Pagina:Werkecarlf02gausrich.djvu/45

Statuamus ${\textstyle r^{2^{2\chi -2}}=\rho }$, eritque ${\textstyle \rho }$ radix aequationis ${\textstyle x^{4q}-1=0}$, et quidem ${\textstyle \rho =\cos {\frac {k}{4q}}360^{\circ }+i\sin {\frac {k}{4q}}360^{\circ };}$ dein fiet

$${\begin{aligned}W&=1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(2^{\chi +1}q-1)^{2}}\\&=2^{\chi -1}(1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(4q-1)^{2}})\end{aligned}}$$

Sed summa seriei ${\textstyle 1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(4q-1)^{2}}}$ per ea, quae de casibus ${\textstyle n=4}$, ${\textstyle n=8}$ explicavimus, determinatur, unde colligimus
in casu eo, ubi ${\textstyle q=1}$, sive ubi ${\textstyle n}$ est potestas numeri 4, fieri

$${\begin{aligned}&W=(1+i)2^{\chi }=(1+i)\surd n,{\text{ si fuerit }}k{\text{ formae }}4\mu +1\\&W=(1-i)2^{\chi }=(1-i)\surd n,{\text{ si fuerit }}k{\text{ formae }}4\mu +3\end{aligned}}$$

quae sunt ipsissimae formulae pro ${\textstyle n=4}$ traditae;
in casu eo autem, ubi ${\textstyle q=2}$, sive ubi ${\textstyle n}$ est potestas binarii cum exponente impari maiori quam 3, fieri

$${\begin{alignedat}{2}W&=(1+i)2^{\chi }\surd 2&&=(1+i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +1\\W&=(-1+i)2^{\chi }\surd 2&&=(-1+i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +3\\W&=(-1-i)2^{\chi }\surd 2&&=(-1-i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +5\\W&=(1-i)2^{\chi }\surd 2&&=(1-i)\surd n,{\text{ si fuerit }}k{\text{ formae }}8\mu +7\end{alignedat}}$$

quae quoque prorsus conveniunt cum iis, quae pro ${\textstyle n=8}$ tradidimus.

24.

Etiam hic operae pretium erit, rationem summae progressionis

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

ad ${\textstyle W}$ determinare, ubi ${\textstyle h}$ integrum quemcunque imparem denotat. Quum ${\textstyle W^{\prime }}$ oriatur ex ${\textstyle W}$, mutando ${\textstyle k}$ in ${\textstyle kh}$, valor ipsius ${\textstyle W^{\prime }}$ perinde a forma numeri ${\textstyle kh}$ pendebit, ut ${\textstyle W}$ a forma ipsius ${\textstyle k}$. Statuamus ${\textstyle {\frac {W^{\prime }}{W}}=l}$, patetque fieri

I. in casu eo, ubi ${\textstyle n=4}$, vel altior potestas binarii cum exponente pari, fieri

$${\begin{array}{l}l=1,{\text{ si fuerit }}h{\text{ formae }}4\mu +1\\l=-i,{\text{ si fuerit }}h{\text{ formae }}4\mu +3,{\text{ atque }}k{\text{ formae }}4\mu +1\\l=+i,{\text{ si fuerit }}h{\text{ formae }}4\mu +3,{\text{ atque }}k{\text{ eiusdem formae}}\\\end{array}}$$