Pagina:Opere matematiche (Cremona) I.djvu/100

86

intorno alle superficie della seconda classe ecc.

desima. Quelle funzioni sono:

Φ = ABC(D — A — B — C)
θ1 = DBC(D — B — C)	Ξ1 = A(D — A)
θ2 = DCA(D — C — A)	Ξ2 = B(D — B)
θ3 = DAB(D — A — B)	Ξ3 = C(D — C)

e sostituendo per A, B, C, D i loro rispettivi valori¹:

9)	Φ ≡ i(α + β + γ)αβγ(λ — i)(μ — i)(ν — i)

10)

${\begin{cases}\Theta _{1}\equiv \beta \gamma (\alpha +\beta +\gamma )(\mu -i)(\nu -i)\left(\alpha +i(\beta '+\gamma ')\right)\\\Theta _{2}\equiv \gamma \alpha (\alpha +\beta +\gamma )(\nu -i)(\lambda -i)\left(\beta +i(\gamma '+\alpha ')\right)\\\Theta _{3}\equiv \alpha \beta (\alpha +\beta +\gamma )(\lambda -i)(\mu -i)\left(\gamma +i(\alpha '+\beta ')\right)\end{cases}}$

11)	${\begin{cases}\Xi _{1}\equiv \alpha (\lambda -i)(\beta +\gamma +i\alpha ')\\\Xi _{2}\equiv \beta (\mu -i)(\gamma +\alpha +i\beta ')\\\Xi _{3}\equiv \gamma (\nu -i)(\alpha +\beta +i\gamma ').\end{cases}}$

Posto per brevità:

12)	$\lambda '=-{\frac {\beta +\gamma }{\alpha '}}$     $\mu '=-{\frac {\gamma +\alpha }{\beta '}}$     $\nu '=-{\frac {\alpha +\beta }{\gamma '}}$

13)	$\lambda ''=-{\frac {\alpha }{\beta '+\gamma '}}$     $\mu ''=-{\frac {\beta }{\gamma '+\alpha '}}$     $\nu ''=-{\frac {\gamma }{\alpha '+\beta '}}$

le espressioni superiori divengono:

14)

${\begin{cases}\Theta _{1}\equiv \beta \gamma (\beta '+\gamma ')(i-\mu )(i-\nu )(i-\lambda '')(\alpha +\beta +\gamma )\\\Theta _{2}\equiv \gamma \alpha (\gamma '+\alpha ')(i-\nu )(i-\lambda )(i-\mu '')(\alpha +\beta +\gamma )\\\Theta _{3}\equiv \alpha \beta (\alpha '+\beta ')(i-\lambda )(i-\mu )(i-\nu '')(\alpha +\beta +\gamma )\end{cases}}$

15)	Ξ1 ≡ — (i — λ)(i — λ’),     Ξ2 ≡ — (i — μ)(i — μ’), Ξ3 ≡ — (i — ν)(i — ν’).

Ciò posto, i criteri per distinguere la specie della superficie rappresentata dalla equazione (5) sono i seguenti².

---

1. Il simbolo ≡ indica l’eguaglianza di segno.
2. Plücker, Op. cit.