  We formulate the problem of the formation and collapse of nonaxisymmetric
protostellar cores in weakly ionized, self-gravitating, magnetic molecular
clouds. In our formulation, molecular clouds are approximated as isothermal,
thin (but with finite thickness) sheets. We present the governing dynamical
equations for the multifluid system of neutral gas and ions, including
ambipolar diffusion, and also a self-consistent treatment of thermal pressure,
gravitational, and magnetic (pressure and tension) forces. The dimensionless
free parameters characterizing model clouds are discussed. The response of
cloud models to linear perturbations is also examined, with particular emphasis
on length and time scales for the growth of gravitational instability in
magnetically subcritical and supercritical clouds. We investigate their
dependence on a cloud's initial mass-to-magnetic-flux ratio (normalized to the
critical value for collapse), the dimensionless initial neutral-ion collision
time, and also the relative external pressure exerted on a model cloud. Among
our results, we find that nearly-critical model clouds have significantly
larger characteristic instability lengthscales than do more distinctly sub- or
supercritical models. Another result is that the effect of a greater external
pressure is to reduce the critical lengthscale for instability. Numerical
simulations showing the evolution of model clouds during the linear regime of
evolution are also presented, and compared to the results of the dispersion
analysis. They are found to be in agreement with the dispersion results, and
confirm the dependence of the characteristic length and time scales on
parameters such as the initial mass-to-flux ratio and relative external
pressure.

Astrophysics