In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in 2n variables. 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 we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae.

The broader context of this result is the geometrization of Fritz John's 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the condition for functions on line space to come from line integrals of functions in Euclidean 3-space, and hence it appears as a compatibility condition for tomographic data.

The introduction of the canonical neutral Kaehler metric on the space of oriented lines clarifies the relationship and broadens the paradigm to allow new insights. In particular, it is proven that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves that arise as the image of John's conjugate circles under a conformal map. These pairs of curves are then shown to be conjugate conics, which include circles and hyperbolae.

John identified conjugate circles with the two rulings of a hyperboloid of 1-sheet. Conjugate hyperbolae are identified with the two rulings of either a piece of a hyperboloid of 1-sheet or a hyperbolic paraboloid.

1937 年，Asgeirsson 为一般超双曲方程在 2 n 个变量中的解建立了平均值属性。在有四个变量的情况下，它表明某个解在某些共轭圆对上的积分是相同的。在本文中，我们将此结果扩展到非退化共轭圆锥曲线，其中包括共轭圆的原始情况并添加了共轭双曲线的新情况。

这一结果的更广泛背景是弗里茨约翰 1938 年对超双曲方程的分析的几何化。方程的解作为线空间上的函数来自欧几里得 3 空间中函数的线积分的条件而出现，因此它作为断层摄影数据的兼容性条件出现。

在定向线空间上引入规范的中性 Kaehler 度量澄清了关系并拓宽了范式以允许新的见解。特别是，证明了超双曲方程的解在任何一对曲线上都具有平均值属性，这些曲线是在等角映射下作为约翰共轭圆的图像出现的。这些曲线对随后显示为共轭圆锥曲线，包括圆和双曲线。

约翰用 1 张双曲面的两条规则确定了共轭圆。共轭双曲线用 1 片双曲面或双曲抛物面的两个规则来标识。