Pagina:Gauss, Carl Friedrich - Werke (1870).djvu/139

Haec pagina emendata est

129

transformatio.

Hinc sequitur, maximum divisorem communem numerorum $a$ , $b(2b)$ , $c$ simul metiri divisorem communem maximum numerorum $a'\!$ , $b'\!(2b'\!)$ , $c'\!$ . Quodsi igitur insuper forma $(a'\!,b'\!,c'\!)$ formam $(a,b,c)$ implicat, i. e. formae sunt aequivalentes, divisor communis maximus numerorum $a$ , $b(2b)$ , $c$ divisori communi maximo numerorum $a'\!$ , $b'\!(2b'\!)$ , $c'\!$ aequalis erit, quoniam tum ille hunc metiri debet, tum hic illum. Si itaque in hoc casu $a$ , $b(2b)$ , $c$ divisorem communem non habent, i. e. si maximus $=1$ , etiam $a'\!$ , $b'\!(2b'\!)$ , $c'\!$ divisorem communem non habebunt.

Nexus omnium transformationum similium formae datae in formam datam.

162.

Problema. Si forma $\quad AXX+2BXY+CYY\quad \ldots \quad F$ formam $\qquad \qquad \qquad axx+2bxy+cyy\quad \ldots \ldots \quad f$ implicat, atque transformatio aliqua illius in hanc est data: ex hac omnes reliquas transformationes ipsi similes deducere.

Solutio. Sit transformatio data haec $X=\alpha x+\beta y$ , $Y=\gamma x+\delta y$ , ponamusque primo aliam huic similem datam esse $X=\alpha '\!x+\beta '\!y$ , $Y=\gamma '\!x+\delta '\!y$ , ut quid inde sequatur, investigemus. Tum positis determinantibus formarum $F$ , $f$ , $=D$ , $d$ , atque $\alpha \delta -\beta \gamma =e$ , $\alpha '\!\delta '\!-\beta '\!\gamma '\!=e'\!$ erit (art. 157) $d=Dee=De'\!e'\!$ et quum ex hyp. $e$ , $e'\!$ eadem signa habeant, $e=e'\!$ . Habebuntur autem sequentes sex aequationes : $A\alpha \alpha \,\,+2B\alpha \gamma \,\,+C\gamma \gamma \,\,=a\$ $\ \ldots \$ $\ \ldots \$ $[1]$
$A\alpha '\!\alpha '\!+2B\alpha '\!\gamma '\!+C\gamma '\!\gamma '\!=a\$ $\ \ldots \$ $\ \ldots \$ $[2]$
$A\alpha \beta \,\,+B(\alpha \delta \,+\,\beta \gamma )\,\,+C\gamma \delta \,\,=b\$ $\ \ldots \$ $\ \ldots \$ $[3]$
$A\alpha '\!\beta '\!+B(\alpha '\!\delta '\!+\beta '\!\gamma '\!)+C\gamma '\!\delta '\!=b\$ $\ \ldots \$ $\ \ldots \$ $[4]$
$A\beta \beta \,\,+2B\beta \delta \,\,+C\delta \delta \,\,=c\$ $\ \ldots \$ $\ \ldots \$ $[5]$
$A\beta '\!\beta '\!+2B\beta '\!\delta '\!+C\delta '\!\delta '\!=c\$ $\ \ldots \$ $\ \ldots \$ $[6]$

Si brevitatis gratia numeros $A\alpha \alpha '\!+B(\alpha \gamma '\!+\gamma \alpha '\!)+C\gamma \gamma '\!$
$A(\alpha \beta '\!+\beta \alpha '\!)+B(\alpha \delta '\!+\beta \gamma '\!+\gamma \beta '\!+\delta \alpha '\!)+C(\gamma \delta '\!+\delta \gamma '\!)$
$A(\beta \beta '\!+B(\beta \delta '\!+\delta \beta '\!)+C\delta \delta '\!$

I. 17