Pagina:Werkecarlf02gausrich.djvu/49

X. Si, manentibus reliquis ut in IX, ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 8\mu +3}$, erit ${\textstyle L=(-1+i)\surd a}$.

XI. Si manentibus reliquis ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 8\mu +5}$, erit ${\textstyle L=(-1-i)\surd a}$.

XII. Si manentibus reliquis ${\textstyle {\frac {kn}{a}}}$ est formae ${\textstyle 8\mu +7}$, erit ${\textstyle L=(1-i)\surd a}$.

29.

Sit exempli caussa ${\textstyle n=2520=8.9.5.7}$, atque ${\textstyle k=13}$. Hic erit

pro ${\textstyle a=8}$, per casum XII, ${\textstyle L=(1-i)\surd 8}$
pro factore 9, per casum V, summa respondens erit ${\textstyle =\surd 9}$
pro factore 5, per casum II, summa respondens erit ${\textstyle =-\surd 5}$

pro factore 7, per casum III, summa respondens erit ${\textstyle =+i\surd 7}$

Hinc fit ${\textstyle W=(1-i)\cdot (-i)\cdot \surd 2520=(-1-i)\surd 2520}$.

Sit pro eodem valore ipsius ${\textstyle n}$, ${\textstyle k=1}$: tunc respondebit

factori 8 summa ${\textstyle (-1+i)\surd 8}$
factori 9 summa ${\textstyle \surd 9}$
factori 5 summa ${\textstyle \surd 5}$

factori 7 summa ${\textstyle -i\surd 7}$

Hinc conflatur productum ${\textstyle W=(1+i)\surd 2520}$.

30.

Methodus alia, summam ${\textstyle W}$ generaliter determinandi, petitur ex iis, quae in artt. 22, 24 exposita sunt. Statuamus ${\textstyle \cos \omega +i\sin \omega =\rho }$, atque

$$\rho ^{\frac {nn}{aa}}=\alpha ,\quad \rho ^{\frac {nn}{bb}}=\beta ,\quad \rho ^{\frac {nn}{cc}}=\gamma {\text{ etc.}}$$

ita ut habeatur ${\textstyle r=\rho ^{k}}$, ${\textstyle A=\alpha ^{k}}$, ${\textstyle B=\beta ^{k}}$, ${\textstyle C=\gamma ^{k}}$ etc. Tunc erit

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

productum e factoribus

$${\begin{aligned}&1+\alpha +\alpha ^{4}+\alpha ^{9}+{\text{etc.}}+\alpha ^{(a-1)^{2}}\\&1+\beta +\beta ^{4}+\beta ^{9}+{\text{etc.}}+\beta ^{(b-1)^{2}}\\&1+\gamma +\gamma ^{4}+\gamma ^{9}+{\text{etc.}}+\gamma ^{(c-1)^{2}}{\text{ etc.}}\end{aligned}}$$