  Although several dozen double white dwarfs (DWDs) have been observed, for
many the exact nature of the evolutionary channel(s) by which they form remains
uncertain. The canonical explanation calls for the progenitor binary system to
undergo two subsequent mass-transfer events, both of which are unstable and
lead to a common envelope (CE). However, it has been shown that if both CE
events obey the standard alpha-prescription (parametrizing energy loss), it is
not possible to reproduce all of the observed systems. The gamma-prescription
was proposed as an alternative to this description, instead parametrizing the
fraction of angular momentum carried away in dynamical-timescale mass loss. In
this paper, we analyze simultaneous energy and angular-momentum conservation,
and show that the gamma-prescription cannot adequately describe a CE event for
an arbitrary binary, nor can the first phase of mass loss always be understood
in general as a dynamical-timescale event. We consider in detail the first
episode of mass transfer in binary systems with initially low companion masses,
with a primary mass in the range 1.0--1.3 solar masses and an initial mass
ratio between the secondary and primary stars of 0.83--0.92. In these systems,
the first episode of dramatic mass loss may be stable, non-conservative mass
transfer. This strips the donor's envelope and dramatically raises the mass
ratio; the considered progenitor binary systems can then evolve into DWDs after
passing through a single CE during the second episode of mass loss. We find
that such a mechanism reproduces the properties of the observed DWD systems
which have an older component with less than approximately 0.46 solar masses
and mass ratios between the younger and older WDs greater than 1.
