Pagina:Werkecarlf02gausrich.djvu/38

16.

Transimus ad casum alterum, ubi ${\textstyle n}$ est numerus par. Sit primo ${\textstyle n}$ formae ${\textstyle 4\mu +2}$ sive impariter par, patetque, numeros ${\textstyle {\frac {1}{4}}nn}$, ${\textstyle ({\frac {1}{2}}n+1)^{2}-1}$, ${\textstyle ({\frac {1}{2}}n+2)^{2}-4}$ etc. sive generaliter ${\textstyle ({\frac {1}{2}}n+\lambda )^{2}-\lambda \lambda }$ per ${\textstyle {\frac {1}{2}}n}$ divisos producere quotientes impares, adeoque secundum modulum ${\textstyle n}$ congruos fieri ipsi ${\textstyle {\frac {1}{2}}n}$. Hinc colligitur, si ${\textstyle r}$ sit radix propria aequationis ${\textstyle x^{n}-1=0}$, adeoque ${\textstyle r^{{\frac {1}{2}}n}=-1}$, fieri

$${\begin{aligned}r^{({\frac {1}{2}}n)^{2}}&=-1\\r^{({\frac {1}{2}}n+1)^{2}}&=-r\\r^{({\frac {1}{2}}n+2)^{2}}&=-r^{4}\\r^{({\frac {1}{2}}n+3)^{2}}&=-r^{9}{\text{ etc.}}\end{aligned}}$$

Hinc in progressione

$$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$

terminus ${\textstyle r^{({\frac {1}{2}}n)^{2}}}$ destruet primum, sequens secundum etc., adeoque erit

$$W=0,\quad T=0,\quad U=0$$

17.

Superest casus, ubi ${\textstyle n}$ est formae ${\textstyle 4\mu }$ sive pariter par. Hic generaliter ${\textstyle ({\frac {1}{2}}n+\lambda )^{2}-\lambda \lambda }$ divisibilis erit per ${\textstyle n}$, adeoque

$$r^{({\frac {1}{2}}n+\lambda )^{2}}=r^{\lambda \lambda }$$

Hinc in serie

$$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$

terminus ${\textstyle r^{({\frac {1}{2}}n)^{2}}}$ aequalis erit primo, sequens secundo etc., ita ut fiat

$$W=2(1+r+r^{4}+r^{9}+{\text{etc.}}+r^{({\frac {1}{2}}n-1)^{2}})$$

Iam supponamus, in aequ. [7] art. 15 statui ${\textstyle m={\frac {1}{2}}n-1}$, et pro ${\textstyle y}$ accipi radicem propriam aequationis ${\textstyle y^{n}-1=0}$, puta ${\textstyle r}$. Tunc perinde ut in art. 15 aequatio sequentem formam obtinet:

$$1+r+r^{4}+{\text{etc.}}+r^{({\frac {1}{2}}n-1)^{2}}=(1-r^{-1})(1+r^{-2})(1-r^{-3})\ldots (1-r^{-{\frac {1}{2}}n+1})$$