  Accretion disks around stars, or other central massive bodies, can support
long-lived, slowly precessing $m=1$ disturbances in which the fluid motion is
nearly Keplerian with non-zero eccentricity. We study such `slow modes' in
disks that are subject to both pressure and self-gravity forces. We derive a
second-order WKB dispersion relation that describes the dynamics quite
accurately, and show that the apparently complicated nature of the various
modes can be understood in a simple way with the help of a graphical method. We
also solve the linearized fluid equations numerically, and show that the
results agree with the theory. We find that when self-gravity is weak
($Q\gtrsim 1/h$, where $Q$ is Toomre's parameter, and $h$ is the disk aspect
ratio) the modes are pressure dominated. But when self-gravity is strong
($1<Q\lesssim 1/h$), two kinds of gravity-dominated modes appear: one is an
aligned elliptical pattern and the other is a one-armed spiral. In the context
of protoplanetary disks, we suggest that if the radial eccentricity profile can
be measured, it could be used to determine the total disk mass.
