Fourth Order Genera Enumeration

Summary: Euler’s case-by-case derivation of the 146 genera of fourth-order lines. The principal member $\alpha y^4+\beta y^{3}x+\gamma y^2{x}^2+\delta y{x}^3+ϵx^4$ falls into one of eight factor types; refinement of each real linear factor through the chapter-8 machinery gives the asymptote species, and the multiset of species defines the genus.

Sources: chapter11 (§§260–270).

Last updated: 2026-05-02.

---

Method recap

For a real linear factor of the principal member, after rotating coordinates so the factor becomes (say) $x$, the asymptote refinement reads off as follows. Substituting $x\to \infty$ in the lower members and balancing against the leading $\cdots \cdot x$ term gives $y\sim A/x^k$ for some $k$, so the species is $u=A/t$, $u=A/t^2$, or $u=A/t^3$ depending on which lower-order coefficients vanish. A doubled real factor instead admits the cases of parabolic-asymptote (parabolic species $u^2=At$, $u^2=A/t$) or splits into two parallel rectilinear asymptotes; a tripled or quadrupled factor brings in the cubic and quartic asymptotic curves $u^3=A{t}^2$, $u^3=At$, $u^4=A{t}^3$, $u^4=At$.

The genus is the multiset of branch species, plus the qualitative pattern (parallel? hyperbolic? bounded?) of any rectilinear or parabolic asymptote it generates.

Case I — all four factors complex (§261)

No real linear factor in the principal member ⇒ no infinite branch. Simplest equation:

$$(y^2+m^2{x}^2)(y^2-2pxy+q^2{x}^2)+a{y}^{2}x+by{x}^2+c{y}^2+dyx+e{x}^2+fy+gx+h=0,p^2<q^2.$$

The principal member must keep $y^4$ and $x^4$, so the $y^3$ and $x^3$ terms can be killed by a translation of the origin.

• GENUS I — no branch goes to infinity.

Case II — two real distinct factors (§262)

By a choice of oblique coordinates the two real linear factors become $y$ and $x$:

$$yx(y^2-2myx+n^2{x}^2)+a{y}^{2}x+by{x}^2+c{y}^2+dyx+e{x}^2+fy+gx+h=0,m^2<n^2.$$

Two straight-line asymptotes, $y=0$ and $x=0$. Refining the $y=0$ asymptote against $n^{2}y{x}^3+e{x}^2+gx+h=0$ gives species $u=A/t$, $u=A/t^2$, or $u=A/t^3$ according as $e$, then $g$, then $h$ is the first non-vanishing coefficient. The $x=0$ asymptote is governed analogously by $c$, $f$, $h$.

The six unordered species pairs from $\{A/t,A/t^2,A/t^3\}$ give six genera:

Genus	Asymptote species	Conditions
II	$A/t,A/t$	$c\ne 0,e\ne 0$
III	$A/t,A/t^2$	$e\ne 0,g\ne 0$ on one factor; $c\ne 0$ on the other
IV	$A/t,A/t^3$	$c\ne 0$ on one; $e=g=0,h\ne 0$ on the other
V	$A/t^2,A/t^2$	$f\ne 0$ and $g\ne 0$
VI	$A/t^2,A/t^3$	$f\ne 0$ on one; $e=g=0$ on the other
VII	$A/t^3,A/t^3$	both factors fully degenerate

Case III — two real equal factors (§263)

The doubled real factor (taken as $y$) plus a complex pair:

$$y^2(y^2-2myx+n^2{x}^2)+ay{x}^2+b{x}^3+c{y}^2+dyx+e{x}^2+fy+gx+h=0,n^2>m^2.$$

If $b\ne 0$, the doubled factor produces a parabolic asymptote $u^2=At$. If $b=0$, the double factor’s behaviour is governed by the residual quadratic in $y$ obtained from $x=\infty$:

$$y^2+\frac{a}{n^2}y+\frac{g}{n^{2}x}+\frac{e}{n^2{x}^2}=0.$$

The discriminant $a^2-4n^{2}e$ then drives a sub-case split:

