An Extension of Asgeirsson's Mean Value Theorem for Solutions of the ultra-hyperbolic Equation in Dimension Four

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 compatibility for functions on line space to come from line integrals of functions in Euclidean 3-space. 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.


Introduction and Results
A twice continuously differentiable function u of four variables is said to be a solution of the ultra-hyperbolic equation if it satisfies In 1937 Asgeirsson's proved [2] that a solution of the ultra-hyperbolic equation satisfies (1.2) 2π 0 u(a + r cos θ, b + r sin θ, c, d)dθ = 2π 0 u(a, b, c + r cos θ, d + r sin θ)dθ, for all a, b, c, d, r ∈ R with r ≥ 0. This mean value theorem was subsequently considered by Fritz John [15] and discussed at length by Courant and Hilbert [6]. Indeed, John showed that the mean value property is sufficient for a function to satisfy the ultra-hyperbolic equation, thus allowing one to define generalised solutions of a second order partial differential equation that were merely continuoussee the discussion on page 301 of, and Theorem 1.1 of [15]. Moreover, John states in passing that the integrals can be taken over "conjugate conics" if a "certain variable of integration is used". He notes that the case of conjugate hyperbolae has not been considered, and that it is not directly amenable to the methods of his proof, which utilizes affine maps (p306 of [15]). In this paper we fill this gap by utilizing the conformal geometry of line space and thus extending the method from affine maps to projective maps.
From a geometric perspective, Asgeirsson's result can be viewed in R 4 endowed with the flat indefinite metric of signature (2,2): Thus, the second order ultra-hyperbolic operator is the Laplacian of g and solutions of the ultra-hyperbolic equation are the harmonic functions. Moreover, the two curves over which the mean value is computed in (1.2) are circles of center (a, b, c, d) and radius r, one of them lying in the x 1 x 2 −plane and the other in the x 3 x 4 −plane.
We refer to these as conjugate circles and throughout we denote the pair (R 4 , g) simply by R 2,2 .
Note that the induced metric on one plane is positive definite, on the other it is negative definite and that the planes are mutually orthogonal. Moreover, both curves are round circles in the metric induced on the planes by g.
In this paper Asgeirsson's theorem is extended to more general pairs of curves, called non-degenerate conjugate conics. Our main result can be stated as follows: Theorem 1.1. Let u be a solution of the ultra-hyperbolic equation (1.1). Let S, S ⊥ be a pair of non-degenerate conjugate conics in R 2,2 . Then where dl is the line element induced on the curves S, S ⊥ by the flat metric g.
To define the pairs of curves, introduce the following nomenclature. Consider a pair of orthogonal planes π, π ⊥ through a point O in R 2,2 , where orthogonality is with respect to the neutral metric on R 2,2 . If either plane is totally null (the induced metric is identically zero on such a plane) then π = π ⊥ . The induced metric on π can be definite or indefinite but is assumed non-degenerate. Thus we exclude all degenerate cases in what follows -these cases will be dealt with in a separate paper [5].
A pair of non-degenerate conjugate conics consists of a pseudo-circle S of center O and square-radius c 2 in π; and a pseudo-circle S ⊥ of center O and square-radius −c 2 in π ⊥ , for any pair of non-degenerate orthogonal planes π, π ⊥ .
A pair of non-degenerate conjugate conics can be either: (i) a pair of circles with a common center and opposite square-radius in orthogonal definite planes, or (ii) a pair of hyperbolae with a common center and opposite square-radius in orthogonal indefinite planes.
integrating over a pair of conjugate hyperbolae. As the integrals are now over unbounded domains one also should assume that u is integrable.
This result can be recast in a more geometric way as follows. In [15] the first link was made between the ultra-hyperbolic equation and straight lines in Euclidean 3space. In particular, R 2,2 is identified locally with the space of straight lines and then solutions of the equation (1.1) arise from line integrals of functions on R 3 .
The space of all oriented lines is diffeomorphic to T S 2 and is endowed with a canonical metric G of signature (2,2) [12]. The metric is conformally flat, and so can be conformally mapped to R 2,2 locally. Thus solutions of the ultra-hyperbolic equation give rise to solutions of the Laplace equation. Moreover, conjugate conics can be defined in T S 2 as discussed in detail in Section 2.3.
