Pagina:Werkecarlf03gausrich.djvu/151

[35]	$t={\cfrac {\sin t\cos t}{1-{\cfrac {{\frac {1.2}{1.3}}{\sin t}^{2}}{1-{\cfrac {{\frac {1.2}{3.5}}{\sin t}^{2}}{1-{\cfrac {{\frac {3.4}{5.7}}{\sin t}^{2}}{1-{\cfrac {{\frac {3.4}{7.9}}{\sin t}^{2}}{1-{\cfrac {{\frac {5.6}{9.11}}{\sin t}^{2}}{1-{\text{etc.}}}}}}}}}}}}}}$

Statuendo $\alpha =3,$ $\gamma ={\tfrac {5}{2}},$ e formula 30 sponte sequitur fractio continua in Theoria motus corporum coelestium art. 90 proposita. Ibidem duae aliae fractiones continuae prolatae sunt, quarum evolutionem hacce occasione supplere visum est. Statuendo

$Q=1-{\cfrac {{\frac {5.8}{7.9}}x}{1-{\cfrac {{\frac {1.4}{9.11}}x}{1-{\frac {7.10}{11.13}}x{\text{ etc.}}}}}}$

fit l.c. $x-\xi ={\frac {x}{1+{\frac {2x}{35Q}}}}={\frac {xQ}{Q+{\frac {2}{35}}x}},$ adeoque

$\xi ={\frac {{\frac {2}{35}}xx}{Q+{\frac {2}{35}}x}}$

quae est formula prior: posterior sequenti modo eruitur. Statuendo

$R=1-{\cfrac {{\frac {1.4}{7.9}}x}{1-{\cfrac {{\frac {5.8}{9.11}}x}{1-{\cfrac {{\frac {3.6}{11.13}}x}{1-{\frac {7.10}{13.15}}x{\text{ etc.}}}}}}}}$

erit per formulam 25

${\frac {1}{R}}=G({\tfrac {1}{2}},{\tfrac {3}{2}},{\tfrac {7}{2}},x),\quad$ atque $\quad {\frac {1}{Q}}=G({\tfrac {5}{2}},-{\tfrac {1}{2}},{\tfrac {7}{2}},x)$

Hinc