  A transfer matrix scaling technique is developed for randomly diluted
systems, and applied to the site-diluted Ising model on a square lattice in two
dimensions. For each allowed disorder configuration between two adjacent
columns, the contribution of the respective transfer matrix to the decay of
correlations is considered only as far as the ratio of its two largest
eigenvalues, allowing an economical calculation of a configuration-averaged
correlation length. Standard phenomenological-renormalisation procedures are
then used to analyse aspects of the phase boundary which are difficult to
assess accurately by alternative methods. For magnetic site concentration $p$
close to $p_c$, the extent of exponential behaviour of the $T_c \times p$ curve
is clearly seen for over two decades of variation of $p - p_c$. Close to the
pure-system limit, the exactly-known reduced slope is reproduced to a very good
approximation, though with non-monotonic convergence. The averaged correlation
lengths are inserted into the exponent-amplitude relationship predicted by
conformal invariance to hold at criticality. The resulting exponent $\eta$
remains near the pure value (1/4) for all intermediate concentrations until it
crosses over to the percolation value at the threshold.
