A study complementary to Part 1 (Smith 1979) is made of the linear stability characteristics, at high Reynolds number (R), of Poiseuille How through tubes with closed cross-sections. The first significant deviation of the upper branch of the neutral stability curve (Part 1 having described the lower bilanch) from that of plane Poiseuille flow arises when the aspect ratio is decreased from infinity to O(R1/11). The axial wavenumber α on the upper branch is then O(R-1/11). A further decrease of the aspect ratio, to a finite value, forces this α to fall sharply to O(R-1). A similar phenomenon occurs for the lower branch (Part 1). Thus the two branches are likely to meet only when the aspect ratio becomes finite, with the neutrally stable disturbances then having very large axial length scales.