This leads to a more geometric statement of our result: Let v : T S 2 → R be a solution of the Laplace equation of the canonical metric ∆ G v = 0. Let S, S ⊥ be a pair of non-degenerate conjugate conics in T S 2 . Then the following integral equation holds where dτ is the line element induced by G.
In Section 2 the background to Asgeirsson's theorem and Fritz John's contributions are discussed. Details are given of the geometric under-pinning of our approach, as well as a proof that Theorems 1.1 and 1.2 are equivalent.
The picture that emerges is that the splitting of R 2,2 into conjugate planes in the original formulation with positive and negative definite metrics, extends to the case where the conjugate planes are both hyperbolic. The curves change from circles to pseudo-circles, but the integral identity remains the same.
In R 3 the picture is richer. The image of a plane in R 2,2 under the conformal map to T S 2 , is a 2-parameter family of lines which twist around each other so that there is no orthogonal surface in R 3 . This line congruence enjoys the property that the images of (pseudo) circles are special ruled surfaces. John showed that when the two planes are definite, the conics are the two rulings of a 1-sheeted hyperboloid. We show that Theorem 1.3. A pair of non-degenerate conjugate conics can be identified with a pair of rulings of the same surface in R 3 .
In the definite case, they are the two rulings of a 1-sheeted hyperboloid, in the hyperbolic case, they are either part of the two rulings of a 1-sheeted hyperboloid, or the two rulings of a hyperbolic paraboloid.
The uniqueness of this situation is well-expressed by the classical result that states that, other than the plane, the only surfaces in R 3 that have more than one ruling by straight lines are the 1-sheeted hyperboloid and the hyperbolic paraboloid [14].
An outline of the proof of Theorem 1.1 contained in this paper is as follows. For convenience, let S 0 and S ⊥ 0 be the unit circles in the conjugate planes R 2 × (0, 0) ⊂ R 2,2 and (0, 0) × R 2 ⊂ R 2,2 respectively. In Section 2 it is shown that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves S, S ⊥ that arise as the image of S 0 , S ⊥ 0 under an arbitrary conformal mapping of R 2,2 . We refer to this as the conformal extension of Asgeirsson's Theorem which can be expressed in oriented line space as: where dτ is the line element induced by the canonical metric G.
Section 3 shows that any pair of non-degenerate conjugate conics comes as f (S 0 ), f (S ⊥ 0 ) for some conformal map f . This completes the proof of Theorem 1.1.
Finally, there are two appendices, one dedicated to numerical checks of the extended Asgeirsson's theorem for a particular solution of the ultra-hyperbolic equation and for specific choices of conjugate conics, and one giving solutions of the ultra-hyperbolic equation generated by certain 3-dimensional step functions.

2.1.
Background. The X-ray transform is defined by taking the integral of a scalar function over (affine) lines of R n . That is, given a real function f : R n → R and a line γ in R n , define the map where dt is the flat line element in R n .
For functions f with appropriate behaviour at infinity (for example, compactly supported), one obtains the X-ray Transform of f, namely χ f : In comparison, the Radon Transform takes a function and integrates it over hypersurfaces in R n . By elementary considerations, the space of affine hypersurfaces in R n is seen to be n-dimensional, the dimension of the underlying space, while the space of affine lines of R n has dimension 2n − 2, which is bigger than the dimension of the underlying space for n > 2.
Thus, by dimension count, if we consider the problem of inverting the two transforms, given a function on hypersurfaces one can reconstruct the original function on R n , while, for n > 2, the problem is over-determined for functions on lines.
In particular, for n = 3, the space of lines is 4-dimensional and the ultrahyperbolic equation is the condition that must be satisfied for a function on line space to come from an integral of a function on R 3 . This was shown in 1938 when Fritz John [15] characterised solutions of equation (1.1) with certain smoothness and decay conditions as the range of the X-ray transform.
