Pagina:Werkecarlf03gausrich.djvu/188

Valor huius functionis pro $t=a$ 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 $M$ aequalis valori ipsius $\frac{\mathrm{d}T}{\mathrm{d}t}$ pro $t=a,$ uti etiam aliunde constat. Perinde $M^{\prime },$ $M^{\prime \prime },$ $M^{\prime \prime \prime },$ etc. erunt valores ipsius $\frac{\mathrm{d}T}{\mathrm{d}t}$ pro $t=a^{\prime },$ $t=a^{\prime \prime },$ $t=a^{\prime \prime \prime }$ etc.

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

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

quos terminos ordine sequenti disponemus:

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

Sed manifesto eadem quantitas prodit, si in producto e multiplicatione functionis $T$ 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 $t$ exponentibus negativis (sive brevius, in producti parte ea, quae est functio integra ipsius $t$) pro $t$ scribitur $a.$ 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 $T,T^{\prime },T^{\prime \prime }$ alio sensu hic accipi, quam in art. 2.