Pagina:Werkecarlf02gausrich.djvu/34

$${\begin{aligned}f(r,n-1)&=1+r^{-1}+r^{-3}+r^{-6}+{\text{etc.}}+r^{-{\frac {1}{2}}(n-1)n}\\&=(1-r)(1-r^{3})(1-r^{5})\ldots (1-r^{n-2})\end{aligned}}$$

Eadem aequatio etiamnum valebit, si pro ${\textstyle r}$ substituitur ${\textstyle r^{\lambda }}$, designante ${\textstyle \lambda }$ integrum quemcunque ad ${\textstyle n}$ primum: tunc enim etiam ${\textstyle r^{\lambda }}$ erit radix propria aequationis ${\textstyle x^{n}-1=0}$. Scribamus itaque pro ${\textstyle r}$, ${\textstyle r^{n-2}}$ sive quod idem est ${\textstyle r^{-2}}$, eritque $$1+r^{2}+r^{6}+r^{12}+{\text{etc.}}+r^{(n-1)n}=(1-r^{-2})(1-r^{-6})(1-r^{-10})\ldots (1-r^{-2(n-2)})$$ Multiplicemus utramque partem huius aequationis per $$r\cdot r^{3}\cdot r^{5}\ldots \cdot r^{(n-2)}=r^{{\frac {1}{4}}(n-1)^{2}}$$ prodibitque, propter $${\begin{alignedat}{4}r^{2+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n-3)^{2}},\quad &r^{(n-1)n+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n+1)^{2}}\\r^{6+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n-5)^{2}},\quad &r^{(n-2)(n-1)+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n+3)^{2}}\\r^{12+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n-7)^{2}},\quad &r^{(n-3)(n-2)+{\frac {1}{4}}(n-1)^{2}}&=r^{{\frac {1}{4}}(n+5)^{2}}{\text{ etc.}}\end{alignedat}}$$ aequatio sequens $${\begin{aligned}r^{{\frac {1}{4}}(n-1)^{2}}&+r^{{\frac {1}{4}}(n-3)^{2}}+r^{{\frac {1}{4}}(n-5)^{2}}+{\text{etc.}}+r+1\\+r^{{\frac {1}{4}}(n+1)^{2}}&+r^{{\frac {1}{4}}(n+3)^{2}}+r^{{\frac {1}{4}}(n+5)^{2}}+{\text{etc.}}+r^{{\frac {1}{4}}(2n-2)^{2}}\\&=(r-r^{-1})(r^{3}-r^{-3})(r^{5}-r^{-5})\ldots (r^{n-2}-r^{-n+2})\end{aligned}}$$ aut, partibus membri primi aliter dispositis, $$1+r+r^{4}+{\text{etc.}}+r^{(n-1)^{2}}=(r-r^{-1})(r^{3}-r^{-3})\ldots (r^{n-2}-r^{-n+2})\dots \dots \dots [5]$$

13.

Factores membri secundi aequationis [5] ita quoque exhiberi possunt $${\begin{aligned}r-r^{-1}&=-(r^{n-1}-r^{-n+1})\\r^{3}-r^{-3}&=-(r^{n-3}-r^{-n+3})\\r^{5}-r^{-5}&=-(r^{n-5}-r^{-n+5}){\text{ etc.}}\end{aligned}}$$ usque ad $$r^{n-2}-r^{-n+2}=-(r^{2}-r^{-2}){\phantom {\text{ etc.}}}$$ quo pacto aequatio ista hanc formam assumit: