Pagina:Werkecarlf02gausrich.djvu/95

$${\begin{array}{c|c}&{\text{denotare multitudinem}}\\{\text{signum}}&{\text{solutionum congruentiae}}\\\hline (00)&1+\alpha +\gamma \equiv 0\\(01)&1+\alpha +\delta \equiv 0\\(02)&1+\alpha +\alpha ^{\prime }\equiv 0\\(03)&1+\alpha +\beta \equiv 0\\(10)&1+\beta +\gamma \equiv 0\\(11)&1+\beta +\delta \equiv 0\\(12)&1+\beta +\alpha \equiv 0\\(13)&1+\beta +\beta ^{\prime }\equiv 0\\(20)&1+\gamma +\gamma ^{\prime }\equiv 0\\(21)&1+\gamma +\delta \equiv 0\\(22)&1+\gamma +\alpha \equiv 0\\(23)&1+\gamma +\beta \equiv 0\\(30)&1+\delta +\gamma \equiv 0\\(31)&1+\delta +\delta ^{\prime }\equiv 0\\(32)&1+\delta +\alpha \equiv 0\\(33)&1+\delta +\beta \equiv 0\\\end{array}}$$

unde statim habentur sex aequationes:

$$(00)=(22),\quad (01)=(32),\quad (03)=(12),\quad (10)=(23),\quad (11)=(33),\quad (21)=(30)$$

Multiplicando congruentiam ${\textstyle 1+\alpha +\gamma \equiv 0}$ per numerum ${\textstyle \gamma ^{\prime }}$ e complexu ${\textstyle C}$ ita electum, ut fiat ${\textstyle \gamma \gamma ^{\prime }\equiv 1}$, accipiendoque pro ${\textstyle \gamma ^{\prime \prime }}$ residuum minimum producti ${\textstyle \alpha \gamma ^{\prime }}$, quod manifesto quoque complexui ${\textstyle C}$ adnumerandum erit, prodit ${\textstyle \gamma ^{\prime }+\gamma ^{\prime \prime }+1\equiv 0}$, unde colligimus ${\textstyle (00)=(20)}$.

Prorsus simili modo habentur aequationes ${\textstyle (01)=(13)}$, ${\textstyle (03)=(31)}$, ${\textstyle (10)=(11)=(21)}$.

Adiumento harum undecim aequationum sedecim incognitas nostras ad quinque reducere, schemaque ${\textstyle S}$ ita exhibere possumus:

$${\begin{array}{llll}h,&i,&k,&l\\m,&m,&l,&i\\h,&m,&h,&m\\m,&l,&i,&m\end{array}}$$