  We consider discs that orbit a central object and are tidally perturbed by a
circular orbit companion. Such discs are sometimes subject to an eccentric
instability due to the effects of certain resonances. Eccentric instabilities
may be present in planetary rings perturbed by satellites, protostellar discs
perturbed by planets, and discs in binary star systems. Although the basic
mechanism for eccentric instability is well understood, the detailed response
of a gaseous disc to such an instability is not understood. We apply a linear
eccentricity evolution equation developed by Goodchild and Ogilvie. We explore
how the eccentricity is distributed in such a disc and how the distribution in
turn affects the instability growth rate for a range of disc properties. We
identify a disc mode, termed the superhump mode, that is likely at work in the
superhump binary star case. The mode results from the excitation of the
fundamental free precession mode. We determine an analytic expression for the
fundamental free mode precession rate that is applicable to a sufficiently cool
disc. Depending on the disc sound speed and disc edge location, other eccentric
modes can grow faster than the superhump mode and dominate.
