Analysis of the linear theory of capillary instability of threads of Maxwell fluids of diameter D is carried out for the unapproximated normal mode solution and for a solution based on viscoelastic potential flow. The analysis here extends the analysis of viscous potential flow [Int. J. Multiphase Flow 28 (2002) 1459] to viscoelastic fluids of Maxwell type. The analysis is framed in dimensionless variables with a velocity scale based on the natural collapse velocity V = gamma/mu (surface tension/liquid viscosity). The collapse is controlled by two dimensionless parameters, a Reynolds number J = VD rho/mu = rhogamma D/mu(2) = (Oh)(2) where Oh is the Ohnesorge number, and a Deborah number Lambda1 = lambda(1) V/D where lambda(1) is the relaxation time. The density ratio rho(a)/rho and mu(a)/mu are nearly zero and do not have a significant effect on growth rates. The dispersion relation for viscoelastic potential flow is cubic in the growth rate a and it can be solved explicitly and computed without restrictions on the Deborah number. On the other hand, the iterative procedure used to solve the dispersion relation for fully viscoelastic flow fails to converge at very high Deborah number. The growth rates in both theories increase with Deborah number at each fixed Reynolds number, and all theories collapse to inviscid potential flow (IPF) for any fixed Deborah number as the Reynolds number tends to infinity. (C) 2003 Elsevier Science B.V. All rights reserved.