Triple-Factor Asymptote Cases

Summary: When the highest member factors as $P=(ay-bx{)}^{3}M$ — multiplicity 3 — the rotated leading equation has four possible alternatives at infinity, depending on which terms of $Q,R,S$ are present: I. $\alpha u^3+\beta t^2=0$ (cubic, figure 41); II. $\alpha u^3+\beta tu+\gamma t=0$ (parabola plus refinable straight line); III. $\alpha u^3+\beta u^2+\gamma t=0$ (cubic, figure 42); IV. $\alpha u^3+\beta u^2+\gamma u+\delta =0$ (up to three parallel straight lines, each refinable). Equal-root sub-cases of IV demand higher refinements: a double root reduces to the double-factor-asymptote-cases family $(u-c{)}^2+A(u-c)/t^p+B/t^q=0$; a triple root produces $(u-c{)}^3+a/t=0$ or the general $(u-c{)}^3+A(u-c{)}^2/t^p+B(u-c)/t^q+C/t^r=0$. The $(u-c{)}^3=K/t^k$ shape follows the same parity rule as the simple-factor case.

Sources: chapter8 (§§210–214), figures40-43 (figures 41, 42), figures35-39 (figures 36, 37 reused for cubic shape).

Last updated: 2026-04-28

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Setup

§210. With $P=(ay-bx{)}^{3}M$, the rotated members in $(t,u)$ are

$P=\alpha t^{n-3}{u}^3+\alpha t^{n-4}{u}^4+\cdots ,$ $Q=\beta t^{n-1}+\beta t^{n-2}u+\beta t^{n-3}{u}^2+\beta t^{n-4}{u}^3+\cdots ,$ $R=\gamma t^{n-2}+\gamma t^{n-3}u+\gamma t^{n-4}{u}^2+\cdots ,$ $S=\delta t^{n-3}+\delta t^{n-4}u+\delta t^{n-5}{u}^2+\cdots ,$

and so on. (Reused letters denote different coefficients in different members.) The leading $u^3$-coefficient of $P$ is $\alpha t^{n-3}$.

The leading equation at infinity has four alternatives, depending on which of the terms in $Q,R,S$ vanish:

I. $Q$‘s top term $\beta t^{n-1}$ present: $\alpha t^{n-3}{u}^3+\beta t^{n-1}=0$, i.e.

$\alpha u^3+\beta t^2=0.$

II. $\beta t^{n-1}$ absent but $\beta t^{n-2}u$ and $\gamma t^{n-2}$ present: $\alpha t^{n-3}{u}^3+\beta t^{n-2}u+\gamma t^{n-2}=0$, i.e.

$\alpha u^3+\beta tu+\gamma t=0.$

III. Above two also absent, but $\beta t^{n-3}{u}^2$ and $\delta t^{n-3}$ (or equivalent) present:

$\alpha u^3+\beta u^2+\gamma t=0.$

IV. All "$t$" contributions in low-power positions absent, leaving a polynomial in $u$ alone:

$\alpha u^3+\beta u^2+\gamma u+\delta =0.$

Equation I: cubic asymptote (§211, figure 41)

$\alpha u^3+\beta t^2=0$ is a third-order line — a semicubical-parabola–like cubic in $(t,u)$. Take the asymptote axis $XY$ with $A$ the origin (figure 41). The cubic $u^3=-\beta t^2/\alpha$ has two branches $E$ (upper) and $F$ (lower) going to infinity in quadrants $P$ and $Q$ — the branch shape is a cusp at the origin opening in the $+t$ direction.

This case generalises the parabolic asymptote of parabolic-asymptote one degree higher and is a representative of the hyperbolic-and-parabolic-branches parabolic family at species $u^3=A{t}^2$.

Equation II: parabola plus refinable straight line (§211)

$\alpha u^3+\beta tu+\gamma t=0$ admits a $u=\infty$ root (taking $\alpha u^3\to \infty$ dominates), but for finite $u$ the equation must also have $t\to \infty$ — and at infinity it factors as

$(\alpha u^2+\beta t)(\beta u+\gamma )=0$

approximately (Euler resolves it as two equations). Concretely:

• $\beta u+\gamma =0⟹u=-\gamma /\beta$ — a straight-line asymptote, parallel to $AX$.
• $\alpha u^2+\beta t=0$ — a parabolic asymptote (as in equation I of double-factor-asymptote-cases), with two branches.

The straight-line component $u=c:=-\gamma /\beta$ admits the refinement procedure of curvilinear-asymptote-refinement. Substituting $\beta u+\gamma =(u-c)\cdot \beta$ and $u=c$ everywhere else:

$t^{n-2}(u-c)+t^{n-3}(\alpha c^3+\beta c^2+\gamma c+\delta )+t^{n-4}(\alpha c^4+\beta c^3+\gamma c^2+\delta c+ϵ)+\cdots =0,$

leading to refinements $(u-c)+A/t^k=0$ for $k=1,2,\dots ,n-2$, exactly as in curvilinear-asymptote-refinement.

So equation II gives a double asymptote: one parabolic and one straight-line (refinable), in different directions through the origin.

Equation III: cubic, two opposite branches (§212, figure 42)

$\alpha u^3+\beta u^2+\gamma t=0$. As $t\to \infty$, the only way to balance the equation is $u\to \infty$, which makes $\beta u^2$ negligible compared to $\alpha u^3$. The dominant equation reduces to

$\alpha u^3+\gamma t=0,$

a third-order (cubic) asymptote (figure 42). Its two branches $AE$ (in quadrant $P$) and $AF$ (in quadrant $S$) go to infinity in opposite quadrants — figure 42.