Genus	Conditions on $b,a^2-4n^{2}e,g,h$	Asymptote species
VIII	$b\ne 0$	parabolic $u^2=At$
IX	$b=0,a^2<4n^{2}e$	none (no infinite branch)
X	$b=0,a^2>4n^{2}e,g\ne 0$	two parallel $u=A/t$
XI	$b=0,a^2>4n^{2}e,g=0$	two parallel $u=A/t^2$
XII	$b=0,a^2=4n^{2}e,g\ne 0$	hyperbolic $u^2=A/t$
XIII	$b=0,a^2=4n^{2}e,g=0,h<0$	hyperbolic $u^2=A/t^2$
XIV	$b=0,a^2=4n^{2}e,g=0,h>0$	none (no infinite branch)

Seven genera in total.

Case IV — four real distinct factors (§264)

$$yx(y-mx)(y-nx)+a{y}^{2}x+by{x}^2+c{y}^2+dyx+e{x}^2+fy+gx+h=0.$$

Each of the four asymptotes refines independently to species $u=A/t$, $u=A/t^2$, or $u=A/t^3$. Of the fifteen multisets of size four from these three species, Euler enumerates ten:

Genus	Multiset (count of $A/t$, $A/t^2$, $A/t^3$)
XV	$(4,0,0)$ — four hyperbolic of $u=A/t$
XVI	$(3,1,0)$
XVII	$(3,0,1)$
XVIII	$(2,2,0)$
XIX	$(2,1,1)$
XX	$(2,0,2)$
XXI	$(0,4,0)$
XXII	$(0,3,1)$
XXIII	$(0,2,2)$
XXIV	$(0,0,4)$

The five not listed — $(1,3,0),(1,2,1),(1,1,2),(1,0,3),(0,1,3)$ — are precisely those in which the count of the least-refined species $A/t$ is positive but is not the maximum (and similarly for $A/t^2$ when no $A/t$ branches are present). Euler does not give an explicit impossibility argument for these; the omission appears to be deliberate but is not justified in the printed text. (Verify against the source if reusing the genus count for downstream work.)

Case V — two equal factors plus two distinct (§265)

$$y^{2}x(y+nx)+ay{x}^2+b{x}^3+c{y}^2+dyx+e{x}^2+fy+gx+h=0.$$

The doubled $y$ factor contributes one of the seven case-III patterns; the two distinct factors $x$ and $y+nx$ contribute one of the six case-II species pairs. Naive total $7\times 6=42$. Two combinations are impossible:

• when the doubled factor degenerates to two parallel asymptotes of species $u=A/t^2$ (case-III genus XI), and the two distinct factors give species $(A/t,A/t^2)$ or $(A/t,A/t^3)$.

This leaves 40 genera (XXV through LXIV). Euler does not list them: “It would take too long to describe all of these here. … Whoever wishes, can take it upon himself to perform this task, and if necessary, emend this classification.”

Case VI — two pairs of equal factors (§266)

$$y^2{x}^2+a{y}^3+b{x}^3+c{y}^2+dyx+e{x}^2+fy+gx+h=0.$$

Each doubled factor contributes one of the seven case-III patterns, naively giving $7\times 7=49$ combinations. Two are impossible because the same coefficient (here $b$) cannot simultaneously satisfy both the positive-discriminant and negative-discriminant conditions required for the two pairs. 47 genera (LXV through CXI). Cumulative total to this point: 111. Again left unenumerated.

Case VII — three equal factors plus one distinct (§267)

$$y^{3}x+ay{x}^2+b{x}^3+c{y}^2+dyx+e{x}^2+fy+gx+h=0.$$

The two factor-blocks contribute independently:

The simple $x$ factor refines to:

• $u=A/t$ when $c\ne 0$;
• $u=A/t^2$ when $c=0,f\ne 0$;
• $u=A/t^3$ when $c=0,f=0$.

Three sub-varieties.

