  Magnetic and energetic properties are presented for 17 dense cores within a
few hundred pc of the Sun. Their plane-of-sky field strengths are estimated
from the dispersion of polarization directions, following Davis, Chandrasekhar
and Fermi (DCF). Their ratio of mass to magnetic critical mass is 0.5-3,
indicating nearly critical field strengths. The field strength B_pos is
correlated with column density N as B_pos~N^p, where p=1.05+-0.08, and with
density n as B_pos~n^q, where q=0.66+-0.05. These magnetic properties are
consistent with those derived from Zeeman studies (Crutcher et al. 2010), with
less scatter. Relations between virial mass M_V, magnetic critical mass M_B,
and Alfven amplitude sigma_B/B match the observed range of M/M_B for cores
observed to be nearly virial, with M/M_V=0.5-2, with moderate Alfven
amplitudes, and with sigma_B/B=0.1-0.4. The B-N and B-n correlations in the DCF
and Zeeman samples can be explained when such bound, Alfvenic, and
nearly-critical cores have central concentration and spheroidal shape. For
these properties, B~N because M/M_B is nearly constant compared to the range of
N, and B~n^(2/3) because M^(1/3) is nearly constant compared to the range of
n^(2/3). The observed core fields which follow B~n^(2/3) need not be much
weaker than gravity, in contrast to core fields which follow B~n^(2/3) due to
spherical contraction at constant mass (Mestel 1966). Instead, the nearly
critical values of M/M_B suggest that the observed core fields are nearly as
strong as possible, among values which allow gravitational contraction.
