Pagina:Werkecarlf02gausrich.djvu/24

unde patet, quomodo summationum $${\begin{aligned}&2\Sigma \sin ak\omega =\pm \surd n\\&2\Sigma \sin bk\omega =\mp \surd n\end{aligned}}$$ altera ab altera pendeat.

3.

Ecce iam computum numericum pro aliquot exemplis:
I. Pro ${\textstyle n=5}$ adest valor unus ipsius ${\textstyle a^{\prime }}$, puta ${\textstyle a^{\prime }=1}$, valorque unus ipsius ${\textstyle b^{\prime }}$, puta ${\textstyle b^{\prime }=2}$; est autem $$\cos \omega =+0{,}3090169944\qquad \qquad \cos 2\omega =-0{,}8090169944$$ adeoque ${\textstyle 1+4\cos \omega =+\surd 5}$, ${\textstyle 1+4\cos 2\omega =-\surd 5}$.
II. Pro ${\textstyle n=13}$ adsunt tres valores ipsius ${\textstyle a^{\prime }}$, puta ${\textstyle 1}$, ${\textstyle 3}$, ${\textstyle 4}$, totidemque valores ipsius ${\textstyle b^{\prime }}$, puta ${\textstyle 2}$, ${\textstyle 5}$, ${\textstyle 6}$, unde computamus $${\begin{array}{r}\cos \omega {\phantom {1}}=+0{,}8854560257\\\cos 3\omega =+0{,}1205366803\\\cos 4\omega =-0{,}3546048870\\\hline {\text{Summa}}=+0{,}6513878190\end{array}}\qquad \qquad {\begin{array}{r}\cos 2\omega =+0{,}5680647467\\\cos 5\omega =-0{,}7485107482\\\cos 6\omega =-0{,}9709418174\\\hline {\text{Summa}}=-1{,}1513878189\end{array}}$$

Hinc ${\textstyle 1+4\sum \cos a^{\prime }\omega =+\surd 13}$, ${\textstyle 1+4\Sigma \cos b^{\prime }\omega =-\surd 13}$.
III. Pro ${\textstyle n=17}$ habemus quatuor valores ipsius ${\textstyle a^{\prime }}$, puta ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 4}$, ${\textstyle 8}$, totidemque valores ipsius ${\textstyle b^{\prime }}$, puta ${\textstyle 3}$, ${\textstyle 5}$, ${\textstyle 6}$, ${\textstyle 7}$. Hinc computantur cosinus $${\begin{array}{r}\cos \omega {\phantom {1}}=+0{,}9324722294\\\cos 2\omega =+0{,}7390089172\\\cos 4\omega =+0{,}0922683595\\\cos 8\omega =-0{,}9829730997\\\hline {\text{Summa}}=+0{,}7807764064\end{array}}\qquad \qquad {\begin{array}{r}\cos 3\omega =+0{,}4457383558\\\cos 5\omega =-0{,}2736629901\\\cos 6\omega =-0{,}6026346364\\\cos 7\omega =-0{,}8502171357\\\hline {\text{Summa}}=-1{,}2807764065\end{array}}$$

Hinc ${\textstyle 1+4\Sigma \cos a^{\prime }\omega =+\surd 17,1+4\Sigma \cos b^{\prime }\omega =-\surd 17}$.