The linearised equations describing the onset of axisymmetric viscoelastic instability in Couette flows with negligible inertia (R.G. Larson, E.S.G. Shaqfeh and S.J. Muller, J. Fluid Mech., 218 (1990) 573) are reconsidered in the narrow gap limit. A simple proof is provided that the bifurcation parameter LAMBDA is purely imaginary, and it is shown how LAMBDA can be evaluated accurately using rapidly convergent series. A new lower bound is found for the behaviour of LAMBDA at large wavenumbers.