  We present the detection of a positive correlation between spin-down rate
$\dot{P}$ and pulsed X-ray luminosity in the BATSE archival data of the bright
hard X-ray pulsar GX 1+4. We have also seen a delay of 5.6 $\pm$ 1.2 days
between the luminosity change and the corresponding change in the spin-down
rate. The observed correlation between $\dot{P}$ and L_X is used to reproduce
the period history of GX 1+4 based on the observed luminosity alone, and it is
found that the spin period can be predicted correct to 0.026% when the
luminosity is adequately sampled. The idea that at a higher luminosity more
matter is accreted and the accretion disk extends closer to the neutron star
thereby transferring more angular momentum to the system, seems not to be the
case with GX 1+4. The observed lag between the spin-down rate and the
luminosity is reported here for the first time in any such binary X-ray pulsar,
and is found to be consistent with the time scale for the core-crust relaxation
in a neutron star.
