Pagina:Werkecarlf02gausrich.djvu/96

Porro habemus aequationes

$${\begin{aligned}&(00)+(01)+(02)+(03)=2n+1\\&(10)+(11)+(12)+(13)=2n+1\\&(20)+(21)+(22)+(23)=2n\\&(30)+(31)+(32)+(33)=2n+1\end{aligned}}$$

sive, adhibendo signa modo introducta, has tres (I):

$${\begin{aligned}h+i+k+l&=2n+1\\2m+i+l&=2n+1\\h+m&=n\end{aligned}}$$

quarum itaque adiumento incognitas nostras iam ad duas reducere licet.

Aequationes reliquas e consideratione multitudinis solutionum congruentiae ${\textstyle 1+\alpha +\beta +\gamma \equiv 0}$ derivabimus (per ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$, etiam hic indefinite numeros e complexibus ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$ resp. denotantes). Scilicet perpendendo primo, ${\textstyle 1+\alpha }$ praebere ${\textstyle h}$, ${\textstyle i}$, ${\textstyle k}$, ${\textstyle l}$ numeros resp. ad ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$, ${\textstyle D}$ pertinentes, et pro quovis valore dato ipsius ${\textstyle \alpha }$ in his quatuor casibus resp. haberi solutiones ${\textstyle m}$, ${\textstyle l}$, ${\textstyle i}$, ${\textstyle m}$, multitudo omnium solutionum erit

$$=hm+il+ik+lm$$

Secundo quum ${\textstyle 1+\beta }$ exhibeat ${\textstyle m}$, ${\textstyle m}$, ${\textstyle l}$, ${\textstyle i}$ numeros ad ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$, ${\textstyle D}$ pertinentes, et pro quovis valore dato ipsius ${\textstyle \beta }$ in his quatuor casibus exstent solutiones ${\textstyle h}$, ${\textstyle m}$, ${\textstyle h}$, ${\textstyle m}$, multitudo omnium solutionum erit

$$=hm+mm+hl+im$$

unde derivamus aequationem

$$0=mm+hl+im-il-ik-lm$$

quae adiumento aequationis ${\textstyle k=2m-h}$, ex (I) petitae, transit in hanc:

$$0=mm+hl+hi-il-im-lm$$

Iam ex aequationibus I habemus etiam ${\textstyle l+i=1+2h}$, unde

$${\begin{aligned}&2i=1+2h+(i-l)\\&2l=1+2h-(i-l)\end{aligned}}$$