The tripled $y^3$ factor gives a parabolic asymptote $u^3=A{t}^2$ when $b\ne 0$. When $b=0$, the limit at $x=\infty$ leaves

$$y^3+ayz+dy+ez+g+\frac{c{y}^2+fy+h}{x}=0,$$

and an analysis of which lower-order terms vanish ($e\ne 0$ vs $e=0$, $a\ne 0$ vs $a=0$, etc.) splits the doubled-factor refinement into eight sub-varieties — combinations of a parabolic ($u^3=A{t}^2$, $u^2=At$, $u^3=At$, or $u^3=A/t$) and an associated hyperbolic asymptote ($u=A/t$, $u=A/t^2$, or $u=A/t^3$), or three hyperbolic asymptotes of one species, etc.

Total $3\times 8=24$ genera (CXII through CXXXV). Cumulative: 135.

Case VIII — all four factors equal (§§268–270)

$$y^4+a{y}^{2}x+by{x}^2+k{x}^3+c{y}^2+dyx+e{x}^2+fy+gx+h=0.$$

Eleven genera; this case Euler does enumerate completely.

§268 — $k\ne 0$ or $b\ne 0$

Genus	Conditions	Asymptote species
CXXXVI	$k\ne 0$	parabolic $u^4=A{t}^3$
CXXXVII	$k=0,b\ne 0$, generic	parabolic $u^3=A{t}^2$ + hyperbolic $u=A/t$
CXXXVIII	$k=0,b\ne 0$, $a{e}^2-bde+b^{2}g=0$	parabolic $u^3=A{t}^2$ + hyperbolic $u=A/t^2$

The straight-line asymptote $by+e=0$ has species $u=A/t$ generically; the listed algebraic relation forces it to refine to $u=A/t^2$.

§269 — $k=0,b=0$

$$y^4+a{y}^{2}x+c{y}^2+dyx+e{x}^2+fy+gx+h=0.$$

If $e\ne 0$, the leading equation $y^4+a{y}^{2}x+e{x}^2=0$ admits two parabolic branches related to a common axis ($u^2=At$). Their reality and coincidence give:

Genus	Conditions	Asymptote species
CXXXIX	$e\ne 0,a^2<4e$	none (impossible)
CXL	$e\ne 0,a^2>4e$	two parabolic $u^2=At$
CXLI	$e\ne 0,a^2=4e$	one (merged) parabolic $u^2=At$

If $e=0$ and $a\ne 0$, the equation splits as $y^2+ax=0$ together with $a{y}^2+dy+g=0$ at $y=$ const. The latter quadratic in $y$ gives 2 / 1 / 0 real values:

Genus	Conditions	Asymptote species
CXLII	$a\ne 0,e=0$, two real $y$-roots	parabolic $u^2=At$ + two parallel $u=A/t$
CXLIII	$a\ne 0,e=0$, equal $y$-roots	parabolic $u^2=At$ + parallel $u^2=A/t$
CXLIV	$a\ne 0,e=0$, no real $y$-roots	parabolic $u^2=At$ alone

§270 — $a=0$ as well

$$y^4+c{y}^2+dyx+fy+gx+h=0.$$

Genus	Conditions	Asymptote species
CXLV	$d\ne 0$	parabolic $u^3=At$ + hyperbolic $u=A/t$
CXLVI	$d=0$	parabolic $u^4=At$

Cumulative count

After case	Genera so far
I	1
II	7
III	14
IV	24
V	64
VI	111
VII	135
VIII	146

Related pages

• chapter-11-on-fourth-order-lines — chapter-level summary and §271’s higher-order remark.
• cubic-species-classification — the case-by-case enumeration template inherited here.
• hyperbolic-and-parabolic-branches — the $u^j=A/t^k$ vs $u^j=A{t}^k$ catalogue from which species labels are drawn.
• curvilinear-asymptote — the cubic and quartic asymptotic curves invoked in cases VII and VIII.
• parabolic-asymptote — the doubled-factor refinement used in cases III, V, VI, VIII.
• double-factor-asymptote-cases, triple-factor-asymptote-cases — the per-multiplicity machinery from chapter 8.