  Part I of this article studied the specific heats of planar alternating
layered Ising models with strips of strong coupling $J_1$ sandwiched between
strips of weak coupling $J_2$, to illustrate qualitatively the effects of
connectivity, proximity, and enhancement in analogy to those seen in extensive
experiments on superfluid helium by Gasparini and coworkers. It was
demonstrated graphically that finite-size scaling descriptions hold in a
variety of temperature regions including in the vicinity of the two specific
heat maxima. Here we provide exact theoretical analyses and asymptotics of the
specific heat that support and confirm the graphical findings. Specifically, at
the overall or bulk critical point, the anticipated (and always present)
logarithmic singularity is shown to vanish exponentially fast as the width of
the stronger strips increases.
