Observed galaxy number counts on the lightcone up to second order: II.
  Derivation
  We present a detailed derivation of the observed galaxy number over-density
on cosmological scales up to second order in perturbation theory. We include
all relativistic effects that arise from observing on the past lightcone. The
derivation is in a general gauge, and applies to all dark energy models
(including interacting dark energy) and many modified gravity models. The
result will be important for accurate cosmological parameter estimation,
including non-Gaussianity, since all projection effects need to be taken into
account. It also offers the potential for new probes of General Relativity,
dark energy and modified gravity. This paper accompanies Paper I which presents
the key results for the concordance model in Poisson gauge.
