We have mapped linearly polarized dust emission from L183 with the JCMT SCUBA polarimeter and have analyzed these and our previously published data for the prestellar cores L183 , L1544 , and L43 in order to estimate magnetic field strengths in the plane of the sky , B _ { pos } . The analysis used the Chandrasekhar-Fermi technique , which relates the dispersion in polarization position angles to B _ { pos } . We have used these estimates of the field strengths ( neglecting the unmeasured line-of-sight component ) to find the mass-to-magnetic flux ratios \lambda ( in units of the critical ratio for magnetic support ) . Results are B _ { pos } \approx 80 \mu G and \lambda \approx 2.6 for L183 , B _ { pos } \approx 140 \mu G and \lambda \approx 2.3 for L1544 , and B _ { pos } \approx 160 \mu G and \lambda \approx 1.9 for L43 . Hence , without correction for geometrical biases , for all three cores the mass-to-flux ratios are supercritical by a factor of \sim 2 , and magnetic support can not prevent collapse . However , a statistical mean correction for geometrical bias may be up to a factor of three ; this correction would reduce the individual \lambda ’ s to \lambda _ { cor } \approx 0.9 , 0.8 , and 0.6 , respectively ; these values are approximately critical or slightly subcritical . These data are consistent with models of star formation driven by ambipolar diffusion in a weakly turbulent medium , but can not rule out models of star formation driven by turbulence .