  We begin by recalling the isothermal, collisionless, disc-halo. The disc
component is the Mestel disc. Subsequently we introduce spiral arms to such an
isothermal disc-halo system that are co-moving in the mean with an
axi-symmetric background. These correspond to a similar disturbance in the
halo, which is comprised of spiral structures on cones. The arms are
necessarily transient due to the differential winding in the disc and their
gradual destruction is described. Although the spiral potentials are weak
compared to the axi-symmetric potential the arms are not propagating waves on
the background, but rather co-move with it. They have an effect
disproportionate to their relative magnitude on the gas distribution in the
disc. The gas accumulates on the outside leading edge of the 'stellar' arm and
an arm-interarm modulation of up to 100% is possible. Compatible isothermal,
scale-free, distribution functions are found either exactly or approximately
for all of the collisionless components of the disc-halo system. Repeated
episodes of winding arms can produce an exponential disc.

Astrophysics of Galaxies