We say that a surface $f(x,y,z)=0$ is ruled if for each point $p$ in the surface there is a line that passes through $p$ and is contained in the surface. See <http://en.wikipedia.org/wiki/Ruled_surface> for more information. 


Does anybody know if there is a partial differential equation whose solutions are all ruled surfaces and only them? And what is the equation? Some reference would be helpful, too (different from Salmon's old book about surfaces which I unfortunately find unreadable).



partial differential equation for ruled surfaces
Here is a test for when a surface of the form $z = f(x,y)$, where $f$ is a sufficiently smooth function of two variables, is ruled.


To begin, set $I\!I = f\_{xx} dx^2 + 2f\_{xy}dxdy + f\_{yy}dy^2$. If $I\!I$ vanishes identically, then the surface is a plane, so it is ruled.


Suppose that $I\!I$ is nonzero. The *discriminant* of $I\!I$ is defined (up to a factor of $(\mathrm{d}x\wedge\mathrm{d}y)^{\otimes 2}$) to be
$$
\Delta(I\!I) = f\_{xx}f\_{yy}- {f\_{xy}}^2.
$$


If $\Delta(I\!I) >0$, then the surface is locally strictly convex and so cannot be ruled.


If $\Delta(I\!I) = 0$, then the surface is ruled. In fact, it has vanishing Gauss curvature. Moreover, $I\!I = \pm \alpha^2$ for some nonzero $1$-form $\alpha$ on the domain of $f$, and the curves in this domain defined by $\alpha = 0$ (which turn out to be straight lines) lift to the graph $z = f(x,y)$ to be straight lines.


If $\Delta(I\!I) < 0$, 
set $I\!I\!I = f\_{xxx}\ dx^3+3f\_{xxy}\ dx^2dy+3f\_{xyy}\ dxdy^2+f\_{yyy}\ dy^3$ 
and let $I\!I\!I\_0$ be the $I\!I$-trace-free part of $I\!I\!I$. Then the surface $z = f(x,y)$ is ruled if and only if the discriminant of $I\!I\!I\_0$ vanishes. 


(Added later: This latter condition (i.e., the vanishing of the discriminant of $I\!I\!I\_0$) turns out to be equivalent to the condition that $I\!I$ and $I\!I\!I$ have a common linear factor, say, $\alpha$ (which will necessarily be real when $\Delta(I\!I) < 0$), and hence is equivalent to the vanishing of the *resultant* of $I\!I$ and $I\!I\!I$, i.e., $\textrm{Reslt}(I\!I,I\!I\!I) = 0$. When such an $\alpha$ exists, the leaves of $\alpha=0$ are lines on the surface.)


*Notes:* 


1. The discriminant of a cubic form $C = p\ dx^3 + 3q\ dx^2dy + 3r\ dxdy^2 + s\ dy^3$ is, by definition,
$$
\Delta(C) = s^2p^2 + 4r^3p + 4 q^3s - 3 r^2q^2 - 6 sqrp.
$$
It is, up to a multiple, the unique polynomial of degree $4$ in the coefficients that vanishes if and only if $C$ has a multiple factor.
2. Given a quadratic form $Q = a\ dx^2 + 2b\ dxdy + c\ dy^2$ with nonvanishing discriminant $D$, the $Q$-trace of a form $C$ of degree $3$ is the linear form
$$
tr\_Q(C ) = \frac{(ar-2bq+cp)\ dx + (as-2br+cq)\ dy}{D}.
$$
Any cubic form $C$ can be uniquely written in the form
$$
C = C\_0 + L\cdot Q
$$
where $L$ is a linear form and $tr\_Q(C\_0) = 0$. (In fact, $L = \tfrac34 tr\_Q(C)$.) The term $C\_0$ is called the $Q$-trace-free part of $C$.
3. The resultant $\textrm{Reslt}(Q,C)$ of a quadratic form $Q$ and a cubic form $C$ is the (unique up to nonzero multiples) polynomial that is cubic in the coefficients of $Q$, quadratic in the coefficients of $C$, and vanishes exactly when they have a common linear divisor.


