  We report on the long term stability of the millisecond oscillations observed
with the Rossi X-ray Timing Explorer (RXTE) during thermonuclear X-ray bursts
from the low mass X-ray binaries (LMXB) 4U 1728-34 and 4U 1636-53. We show that
bursts from 4U 1728-34 spanning more than 1.5 years have observed asymptotic
oscillation periods which are within 0.2 microsec. of each other, well within
the magnitude which could be produced by the orbital motion of the neutron star
in a typical LMXB. This stability implies a timescale to change the oscillation
period of > 23,000 years, suggesting a highly stable process such as stellar
rotation as the oscillation mechanism. We show that period offsets in three
distinct bursts from 4U 1636-53 can be plausibly interpreted as due to orbital
motion of the neutron star in this 3.8 hour binary system. We discuss the
constraints on the mass function which can in principle be derived using this
technique.
