  The study of long wavelength scalar perturbations, in particular the
existence of conserved quantities when the perturbations are adiabatic, plays
an important role in e.g. inflationary cosmology. In this paper we present some
new conserved quantities at second order and relate them to the curvature
perturbation in the uniform density gauge, $\zeta$, and the comoving curvature
perturbation, ${\cal R}$. We also, for the first time, derive the general
solution of the perturbed Einstein equations at second order, which thereby
contains both growing and decaying modes, for adiabatic long wavelength
perturbations for a stress-energy tensor with zero anisotropic stresses and
zero heat flux. The derivation uses the total matter gauge, but results are
subsequently translated to the uniform curvature and Poisson (longitudinal,
zero shear) gauges.
