Pagina:Werkecarlf03gausrich.djvu/199

17.

Proposita fractione continua

$$\varphi ={\tfrac {v}{w+{\tfrac {v^{\prime }}{w^{\prime }+{\tfrac {v^{\prime \prime }}{w^{\prime \prime }+{\tfrac {v^{\prime \prime \prime }}{w^{\prime \prime \prime }+{\text{ etc.}}}}}}}}}}$$

constat, fractiones continuo magis appropinquantes inveniri per algorithmum sequentem. Formentur duae quantitatum series, ${\textstyle V,}$ ${\textstyle V^{\prime },}$ ${\textstyle V^{\prime \prime },}$ ${\textstyle V^{\prime \prime \prime }}$ etc., ${\textstyle W,}$ ${\textstyle W^{\prime },}$ ${\textstyle W^{\prime \prime },}$ ${\textstyle W^{\prime \prime \prime }}$ etc. per hasce formulas

$${\begin{array}{ll}V=0&\quad W=1\\V^{\prime }=v&\quad W^{\prime }=wW\\V^{\prime \prime }=w^{\prime }V^{\prime }+v^{\prime }V&\quad W^{\prime \prime }=w^{\prime }W^{\prime }+v^{\prime }W\\V^{\prime \prime \prime }=w^{\prime \prime }V^{\prime \prime }+v^{\prime \prime }V^{\prime }&\quad W^{\prime \prime \prime }=w^{\prime \prime }W^{\prime \prime }+v^{\prime \prime }W^{\prime }\\V^{\prime \prime \prime }=w^{\prime \prime \prime }V^{\prime \prime \prime }+v^{\prime \prime \prime }V^{\prime \prime }&\quad W^{\prime \prime \prime \prime }=w^{\prime \prime \prime }W^{\prime \prime \prime }+v^{\prime \prime \prime }W^{\prime \prime }\end{array}}$$

etc. eritque

$${\begin{aligned}&{\frac {V}{W}}=0\\&{\frac {V^{\prime }}{W^{\prime }}}={\frac {v}{w}}\\&{\frac {V^{\prime \prime }}{W^{\prime \prime }}}={\tfrac {v}{w+{\tfrac {v^{\prime }}{w^{\prime }}}}}\\&{\frac {V^{\prime \prime \prime }}{W^{\prime \prime \prime }}}={\tfrac {v}{w+{\tfrac {v^{\prime }}{w^{\prime }+{\tfrac {v^{\prime \prime }}{w^{\prime \prime }}}}}}}\end{aligned}}$$

et sic porro. Praeterea constat, vel facile ex ipsis aequationibus praecedentibus confirmatur, esse

$${\begin{alignedat}{3}&VW^{\prime }&&-V^{\prime }W&&=-v\\&V^{\prime }W^{\prime \prime }&&-V^{\prime \prime }W^{\prime }&&=+vv^{\prime }\\&V^{\prime \prime }W^{\prime \prime \prime }&&-V^{\prime \prime \prime }W^{\prime \prime }&&=-vv^{\prime }v^{\prime \prime }\\&V^{\prime \prime \prime }W^{\prime \prime \prime \prime }&&-V^{\prime \prime \prime \prime }W^{\prime \prime \prime }&&=+vv^{\prime }v^{\prime \prime }v^{\prime \prime \prime }\end{alignedat}}$$

etc. Hinc perspicuum est, seriei

$${\frac {v}{WW^{\prime }}}-{\frac {vv^{\prime }}{W^{\prime }W^{\prime \prime }}}+{\frac {vv^{\prime }v^{\prime \prime }}{W^{\prime \prime }W^{\prime \prime \prime }}}-{\frac {vv^{\prime }v^{\prime \prime }v^{\prime \prime \prime }}{W^{\prime \prime \prime }W^{\prime \prime \prime \prime }}}+{\text{etc. }}$$