§1. In this paper we find solutions of Einstein’s gravitational equations G
μν
= 0 which give the field due to any static distribution of matter symmetrical about an axis; in the later part of the paper an angular velocity about the axis is introduced. We take the ground form
ds
2
= -
e
λ
(
dx
2
+
dr
2
) -
e
-ρ
r
2
d
θ
2
+
e
ρ
dt
2
, (1) where λ, ρ are functions of
x
and
r
. Further we take ρ to be the Newtonian potential of an auxiliary distribution of matter of density σ (
x, r
), the potential being calculated as though our co-ordinates were Euclidean. We find that it is then possible to determine λ, so that the equations G
μν
= 0 are exactly satisfied everywhere outside the auxiliary body. λ is nearly equal to —ρ, the quantity μ = λ + ρ being of the second order in terms of σ.
Testing axial symmetry by means of directional regression quantiles
