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We study translational correlations of the vortex {\em center of mass}positions of the Abrikosov flux-line lattice in superconducting samples offinite thickness $L$ (along the direction of flux-lines). The Larkincorrelation lengths for the center of mass mode of the flux lines in thepresence of point and correlated disorder are computed, and we find that in thecase of point disorder the average (i.e. center of mass) position of flux linesmaintains positional order on length scales which scale like $\sim\sqrt{L}$ in2+1 dimensions. On still longer length scales, however, we find using a replicaGaussian variational approach that center of mass correlations cross over to apower law growth of the form $r_\perp/L$, which should be observable insuperconducting thin films.Comment: Introduction significantly expanded, some typos correcte

POSITIONAL ORDER OF THE PINNED ABRIKOSOV FLUX LINE LATTICE IN SAMPLES OF FINITE THICKNESS