  We revisit the measurement of the polarization fraction, $P$, and the
measurement of the polarization angle of partially linearly-polarized gamma
rays using their conversion to $e^+e^-$ pairs in the field of a nucleus. We
show that an inappropriate definition of the azimuthal angle, $\varphi$, used
to reference the orientation of the final state degrades the precision of the
measurement of $P$, by comparison to the optimal case where the bisector angle
of the electron and of the positron momenta is used. We then focus on the
lowest part of the energy spectrum, below $\approx 10\,$MeV, where a large part
of the statistics lie for a cosmic source. We obtain the value of the
polarization asymmetry, $A$, of pair conversion at threshold and we show that
in the case where the correct expression is used for $\varphi$, the measured
value of $A$ tends to the limit.
