Pagina:Werkecarlf03gausrich.djvu/200

terminum primum esse ${\textstyle ={\frac {V^{\prime }}{W^{\prime }}}}$
summam duorum terminorum primorum ${\textstyle ={\frac {V^{\prime \prime }}{W^{\prime \prime }}}}$
summam trium terminorum primorum ${\textstyle ={\frac {V^{\prime \prime \prime }}{W^{\prime \prime \prime }}}}$
summam quatuor terminorum primorum ${\textstyle ={\frac {V^{\prime \prime \prime \prime }}{W^{\prime \prime \prime \prime }}}}$
et sic porro; quocirca series ipsa vel in infinitum vel usque dum abrumpatur continuata ipsam fractionem continuam ${\textstyle \varphi }$ exprimet. Simul hinc habetur differentia inter ${\textstyle \varphi }$ atque singulas fractiones appropinquantes ${\textstyle {\frac {V^{\prime }}{W^{\prime }}},}$ ${\textstyle {\frac {V^{\prime \prime }}{W^{\prime \prime }}},}$ ${\textstyle {\frac {V^{\prime \prime \prime }}{W^{\prime \prime \prime }}}}$ etc.

E formula 33 art. 14 Disquisitionum generalium circa seriem infinitam mutando ${\textstyle t}$ in ${\textstyle {\frac {1}{u}},}$ facile obtinemus transformationem seriei

$$\varphi =u^{-1}+{\frac {1}{3}}u^{-3}+{\frac {1}{5}}u^{-5}+{\frac {1}{7}}u^{-7}+{\text{etc. }}$$

in fractionem continuam sequentem

$${\cfrac {1}{u-{\cfrac {\frac {1}{3}}{u-{\cfrac {\frac {2.2}{3.5}}{u-{\cfrac {\frac {3.3}{5.7}}{u-{\cfrac {\frac {4.4}{7.9}}{u-{\text{ etc.}}}}}}}}}}}}$$

ita ut habeatur

$${\begin{aligned}&v=1,v^{\prime }=-{\frac {1}{3}},v^{\prime \prime }=-{\frac {4}{15}},v^{\prime \prime \prime }=-{\frac {9}{35}},v^{\prime \prime \prime \prime }=-{\frac {16}{63}}{\text{ etc. }}\\&w=w^{\prime }=w^{\prime \prime }=w^{\prime \prime \prime }=w^{\prime \prime \prime \prime }{\text{ etc.}}=u.\end{aligned}}$$

Hinc pro ${\textstyle V,}$ ${\textstyle V^{\prime },}$ ${\textstyle V^{\prime \prime },}$ ${\textstyle V^{\prime \prime \prime }}$ etc. ${\textstyle W,}$ ${\textstyle W^{\prime },}$ ${\textstyle W^{\prime \prime },}$ ${\textstyle W^{\prime \prime \prime }}$ etc. nanciscimur valores sequentes

$${\begin{aligned}&V=0&&W=1\\&V^{\prime }=1&&W^{\prime }=u\\&V^{\prime \prime }=u&&W^{\prime \prime }=uu-{\frac {1}{3}}\\&V^{\prime \prime \prime }=uu-{\frac {4}{15}},&&W^{\prime \prime \prime }=u^{3}-{\frac {3}{5}}u\\&V^{\prime \prime \prime \prime }=u^{3}-{\frac {11}{21}}u,&&W^{\prime \prime \prime \prime }=u^{4}-{\frac {6}{7}}uu+{\frac {3}{35}}\\&V^{\scriptscriptstyle V}=u^{4}-{\frac {7}{9}}uu+{\frac {64}{945}}&&W^{\scriptscriptstyle V}=u^{5}-{\frac {10}{9}}u^{3}+{\frac {5}{21}}u\\&V^{\scriptscriptstyle VI}=u^{5}-{\frac {34}{3}}{\frac {4}{3}}u^{3}+{\frac {1}{5}}u,&&W^{\scriptscriptstyle VI}=u^{6}-{\frac {15}{11}}u^{4}+{\frac {5}{11}}uu-{\frac {5}{231}}\\&V^{\scriptscriptstyle VII}=u^{6}-{\frac {50}{39}}u^{4}+{\frac {283}{715}}uu-{\frac {256}{15015}},&&W^{\scriptscriptstyle VII}=u^{7}-{\frac {21}{13}}u^{5}+{\frac {105}{143}}u^{3}-{\frac {35}{429}}u{\text{ etc }}\end{aligned}}$$