Pagina:Werkecarlf03gausrich.djvu/188

Valor huius functionis pro ${\textstyle t={a}}$ colligitur

$$=(n+1)a^{n}+n\alpha a^{n-1}+(n-1)\alpha ^{\prime }a^{n-2}+(n-2)\alpha ^{\prime \prime }a^{n-3}+{\text{etc.}}+\alpha ^{(n-1)}$$

Hinc ${\textstyle M}$ aequalis valori ipsius ${\textstyle {\frac {\operatorname {d} T}{\operatorname {d} t}}}$ pro ${\textstyle t=a,}$ uti etiam aliunde constat. Perinde ${\textstyle M^{\prime },}$ ${\textstyle M^{\prime \prime },}$ ${\textstyle M^{\prime \prime \prime },}$ etc. erunt valores ipsius ${\textstyle {\frac {\operatorname {d} T}{\operatorname {d} t}}}$ pro ${\textstyle t=a^{\prime },}$ ${\textstyle t=a^{\prime \prime },}$ ${\textstyle t=a^{\prime \prime \prime }}$ etc.

Porro invenimus valorem integralis ${\textstyle \int {\frac {T\operatorname {d} t}{t-a}},}$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1,}$

$${\begin{alignedat}{6}={\frac {1}{n+1}}&+{\frac {a}{n}}&&+{\frac {aa}{n-1}}&&+{\frac {a^{3}}{n-2}}&&+{\text{etc.}}&&+a^{n}\\&+{\frac {\alpha }{n}}&&+{\frac {\alpha a}{n-1}}&&+{\frac {\alpha aa}{n-2}}&&+{\text{etc.}}&&+\alpha a^{n-1}\\&&&+{\frac {\alpha ^{\prime }}{n-1}}&&+{\frac {\alpha ^{\prime }a}{n-2}}&&+{\text{etc.}}&&+\alpha ^{\prime }a^{n-2}\\&&&&&+{\frac {\alpha ^{\prime \prime }}{n-2}}&&+{\text{etc.}}&&+\alpha ^{\prime \prime }a^{n-3}\\&&&&&&&+{\text{etc.}}&&{\text{ etc.}}\\&&&&&&&&&+\alpha ^{(n-1)}\end{alignedat}}$$

quos terminos ordine sequenti disponemus:

$${\begin{aligned}a^{n}&+\alpha a^{n-1}+\alpha ^{\prime }a^{n-2}+\alpha ^{\prime \prime }a^{n-3}+{\text{etc.}}+\alpha ^{(n-1)}\\&+{\frac {1}{2}}(a^{n-1}+\alpha a^{n-2}+\alpha ^{\prime }a^{n-3}+{\text{etc.}}+\alpha ^{(n-2)})\\&+{\frac {1}{3}}(a^{n-2}+\alpha a^{n-3}+\alpha ^{\prime }a^{n-4}+{\text{etc.}}+\alpha ^{(n-3)})\\&+{\frac {1}{4}}(a^{n-3}+\alpha a^{n-4}+\alpha ^{\prime }a^{n-5}+{\text{etc.}}+\alpha ^{(n-4)})\\&+{\text{etc.}}\\&+{\frac {1}{n-1}}(aa+\alpha a+\alpha ^{\prime })\\&+{\frac {1}{n}}(a+\alpha )\\&+{\frac {1}{n+1}}\end{aligned}}$$

Sed manifesto eadem quantitas prodit, si in producto e multiplicatione functionis ${\textstyle T}$ in seriem infinitam

$$t^{-1}+{\frac {1}{2}}t^{-2}+{\frac {1}{3}}t^{-3}+{\frac {1}{4}}t^{-4}{\text{ etc. }}$$

orto, reiectis omnibus terminis, qui implicant potestates ipsius ${\textstyle t}$ exponentibus negativis (sive brevius, in producti parte ea, quae est functio integra ipsius ${\textstyle t}$) pro ${\textstyle t}$ scribitur ${\textstyle a.}$ Supponamus itaque, fieri^[1]

$$T(t^{-1}+{\frac {1}{2}}t^{-2}+{\frac {1}{3}}t^{-3}+{\frac {1}{4}}t^{-4}+{\text{etc.}})=T^{\prime }+T^{\prime \prime }$$

1. Vix opus erit monere, characteres ${\textstyle T,T^{\prime },T^{\prime \prime }}$ alio sensu hic accipi, quam in art. 2.