  In this work I present a numerical study of the Finite Size Scaling (FSS) of
a correlation length in the framework of the $CP ^{N-1}$ model by means of the
1/N expansion. This study has been thought as propedeutical to the application
of FSS to the measure on the lattice of a new coupling constant $f_{x}(1/R)$,
defined in terms or rectangular Wilson Loops. I give also a perturbative
expansion of $f_{x}(1/R)$ in powers of the corresponding coupling constant in
the $\overline{MS}$ scheme together with some preliminary numerical results
obtained from the Polyakov ratio and I point out the conceptual problems that
limit this approach.
