  We present a measurement of the coefficient alpha_2 of the leptonic
polar-angle distribution from W boson decays, as a function of the W transverse
momentum. The measurement uses an 80+/-4 pb^{-1} sample of proton-antiproton
collisions at sqrt{s}=1.8 TeV collected by the CDF detector and includes data
from both the W->e+nu and W->mu+nu decay channels. We fit the W boson
transverse mass distribution to a set of templates from a Monte Carlo event
generator and detector simulation in several ranges of the W transverse
momentum. The measurement agrees with the Standard Model expectation, whereby
the ratio of longitudinally to transversely polarized W bosons, in the
Collins-Soper W rest frame, increases with the W transverse momentum at a rate
of approximately 15% per 10 GeV/c.