This impressive result has had a huge impact on fields such as tomography, where information of the interior of a 3D body is deduced from attenuation data of rays traversing the body. The ultra-hyperbolic equation, or an equivalent system of second order partial differential equations known as John's equations, represent a compatibility condition that tomographic data must satisfy, and so it has been exploited practically to obtain image reconstruction algorithms [4] [9] [10] [17].
Asgeirsson's theorem for the ultrahyperbolic equation has also seen applications in imaging and analysis of lightfields. T. Georgiev, H. Qin and H. Li recently used it to derive new kernels for processing lightfields which provide nice features like accurate depth estimation of planar images [7] [8].
More generally, one can consider geodesics on general Riemannian manifolds [22] [23] and an active line of work has been finding the range of the X-ray transform of rank m symmetric tensor fields [11] [16] [18].
The foundation of John's work lies in Asgeirsson's mean value theorem for solutions of second order linear partial differential equations with constant coefficients (see [2] and [6]). While Asgeirsson's original result holds for solutions of the (n, n)dimensional ultra-hyperbolic equation, John exploited the geometry of the (2, 2) case to obtain it in a more general form.
The link between line space and the ultra-hyperbolic equation is made concrete by Plücker coordinates. That is, to describe a line in R 3 John chooses two distinct points (s 1 , s 2 , s 3 ) and (t 1 , t 2 , t 3 ) on the line and forms the sextet (p 1 , p 2 , p 3 , q 1 , q 2 , q 3 ) by Flat coordinates on R 2,2 are related to the Plücker coordinates away from q 3 = 0 by The freedom to choose different representative points on a given line means that the above description has a 2 dimensional redundancy in (p 1 , p 2 , p 3 , q 1 , q 2 , q 3 ). One dimension is generated by scaling, the other by the fact that the embedding satisfies the quadric equation p 1 q 1 + p 2 q 2 + p 3 q 3 = 0. John remarks that the deeper connection between the ultra-hyperbolic equation and line space is given by this embedding of line space in the quadric hypersurface in projective 5-space. In what follows we will explore this further from the point of view of conformal differential geometry.

Geometrization of line space.
To remove the extra degrees of freedom inherent in Plücker coordinates, proceed as in [12] and identify the space of oriented lines in R 3 with T S 2 , the total space of the tangent bundle of the 2-sphere. This double covers the space of unoriented lines, but this makes no difference in what follows as all our calculations are local.
Starting with the usual holomorphic coordinate ξ on S 2 obtained by stereographic projection, construct complex coordinates (ξ, η) on T S 2 by identifying Thus ξ ∈ S 2 is the direction of the oriented line and the complex number η measures the displacement of the line from the origin.
Theorem 2.1. [12] The space of oriented lines T S 2 admits a canonical metric G that is invariant under the Euclidean group acting on lines. The metric is of neutral signature (2,2), is conformally flat and scalar flat, but not Einstein. It can be supplemented by a complex structure J and symplectic structure ω, so that (T S 2 , G, J, ω) is a neutral Kähler 4-manifold.
In terms of the local coordinates (ξ, η) the neutral metric is Recall that a (pseudo-) Riemannian manifold (M, G) is conformally flat if it can be covered by local coordinate systems {x j } n 1 such that the metric is ds 2 = Ω 2 j ±dx 2 j , Ω being a non-zero function defined on the coordinate neighbourhood.
The following result supplies local conformal coordinates for line space: Proof. The result follows from pulling back the metric (2.1) by the transformation (2.4) and (2.5) and the result is Note that these coordinates are only local and exclude lines whose direction is parallel to the xy-plane in R 3 (q 3 = 0 or |ξ| = 1). This comes from our original choice of holomorphic coordinate ξ on S 2 via stereographic projection from the south pole. The whole of T S 2 can be covered by such coordinate patches, each of which is diffeomorphic to the product of an open hemisphere and R 2 .
As John never refers to a metric on the space of lines, many of his computations involve the appearance of a mysterious factor -see e.g. equations (6) or (7), or Theorem 2.2 of [15].
We will now demonstrate that the origin of these factors is precisely the conformal factor in this canonical metric. First recall some elementary facts about conformal maps of scalar flat pseudo-Riemannian manifolds.
