  Molecular-line observations of star-forming cloud cores indicate that they
are not the flattened structures traditionally considered by theory. Rather,
they are elongated, perhaps in the direction of their internal magnetic field.
We are thus motivated to consider the structure and evolution of axisymmetric,
magnetized clouds that start from a variety of initial states, both flattened
(oblate) and elongated (prolate). We devise a new technique, dubbed the
$q$-method, that allows us to construct magnetostatic equilibria of any
specified shape. We find, in agreement with previous authors, that the field
lines in oblate clouds bend inward. However, those in prolate clouds bow
outward, confining the structures through magnetic tension.
  We next follow the quasi-static evolution of these clouds via ambipolar
diffusion, under the assumption of constant core mass. An oblate cloud either
relaxes to a magnetically force-free sphere or, if sufficiently massive,
flattens along its polar axis as its central density runs away. A prolate cloud
always relaxes to a sphere of modest central density. We finally consider the
evolution of an initially spherical cloud subject to the tidal gravity of
neighboring bodies. Although the structure constricts equatorially, it also
shortens along the pole, so that it ultimately flattens on the way to collapse.
In summary, none of our initial states can evolve to the point of collapse
while maintaining an elongated shape. We speculate that this situation will
change once we allow the cloud to gain mass from its environment.
