  The reduced proper motion diagram (RPMD) for a complete sample of faint stars
with high accuracy proper motions in the North Galactic Pole field SA57 is
investigated. Eight stars with very large reduced proper motions are identified
as faint white dwarf candidates. We discriminate these white dwarf candidates
from the several times more numerous QSOs based on proper motion and
variability.
  We discuss the implausibility that these stars could be any kind of survey
contaminant. If {\it bona fide} white dwarfs, the eight candidates found here
represent a portion of the white dwarf population hitherto uninvestigated by
previous surveys by virtue of the faint magnitudes and low proper motions. The
newly discovered stars suggest a disk white dwarf scaleheight larger than the
values of 250-350 pc typically assumed in assessments of the local white dwarf
density. Both a <V/V_{max}> and a more complex maximum likelihood analysis of
the spatial distribution of our likely thin disk white dwarfs yield
scaleheights of 400-600 pc while at the same time give a reasonable match to
the local white dwarf volume density found in other surveys.
  Our results could have interesting implications for white dwarfs as potential
MACHO objects. We can place some direct constraints (albeit weak ones) on the
contribution of halo white dwarfs to the dark matter of the Galaxy. Moreover,
the elevated scale height that we measure for the thin disk could alter the
interpretation of microlensing results to the extent of making white dwarfs
untenable as the dominant MACHO contributor. (Abridged)