Let (M, g M ) and (N, g N ) be pseudo-Riemmanian manifolds of dimension n with vanishing scalar curvatures.
Then Ω n−2 2 is a solution of Laplace's equation on M , that is where ∇ is the Levi-Civita connection associated with g M .
Proof. Denote the pulled back metric byg N = f * g N so thatg N = Ω 2 g M . The change of the scalar curvature on M under a conformal change of the metric is given by the well-known relation [3] (2.6)R = Ω −2 R + 4(n − 1) Since the scalar curvatures of both g M and g N are assumed to be zero, R =R = 0, we conclude that ∆ g M Ω Then, for any u : Proof. As before letg N be the pullback metric of g N under f . From the conformallity of f we haveg N = Ω 2 g M and by the standard conformal change of the Laplace operator (see e.g. [3]) By Proposition 2.3 the second term vanishes and so we are left with ∆ g M Ω n−2 Apply this to the neutral 4-manifold (T S 2 , G) of oriented lines in R 3 . By Theorem 2.1 and Proposition 2.2 the metric is locally conformal to the flat metric with conformal factor Ω = 1 Denote this local conformal map by f : Applying the previous Proposition we have Then u is a solution of the ultra-hyperbolic equation In fact this can be seen directly by writing the Laplacian acting on functions v : T S 2 → R in the two sets of coordinates: Note that the conformal factor Ω is precisely the mysterious factor on the right hand side of equation (6) of John's paper [15].
We can now see that Theorem 1.2 follows from Theorem 1.1 with u = Ωv as the line elements of G and g being related by dτ = Ωdl. The conformal factor in the line element appears in Theorem 2.2 of [15] in an ad hoc manner as the cosine of the angle formed by the ruling of the general 1-sheeted hyperboloid with its axis.
Here we gain an insight into its geometric significance as the line element of the neutral metric.
For brevity we denote both the plane in R 2,2 and its image in T S 2 by π. By a translation in R 2,2 (which corresponds to a translation and rotation in R 3 ) set γ to zero. Thus π contains the vertical line through the origin in R 3 (ξ, η) = (0, 0).
A conformal plane can be one of two types. Consider the canonical projection T S 2 → S 2 which takes an oriented line to its direction, in coordinates (ξ, η) → ξ.
Definition 2.7. A conformal plane π ⊂ T S 2 is graphical if its projection onto S 2 has maximal rank (real dimension two). A conformal plane is non-graphical if its projection onto S 2 has rank one or zero.
In the case of rank zero, the conformal plane is the whole fibre of the bundle, which is totally null, and since such a plane is self-orthogonal, in what follows we will ignore this case.

Substituting equations (2.2) and (2.3) in equation (2.7) with γ = 0 and these simplifications yields a pair of real equations with solutions (2.9).
This means in particular that a conformal plane is a section over a hemisphere and that it goes out to infinity along the fibre above the equator. Proposition 2.9. The metric induced on a conformal plane has fixed signature throughout. For a graphical conformal plane the metric can be definite, indefinite or degenerate, while on a non-graphical plane the induced metric can only be indefinite or degenerate.
A similar calculation for non-graphical planes utilizing equation (2.9) leads to The determinant is zero or negative and so the metric is degenerate or indefinite according to whether θ = 0, π or θ = 0, π, respectively.
Graphical indefinite conformal planes in T S 2 are totally real surfaces, having no complex points. Here, complex is with respect to the canonical complex structure J on T S 2 [12]. In fact, as there are no complex points at infinity (|ξ| = 1) either, these twisting line congruences are examples of complete strictly totally real sections of T S 2 → S 2 that, were they Lagrangian with respect to the canonical symplectic structure ω, could not exist by a conjecture of Toponogov [13] [21].
Being symplectic, the 2-parameter family of lines in R 3 defined by a graphical definite conformal plane in T S 2 twist around each other in such a way as to foliate R 3 . Conversely, foliations of R 3 by lines are exactly the definite surfaces over a hemisphere in T S 2 [20].
Given a non-degenerate plane π in R 2,2 , we defined its conjugate plane π ⊥ to be the plane orthogonal to π with respect to the flat neutral metric. This definition of orthogonality is conformally invariant and thus, given a conformal plane in T S 2 we can define its conjugate conformal plane.
