}{2}}+{\frac {a^{3}\operatorname {Sin} .^{3}3t}{3}}-{\frac {a^{4}\operatorname {Sin} .^{4}4t}{4}}+\ldots$

$=-{\frac {1}{4}}\left\{\operatorname {Arc} .\left(\operatorname {Tang} .={\frac {a\operatorname {Sin} .3t}{1+a\operatorname {Cos} .3t}}\right)-3\operatorname {Arc} .\left(\operatorname {Tang} .={\frac {a\operatorname {Sin} .t}{1+a\operatorname {Cos} .t}}\right)\right\}.$

20. Nous ne pousserons pas plus loin, pour le moment, ces applications qui n’ont, comme l’on voit, rien de bien difficile, et qui conduisent à des résultats très-remarquables. Il nous suf-