  The specific heats of exactly solvable alternating layered planar Ising
models with strips of width $m_1$ lattice spacings and ``strong'' couplings
$J_1$ sandwiched between strips of width $m_2$ and ``weak'' coupling $J_2$,
have been studied numerically to investigate the effects of connectivity and
proximity. We find that the enhancements of the specific heats of the strong
layers and of the overall or `bulk' critical temperature,
$T_c(J_1,J_2;m_1,m_2)$, arising from the collective effects reflect the
observations of Gasparini and coworkers in experiments on confined superfluid
helium. Explicitly, we demonstrate that finite-size scaling holds in the
vicinity of the upper limiting critical point $T_{1c}$ ($\propto J_1/k_B$) and
close to the corresponding lower critical limit $T_{2c}$ ($\propto J_2/k_B$)
when $m_1$ and $m_2$ increase. However, the residual {\it enhancement}, defined
via appropriate subtractions of leading contributions from the total specific
heat, is dominated (away from $T_{1c}$ and $T_{2c}$) by a decay factor
$1/(m_1+m_2)$ arising from the {\it seams} (or boundaries) separating the
strips; close to $T_{1c}$ and $T_{2c}$ the decay is slower by a factor $\ln
m_1$ and $\ln m_2$, respectively.
