Pagina:Werkecarlf03gausrich.djvu/187

inde ut in art. 2. concludimus, ${\textstyle Y}$ cum ${\textstyle y}$ identicam esse, quoties ${\textstyle y}$ quoque sit functio algebraica integra ordinem ${\textstyle n}$ non transscendens, aut saltem ipsius ${\textstyle y}$ vice fungi posse, si ${\textstyle y}$ in seriem secundum potestates ipsius ${\textstyle t}$ progredientem conversa tantam convergentiam exhibeat, ut terminos altiorum ordinum negligere liceat.

8.

Iam ad eruendum integrale ${\textstyle \int {Y}\operatorname {d} t}$ singulas partes ipsius ${\textstyle {Y}}$ consideremus. Designemus productum

$$(t-a)(t-a^{\prime })(t-a^{\prime \prime })(t-a^{\prime \prime \prime })\ldots (t-a^{(n)})$$

per ${\textstyle T,}$ fiatque per evolutionem huius producti

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

Numerator fractionis, per quam, in parte prima ipsius ${\textstyle {Y},}$ multiplicata est ${\textstyle A,}$ fit ${\textstyle ={\frac {T}{t-a}};}$ numeratores in partibus sequentibus perinde sunt ${\textstyle {\frac {T}{t-a^{\prime }}},}$ ${\textstyle {\frac {T}{t-a^{\prime \prime }}},}$ ${\textstyle {\frac {T}{t-a^{\prime \prime \prime }}}}$ etc. Denominatores vero nihil aliud sunt, nisi valores determinati horum numeratorum, si resp. statuitur ${\textstyle t=a,}$ ${\textstyle t=a^{\prime },}$ ${\textstyle t=a^{\prime \prime },}$ ${\textstyle t=a^{\prime \prime \prime }}$ etc.: denotemus hos denominatores resp. per ${\textstyle M,}$ ${\textstyle M^{\prime },}$ ${\textstyle M^{\prime \prime },}$ ${\textstyle M^{\prime \prime \prime }}$ etc., ita ut sit

$$Y={\frac {AT}{M(t-a)}}+{\frac {A^{\prime }T}{M^{\prime }(t-a^{\prime })}}+{\frac {A^{\prime \prime }T}{M^{\prime \prime }(t-a^{\prime \prime })}}+{\text{etc.}}+{\frac {A^{(n)}T}{M^{(n)}(t-a^{(n)})}}$$

Quum fiat ${\textstyle T=0,}$ pro ${\textstyle t=a,}$ habemus aequationem identicam

$$a^{n+1}+\alpha a^{n}+\alpha ^{\prime }a^{n-1}+\alpha ^{\prime \prime }a^{n-2}+{\text{etc.}}+\alpha ^{(n)}=0$$

adeoque

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

Hinc dividendo per ${\textstyle t-a}$ fit

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