  Cometary globule (CG) 1 and CG 2 are "classic" CGs in the Gum Nebula. They
have compact heads and long dusty tails that point away from the centre of the
Gum Nebula. We study the structure of CG 1 and CG 2 and the star formation in
them to find clues to the CG formation mechanism. The two possible mechanisms,
radiation-driven implosion (RDI) and a supernova (SN) blast wave, produce a
characteristic mass distribution where the major part of the mass is situated
in either the head (RDI) or the tail (SN). CG 1 and CG 2 were imaged in the
near infrared (NIR) JsHKs bands. NIR photometry was used to locate NIR excess
objects and to create extinction maps of the CGs. The A_V maps allow us to
analyse the large-scale structure of CG 1 and CG 2. Archival images from the
WISE and Spitzer satellites and HIRES-processed IRAS images were used to study
the small-scale structure. In addition to the previously known CG 1 IRS 1 we
discovered three new NIR-excess objects, two in CG 1 and one in CG 2. CG 2 IRS
1 is the first detection of star formation in CG 2. Spectral energy
distribution (SED) fitting suggests the NIR-excess objects are young low-mass
stars. CG 1 IRS 1 is probably a class I protostar in the head of CG 1. CG 1 IRS
1 drives a bipolar outflow, which is very weak in CO, but the cavity walls are
seen in reflected light in our NIR and in the Spitzer 3.6 and 4.5 mum images.
Strong emission from excited polycyclic aromatic hydrocarbon particles and very
small grains were detected in the CG 1 tail. The total mass of CG 1 in the
observed area is 41.9 Msun of which 16.8 Msun lies in the head. For CG 2 these
values are 31.0 Msun total and 19.1 Msun in the head. The observed mass
distribution does not offer a firm conclusion for the formation mechanism of
these CGs: CG 1 is in too evolved a state, and in CG 2 part of the globule tail
was outside the observed area. (abridged)
