  Asgeirsson's theorem establishes a mean value property for solutions of the
ultrahyperbolic equation. In the case of four variables, it states that the
integrals of a solution over certain pairs of conjugate circles are the same.
In this paper, the invariance of the four dimensional ultrahyperbolic equation
under conformal maps of the pseudo-Euclidean space of signature 2+2 is used to
get the most general version of the mean value theorem.
  The name non-degenerate conjugate conics is used for the most general pairs
of curves over which solutions of the ultrahyperbolic equation enjoy the mean
value property. These are proven to be pairs of conic sections, so that, in
addition to conjugate circles which were known to exist, the picture is
completed by finding mean value theorems over conjugate hyperbolae, conjugate
parabolae, and line-empty pairs.
  In addition, Fritz John established a link between conjugate circles and the
two rulings of a hyperboloid of revolution. The line incidence property of
doubly ruled surfaces is used to prove a one-to-one correspondence between
non-degenerate conjugate conics and pairs of rulings of doubly ruled surfaces
in Euclidean 3-space.

Differential Geometry;Analysis of PDEs