Pagina:Werkecarlf03gausrich.djvu/195

$${\begin{aligned}&{\frac {R}{t-a}}+{\frac {R^{\prime }}{t-a^{\prime }}}+{\frac {R^{\prime \prime }}{t-a^{\prime \prime }}}+{\text{etc.}}+{\frac {R^{(n)}}{t-a^{(n)}}}\\&\quad =(1-k)t^{-1}+({\frac {1}{2}}-k^{\prime })t^{-2}+({\frac {1}{3}}-k^{\prime \prime })t^{-3}+({\frac {1}{4}}-k^{\prime \prime \prime })t^{-4}+{\text{etc. }}\\&\quad =t^{-1}+{\frac {1}{2}}t^{-2}+{\frac {1}{3}}t^{-3}+{\frac {1}{4}}t^{-4}+{\text{etc.}}-\Theta \end{aligned}}$$

si statuimus

$$\Theta =kt^{-1}+k^{\prime }t^{-2}+k^{\prime \prime }t^{-3}+k^{\prime \prime \prime }t^{-4}+{\text{etc. }}$$

sive potius (quum iam sciamus, ${\textstyle k,}$ ${\textstyle k^{\prime },}$ ${\textstyle k^{\prime \prime },}$ ${\textstyle k^{\prime \prime \prime }}$ etc. usque ad ${\textstyle k^{(n)}}$ sponte evanescere debere)

$$\Theta =k^{(n+1)}t^{-(n+2)}+k^{(n+2)}t^{-(n+3)}+k^{(n+3)}t^{-(n+4)}+{\text{etc.}}$$

Multiplicando per ${\textstyle T}$ fit

$$T({\frac {R}{t-a}}+{\frac {R^{\prime }}{t-a^{\prime }}}+{\frac {R^{\prime \prime }}{t-a^{\prime \prime }}}+{\text{etc.}}+{\frac {R^{(n)}}{t-a^{(n)}}})=T^{\prime }+T^{\prime \prime }-T\Theta$$

Pars prior huius aequationis est functio integra ipsius ${\textstyle t}$ ordinis ${\textstyle n,}$ eiusque valores determinati pro ${\textstyle t=a,}$ ${\textstyle t=a^{\prime },}$ ${\textstyle t=a^{\prime \prime }}$ etc. resp. fiunt ${\textstyle MR,}$ ${\textstyle M^{\prime }R^{\prime },}$ ${\textstyle M^{\prime \prime }R^{\prime \prime }}$ etc.: quapropter, quum eadem valeant de functione ${\textstyle T^{\prime },}$ uti ex ipso modo numeros ${\textstyle R,}$ ${\textstyle R^{\prime },}$ ${\textstyle R^{\prime \prime }}$ etc. determinandi perspicuum est, necessario illa pars prior aequationis identica esse debet cum ${\textstyle T^{\prime },}$ adeoque ${\textstyle T^{\prime \prime }=T\Theta .}$ Oritur itaque ${\textstyle \Theta }$ ex evolutione fractionis ${\textstyle {\frac {T^{\prime \prime }}{T}},}$ quo pacto coëfficientes ${\textstyle k^{(n+1)},}$ ${\textstyle k^{(n+2)}}$ etc. quousque libet determinari poterunt. Quibus inventis correctio valoris nostri approximati integralis ${\textstyle \int y\operatorname {d} t}$ erit

$$=k^{(n+1)}K^{(n+1)}+k^{(n+2)}K^{(n+2)}+{\text{etc. }}$$

si series, in quam evolvitur ${\textstyle y,}$ est

$$y=K+K^{\prime }t+K^{\prime \prime }tt+K^{\prime \prime \prime }t^{3}+{\text{etc. }}$$

14.

Si magis placet, correctionem exprimere per coëfficientes seriei secundum potestates ipsius ${\textstyle t-{\frac {1}{2}}}$ progredientis

$$y=L+L^{\prime }(t-{\frac {1}{2}})+L^{\prime \prime }(t-{\frac {1}{2}})^{2}+L^{\prime \prime \prime }(t-{\frac {1}{2}})^{3}+{\text{etc. }}$$

illa erit