We present high-quality H alpha spectra of the massive X-ray binary HDE 226
868 = Cyg X-1 that were made in 1985 and 1986. We measured radial velocitie
s using the He I lambda 6678 line, and we use the resulting orbital solutio
n together with a Hipparcos light curve of the ellipsoidal variation to arr
ive at a revised period estimate of P = 5.59977 +/- 0.00002 days. We analyz
e the orbital phase-related variations in H alpha emission using a Doppler
tomography algorithm, and we show that the profile variations are well matc
hed by a linear combination of two components: a P Cygni profile that moves
with the motion of the supergiant star and an emission component that foll
ows an antiphase but phase-shifted velocity curve (semiamplitude K-em = 68
km s(-1) and radial velocity maximum at phi(0) = 0.86; these parameters are
similar to those for the He II lambda 4686 emission line). We use a vector
decomposition of this motion to argue that the latter emission component f
orms between the stars in a focused wind flow from the supergiant to the un
seen companion. We searched for but found no clear evidence of H alpha emis
sion from the accretion disk surrounding the companion (any such emission c
ontributes less than 3% of the continuum intensity). The results demonstrat
e that the H alpha line provides an important probe of the mass transfer pr
ocess that eventually fuels the X-ray source.