Consider the associated conformal planes in T S 2 . When a conformal plane is graphical it is given by equation (2.8), and the tangent space at the origin is spanned by vectors of the form for A ∈ C. A short computation shows that the tangent spaces of two graphical planes intersecting at the origin are orthogonal, G(X,X)| 0 = 0 for all X ∈ T 0 π, X ∈ T 0π , iffα =ᾱ andβ = −β.
A similar calculation for non-graphical planes yields the stated result. The answer is yes, as we now prove.

Theorem 1.3
A pair of non-degenerate conjugate conics can be identified with a pair of rulings of the same surface in R 3 .
In the definite case, they are the two rulings of a 1-sheeted hyperboloid, in the hyperbolic case, they are either parts of the two rulings of a 1-sheeted hyperboloid or the two rulings of a hyperbolic paraboloid.
Proof. The case of definite (and hence graphical) planes is originally John's result -see Theorem 2.1 of [15].
The link between T S 2 and R 3 is given by the map Φ : T S 2 × R → R 3 which takes an oriented line and a number, to the point in R 3 on the line that lies the given distance from the point on the line closest to the origin.
In coordinates (ξ, η) on T S 2 and (X 1 , X 2 , X 3 ) on R 3 , Φ((ξ, η), r) can be written explicitly (2.10) For graphical conformal planes, our starting point is equation (2.8) which can be simplified by rotation and translation so that α = −ai and β = b for a, b ∈ R. To compute the unit (pseudo)-circle, note that the distance to the origin in flat coordinates via equation (2.2) is where we have introduced polar coordinates ξ = Re iθ .
Here one sees again that the metric is definite on the plane if a > b > 0 and so, in that case the unit circle in the conformal plane is given by Substituting equation (2.8) in (2.10) and restricting to the curve (2.11) one finds after a lengthy calculation that the ruled surface in R 3 determined by the unit (pseudo)-circle in a graphical conformal plane satisfies , for the given constants a, b ∈ R. This a one-sheeted hyperboloid, as claimed.
Moreover, the conjugate conic, obtained by switching the signs of a and b, and the sign of the unit radius, satisfies exactly the same quadratic, as can be confirmed by the same calculation with these signs flipped. Thus we have reproven John's result that a pair of conjugate circles in definite planes generate a pair of rulings of the hyperboloid.
When a 2 < b 2 , the metric is indefinite and consider the curve defined by equation (2.11) only for values of θ for which R is real. These are the pseudo-circles of the Lorentz metric and the two rulings are of the same surface given by equation (2.12).
For a non-graphical conformal plane with parameters (θ, φ, H) as given in equation (2.9), the distance to the origin in flat coordinates via equation (2.2) is Non-degeneracy of the induced metric means that θ = 0, π and so one restricts to the unit pseudo-circle by setting The resulting 1-parameter family of lines defines a ruled surface in R 3 that, upon substitution in equations (2.10), is found to satisfy sin θX 3 + 2(cos θ + cos φ)X 2 1 + 4 sin φX 1 X 2 + 2(cos θ − cos φ)X 2 2 = 0. This is a hyperbolic paraboloid for θ = 0, π and flipping to the conjugate pseudocircle, we find it satisfies the same equation. This completes the proof.
Proof. Let Ω be the conformal factor of f . Let α : I → S 0 be a parametrization of S 0 . Then Similarly we would get By Proposition 2.4 we have that ∆Ω(u • f ) = 0, so we can apply Asgeirsson's theorem (Theorem 2.11) to the integrand Ω(u • f ). This can also be expressed more geometrically as Let v : T S 2 → R be a solution of ∆ G v = 0 and f : T S 2 → T S 2 be a conformal map. Then where dτ is the line element induced by the canonical metric G.

