In this paper, we study the linearized stability of three symmetric arrangements of two liquids in core-annular Poiseuille flow in round pipes. Deferring to one important application, we say oil and water when we mean more viscous and less viscous liquids. The three arrangements are (i) oil is in the core and water on the wall, (ii) water is in the core and oil is outside and (iii) three layers, oil inside and outside with water in between. The arrangement in (iii) is our model for lubricated pipelining when the pipe walls are hydrophobic and i t has not been studied before. The arrangement in (ii) was studied by Hickox (1971) who treated the problem as a perturbation of long waves, effectively suppressing surface tension and other essential effects which are necessary to explain the flows observed, say, in recent experiments of W. L. Olbricht and R. W. Aul. The arrangement in (i) was studied in Part 1 of this paper (Preziosi, Chen & Joseph 1987). We have confirmed and extended their pseudo-spectral calculation by introducing a more efficient finite-element code. We have calculated neutral curves, growth rates, maximum growth rate, wavenumbers for maximum growth and the various terms which enter into the analysis of the equation for the evolution of the energy of a small disturbance. The energy analysis allows us to identify the three competing mechanisms underway : interfacial tension, interfacial friction and Reynolds stress. Many results are presented.