Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/394

382

SOMMATION

 Les corrections sont expliquées en page de discussion

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

$=\pm {\frac {1}{2^{n}}}\left\{\operatorname {Arc} .\left[\operatorname {Tang} .={\frac {a\operatorname {Sin} .nt}{1+a\operatorname {Cos} .nt}}\right]-{\frac {n}{1}}\operatorname {Arc} .\left[\operatorname {Tang} .={\frac {a\operatorname {Sin} .(n-1)t}{1+a\operatorname {Cos} .(n-1)t}}\right]+\ldots \right\}$

18. Si, par exemple, on suppose $n=2,$ les première, deuxième, quatrième et cinquième formules deviendront, en ayant toujours égard aux limitations prescrites pour les seconds membres,

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

$=+{\frac {1}{2}}\left\{\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(t+u)+a^{2}}}+\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(t-u)+a^{2}}}\right\}.$

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

$=-{\frac {1}{2}}\left\{\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(t+u)+a^{2}}}-\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .(t-u)+a^{2}}}\right\}.$

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

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

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

$=-{\frac {1}{2}}\left\{\operatorname {Log} .{\sqrt {1+2a\operatorname {Cos} .2t+a^{2}}}-\operatorname {Log} .(1+a)\right\}.$

19. Si l’on suppose $n=3,$ les première, troisième, quatrième et sixième formules donneront

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