  We report on a three-year program of coordinated X-ray and optical monitoring
of the Narrow-Line Seyfert 1 galaxy NGC 4051. The rapid continuum variations
observed in the X-ray spectra are not detected in the optical, although the
X-ray and optical continuum fluxes are correlated on time scales of many weeks
and longer. Variations in the flux of the broad H-beta line are found to lag
behind the optical continuum variations by approximately 6 days (with an
uncertainty of 2-3 days), and combining this with the line width yields a
virial mass estimate of about 1.1e6 solar masses, at the very low end of the
distribution of AGN masses measured by line reverberation. Strong variability
of He II lambda 4686 is also detected, and the response time measured is
similar to that of H-beta, but with a much larger uncertainty. The He II lambda
4686 line is almost five times broader than H-beta, and it is strongly blueward
asymmetric, as are the high-ionization UV lines recorded in archive spectra of
NGC 4051. The data are consistent with the Balmer lines arising in a
low-inclination (nearly face-on) disk-like configuration, and the
high-ionization lines arising in an outflowing wind, of which we observe
preferentially the near side. During the third year of monitoring, both the
X-ray continuum and the He II lambda 4686 line went into extremely low states,
although the optical continuum and the H-beta broad line were both still
present and variable. We suggest that the inner part of the accretion disk may
have gone into an advection-dominated state, yielding little radiation from the
hotter inner disk.
