Pagina:Werkecarlf02gausrich.djvu/36

$${\begin{aligned}r-r^{-1}&=2i\sin \omega \\r^{3}-r^{-3}&=2i\sin 3\omega \\r^{5}-r^{-5}&=2i\sin 5\omega {\text{ etc., }}\end{aligned}}$$

aequatio ista transmutatur in

$$W=(2i)^{{\frac {1}{2}}(n-1)}\sin \omega \sin 3\omega \sin 5\omega \ldots \sin(n-2)\omega$$

Iam in casu eo, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, in serie numerorum imparium

$$1,3,5,7\ldots {\frac {1}{2}}(n-3),{\frac {1}{2}}(n+1)\ldots (n-2)$$

reperiuntur ${\textstyle {\frac {1}{4}}(n-1)}$, qui sunt minores quam ${\textstyle {\frac {1}{2}}n}$, hisque manifesto respondent sinus positivi; contra reliqui ${\textstyle {\frac {1}{4}}(n-1)}$ erunt maiores quam ${\textstyle {\frac {1}{2}}n}$, hisque sinus negativi respondebunt: quapropter productum omnium sinuum statuendum est aequale producto e quantitate positiva in multiplicatorem ${\textstyle (-1)^{{\frac {1}{4}}(n-1)}}$, adeoque ${\textstyle W}$ aequalis erit producto e quantitate reali positiva in ${\textstyle i^{n-1}}$ sive in 1, quoniam ${\textstyle i^{4}=1}$, atque ${\textstyle n-1}$ per 4 divisibilis: i.e. quantitas ${\textstyle W}$ erit realis positiva, unde necessario esse debebit

$$W=+\surd n,\quad T=+\surd n$$

In casu altero, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$ in serie numerorum imparium

$$1,3,5,7\ldots {\frac {1}{2}}(n-1),{\frac {1}{2}}(n+3)\ldots (n-2)$$

priores ${\textstyle {\frac {1}{4}}(n+1)}$ erunt minores quam ${\textstyle {\frac {1}{2}}n}$, reliqui ${\textstyle {\frac {1}{4}}(n-3)}$ autem maiores. Hinc inter sinus arcuum ${\textstyle \omega ,3\omega ,5\omega \ldots (n-2)\omega }$ negativi erunt ${\textstyle {\frac {1}{4}}(n-3)}$, adeoque ${\textstyle W}$ erit productum ex ${\textstyle i^{{\frac {1}{2}}(n-1)}}$ in quantitatem realem positivam in ${\textstyle (-1)^{{\frac {1}{4}}(n-3)}}$; factor tertius est ${\textstyle =i^{{\frac {1}{2}}(n-3)}}$, qui cum primo iunctus producit ${\textstyle i^{n-2}=i}$, quoniam ${\textstyle i^{n-3}=1}$. Quamobrem necessario erit

$$W=+i\surd n,{\text{ atque }}U=+\surd n$$

15.

Iam ostendemus, quo pacto eaedem conclusiones e progressione in art. 9 considerata deduci possint. Scribamus in aequ. [4] pro ${\textstyle x^{\frac {1}{2}}}$, ${\textstyle -y^{-1}}$, eritque