Pagina:Werkecarlf02gausrich.djvu/24

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

3.

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

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

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