Pagina:Werkecarlf02gausrich.djvu/34

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

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

13.

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