Equation IV: parallel-line family (§§212–214)

$\alpha u^3+\beta u^2+\gamma u+\delta =0$ is a cubic in $u$ alone, with one or three real roots. Three sub-cases, corresponding to the double-factor-asymptote-cases discriminant analysis but at one degree higher.

Three distinct real roots: three parallel straight lines

Three roots $u=c,c^{\prime },c^{\prime \prime }$ give three parallel-line asymptotes. Each is refined individually by the curvilinear-asymptote-refinement procedure. Substituting $\alpha u^3+\beta u^2+\gamma u+\delta =(u-c)(f{u}^2+gu+h)$ and $u=c$ everywhere else, the refinement is

$t^{n-3}(u-c)+A{t}^{n-4}+B{t}^{n-5}+C{t}^{n-6}+\cdots =0,$

whence $(u-c)+K/t^k=0$ as before.

Double root $(u-c{)}^2(fu+g)=0$ (§213)

The factor $(u-c)$ has multiplicity 2, the other root is $u=-g/f$ (a single straight-line asymptote, refinable as in curvilinear-asymptote-refinement).

For the double root: substituting $u=c$ everywhere except in $(u-c{)}^2$:

$(u-c{)}^2+\frac{A(u-c)}{t^p}+\frac{B}{t^q}=0$

— exactly the form analysed in double-factor-asymptote-cases §207–208. The three sub-cases on $q$ vs $2p$ apply.

Triple root $(u-c{)}^3=0$ (§213)

All three roots coincide at $u=c$. The leading equation $\alpha u^3+\beta u^2+\gamma u+\delta =\alpha (u-c{)}^3$ collapses, and the refined equation has the form

$(u-c{)}^3{t}^{n-3}+P{t}^{n-4}+Q{t}^{n-5}+R{t}^{n-6}+S{t}^{n-7}+\cdots =0,$

where $P,Q,R,S,\dots$ are coefficients of the lower members evaluated at $u=c$. Substituting $u=c$ in the lower members may again reveal $(u-c)$ factors. Several refined forms can arise:

• If $P\ne 0$: simplest case, $(u-c{)}^3+a/t=0$.
• If $P$ is divisible by $u-c$ (i.e. $u=c$ is also a root of the next coefficient): keep one factor $u-c$ and add the next term, giving $(u-c{)}^3+A(u-c)/t+B/t^q=0$.
• If $P$ is divisible by $(u-c{)}^2$: keep $(u-c{)}^2$ and add the next:

$(u-c{)}^3+\frac{A(u-c{)}^2}{t}+\frac{B}{t^2}=0.$

• The general form, after exhausting all such cascading divisibilities:

$(u-c{)}^3+\frac{A(u-c{)}^2}{t^p}+\frac{B(u-c)}{t^q}+\frac{C}{t^r}=0,$

where $r$ is less than $n-2$, $q$ is less than $r$, and $p$ is less than $q$.

Shape catalogue for triple-root refinement (§214)

§214. The general triple-root refinement produces three roots in $u-c$, each of one of the three forms:

• Three of the form $(u-c)=K/t^k$ (three hyperbolic-rate refinements);
• Or one of that form and one of the form $(u-c{)}^2=K/t^k$ (mixed);
• Or a single $(u-c{)}^3=K/t^k$ (all three roots merge at the same rate). This last occurs when both $3p\ge r$ and $3q\ge 2r$ in the general form above. Two of the three could also be complex, giving no asymptote in those rates.

For the cubic shape $(u-c{)}^3=K/t^k$: setting $z=u-c$ and choosing the asymptote line $u=c$ as axis $XY$ (origin at $A$), the parity of $k$ determines the quadrant pattern, exactly as in curvilinear-asymptote-refinement §203:

• $k$ odd: two branches $EX$, $FY$ in opposite quadrants $P,S$ — figure 36 reused.
• $k$ even: two branches $EX$, $EY$ in adjacent quadrants $P,Q$ (both above the asymptote, say) — figure 37 reused.

The same parity rule that applied to the simple-factor refinement applies here, simply lifted to the cubic exponent on $z$.

Summary of cases

Equation	Asymptote(s)	Branches	Figure
I. $\alpha u^3+\beta t^2=0$	cubic	2 in $P,Q$	41
II. $\alpha u^3+\beta tu+\gamma t=0$	parabola + straight line (refinable)	2 + 2	—
III. $\alpha u^3+\beta u^2+\gamma t=0$	cubic	2 in $P,S$	42
IV. $\alpha u^3+\beta u^2+\gamma u+\delta =0$, distinct real roots	three parallel lines, each refinable	2 + 2 + 2	—
IV. double root $(u-c{)}^2(fu+g)$	line + double-factor case	per double-factor-asymptote-cases	—
IV. triple root $(u-c{)}^3$	$(u-c{)}^3=K/t^k$	per parity (§214)	36, 37

Beyond multiplicity 3

§215. Quadruple, quintuple, and higher coincident factors follow the same logic — leading equation is a polynomial of the corresponding degree in $u$, and refined forms are $(u-c{)}^k=K/t^j$ — but Euler does not develop them in detail. The chapter ends with the example in example-curve-eight-branches.

Figures

Figures 35–39

Figures 40–43

Related pages

• chapter-8-concerning-asymptotes
• curvilinear-asymptote-refinement
• double-factor-asymptote-cases
• parabolic-asymptote
• curvilinear-asymptote
• hyperbolic-and-parabolic-branches
• example-curve-eight-branches
• branches-at-infinity