Conformal Images of Conjugate Circles
3.1. Pseudo-Euclidean affine space. Let R 2,2 be regarded as an affine space, denoting the underlying vector space by − − → R 2,21 . Let Q be the indefinite distance function associated with the metric, viewed as a quadratic form acting on vectors in − − → R 2,2 : We use ||x|| 2 to mean Q(x) and say that a vector x ∈ − − → R 2,2 is positive if ||x|| 2 > 0, negative if ||x|| 2 < 0, and null whenever ||x|| 2 = 0. Let denote the associate symmetric bilinear form to Q for all x, y ∈ − − → R 2,2 .
We use the symbol ⊥ to denote orthogonality with respect to , . Given a linear subspace W ⊂ − − → R 2,2 we define its orthogonal complement as usual Let P be a 2-subspace of R 2,2 through the origin. Define the degeneracy of the restriction , P to be The same can be defined for an affine plane π. This way, π will be non-degenerate iff its direction − → π is non-degenerate, π will be positive definite iff − → π is positive definite, etc. Note that π is the same as a conformal plane defined in Section 2.3.
In this section we want to show that S ⊥ can be obtained as the locus of points which lie 0 distance (measured by Q) from any three points in S. Similarly, we will see that S can be obtained as the locus of points lying 0−distance from any three points in S ⊥ . Lemma 3.9. Let q, q , q be three skew points on a non-degenerate affine plane π. Then there exists a unique point O ∈ π and a real number c 2 so that || − → Proof. The locus of points in π which are equidistant to q and q is Similarly, let m qq be the locus of points equidistant to q and q . Since q, q , q are assumed to be in general position we have span Hence the affine lines m qq and m qq must intersect on a unique point O in π. Then and we define the radius c 2 to be its value.
For p, p , p distinct points in S ⊥ , we prove S = N (p, p , p ). ⊆: Let q ∈ S, a similar argumentation as before works.
One can assume q = 0 by use of a translation mapping to the origin. One can also assume q = ∞ since an inversion with respect to the hypersphere centered at q going through 0 maps q to ∞ while fixing 0. We show that there is a conformal mapping fixing 0 and ∞, which sends q to 1.
The point q is non-null as q and q were assumed to be skew, and conformal transformations preserve skewness of points. Assume q is positive and let u 1 := q , . . . , u 4 a basis of R 2,2 with Gram matrix I 2,2 := diag (1, 1, −1, −1). Let M be the 4 × 4 real matrix whose columns are the coordinates of the vectors u i with respect to the canonical basis e i . Then M T I 2,2 M = I 2,2 and M e 1 = q , so M is the matrix of an orthogonal transformation mapping e 1 to q . Orthogonal transformations are conformal, and so we are done.
If q is negative then we can find a basis v 1 , v 2 , v 3 := q , v 4 of R 2,2 with Gram matrix I 2,2 . By a similar argument as before we can find an orthogonal transformation of sending e 3 to q . Precomposing with the anti-orthogonal (and therefore conformal) transformation (a, b, c, d) → (c, d, a, b) we get a conformal map sending q to e 1 . Proposition 3.12. Let S, S ⊥ be a pair of non-degenerate conjugate conics. There exists a conformal transformation f : R 2,2 → R 2,2 such that S = f (S 0 ) and S ⊥ = f (S ⊥ 0 ). Proof. Let p, p , p ∈ S ⊥ and p 0 , p 0 , p 0 ∈ S ⊥ 0 . Any three distinct points on a nondegenerate pseudo-circle must be skew, so using the transitivity property we can find an f such that f (p 0 ) = p, f (p 0 ) = p and f (p 0 ) = p . Then f (S 0 ) = f (N (p 0 , p 0 , p 0 )) = N (p, p , p ) = S. Now choose q 0 , q 0 , q 0 ∈ S, and observe that q := f (q 0 ), q := f (q 0 ), q := f (q 0 ) ∈ S as we have already seen f (S 0 ) = S. f (S ⊥ 0 ) = f (N (q 0 , q 0 , q 0 )) = N (q, q , q ) = S ⊥ .
Thus, every non-degenerate pair of conics can be mapped by a conformal map to the standard conjugate circles. By Theorem 1.4 conformal maps preserve the mean value property and so the mean value property extends to any non-degenerate conics and we have completed the proof of Theorem 1.1. We can therefore interpret the linearity of the ultra-hyperbolic equation as the non-interaction of densities at different points.