Canonical solution of the equations of axisymmetric gravitation including time dependence

The Einstein vacuum-field equations for axially symmetric gravitational fields are solved by means of a convergent series in powers of the cylindrical radius. The principles of the method of axial expansions are explained and illustrated; the general line element, main field equations, and subsidiary field equations are derived; the canonical solution for the case of time-dependent azimuthally invariant fields is obtained; two special cases of generators giving the Einstein-Rosen cylindrical-wave and the Weyl static solutions, respectively, are examined; and the extension of the line element to rotating solutions is considered.