Pagina:Werkecarlf03gausrich.djvu/187

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

8.

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

$$(t-a)(t-a^{\prime })(t-a^{″})(t-a^{‴})\dots (t-a^{(n)})$$

per $T,$ fiatque per evolutionem huius producti

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

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

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

Quum fiat $T=0,$ pro $t=a,$ habemus aequationem identicam

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

adeoque

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

Hinc dividendo per $t-a$ fit

$$\begin{array}{cccccccccc}\frac{T}{t-a}=t^n& +a{t}^{n-1}& & +aa{t}^{n-2}& & +a^3{t}^{n-3}& & +\text{etc.}& & +a^n\\ & +\alpha t^{n-1}& & +\alpha a{t}^{(n-2)}& & +\alpha aa{t}^{n-3}& & +\text{etc.}& & +\alpha a^{n-1}\\ & & & +{\alpha }^{\prime }{t}^{n-2}& & +{\alpha }^{\prime }a{t}^{n-3}& & +\text{etc.}& & +{\alpha }^{\prime }{a}^{n-2}\\ & & & & & +{\alpha }^{″}{t}^{n-3^{-}}& & +\text{etc.}& & +{\alpha }^{″}{a}^{n-3}\\ & & & & & & & +\text{etc.}& & \text{ etc. }\\ & & & & & & & & & +{\alpha }^{(n-1)}\end{array}$$