Pagina:Werkecarlf02gausrich.djvu/68

$${\begin{aligned}&x^{q\alpha ^{p-\mu -2}}-x^{\alpha ^{p-2}}\\&x^{q\alpha ^{p-\mu -1}}-x\\&x^{q\alpha ^{p-\mu }}-x^{\alpha }\\&x^{q\alpha ^{p-\mu +1}}-x^{\alpha \alpha }\\&\quad \vdots \\&x^{q\alpha ^{p-2}}-x^{\alpha ^{\mu -1}}\end{aligned}}$$

per ${\textstyle 1-x^{p}}$ divisibiles. Quibus quantitatibus, alternis vicibus positive et negative sumtis atque summatis, patet, per ${\textstyle 1-x^{p}}$ divisibilem esse functionem

$$x^{q}-x^{q\alpha }+x^{q\alpha \alpha }-x^{q\alpha \alpha ^{3}}+{\text{etc.}}-x^{q\alpha ^{p-2}}\mp \xi$$

valente signo superiori vel inferiori, prout ${\textstyle \mu }$ par sit vel impar, i.e. prout ${\textstyle q}$ sit residuum quadraticum ipsius ${\textstyle p}$ vel non-residuum. Statuemus itaque

$$x^{q}-x^{q\alpha }+x^{q\alpha }-x^{q\alpha ^{3}}+{\text{etc.}}-x^{q\alpha ^{p-2}}-\gamma \xi =(1-x^{p})W$$

faciendo ${\textstyle \gamma =+1}$, vel ${\textstyle \gamma =-1}$, prout ${\textstyle q}$ est residuum quadraticum ipsius ${\textstyle p}$ vel non-residuum, patetque, ${\textstyle W}$ fieri functionem integram.

6.

His ita praeparatis, e combinatione aequationum praecedentium deducimus

$$q\xi {X}=\varepsilon p(\delta p^{{\frac {1}{2}}(q-1)}-\gamma )+{\frac {1-x^{p}}{1-x}}\cdot (Z(\delta p^{{\frac {1}{2}}(q-1)}-\gamma )+Y\xi \xi -W\xi (1-x))$$

Supponamus, ex divisione functionis ${\textstyle \xi X}$ per

$$x^{p-1}+x^{p-2}+x^{p-3}+{\text{etc.}}+x+1$$

oriri quotientem ${\textstyle U}$ cum residuo ${\textstyle T}$, sive haberi

$$\xi X={\frac {1-x^{p}}{1-x}}\cdot U+T$$

ita ut ${\textstyle U}$, ${\textstyle T}$ sint functiones integrae, etiam respectu coëfficientium numericorum, et quidem ${\textstyle T}$ ordinis certe inferioris, quam divisor. Erit itaque

$$qT-\varepsilon p(\delta p^{{\frac {1}{2}}(q-1)}-\gamma )={\frac {1-x^{p}}{1-x}}\cdot (Z(\delta p^{{\frac {1}{2}}(q-1)}-\gamma )+Y\xi \xi -W\xi (1-x)-qU)$$

quae aequatio manifesto subsistere nequit, nisi tum membrum a laeva tum membrum a dextra per se evanescat. Erit itaque ${\textstyle \varepsilon p(\delta p^{{\frac {1}{2}}(q-1)}-\gamma )}$ per ${\textstyle q}$ divisibi-