  We show that it is possible to reconcile NMR and neutron scattering
experiments on both LSCO and YBCO, by making use of the Millis-Monien-Pines
mean field phenomenological expression for the dynamic spin-spin response
function, and reexamining the standard Shastry-Mila-Rice hyperfine Hamiltonian
for NMR experiments. The recent neutron scattering results of Aeppli et al on
LSCO (x=14%) are shown to agree quantitatively with the NMR measurements of
$^{63}T_1$ and the magnetic scaling behavior proposed by Barzykin and Pines.
The reconciliation of the $^{17}T_1$ relaxation rates with the degree of
incommensuration in the spin fluctuation spectrum seen in neutron experiments
is achieved by introducing a new transferred hyperfine coupling $C'$ between
oxygen nuclei and their next nearest neighbor $Cu^{2+}$ spins; this leads to a
near-perfect cancellation of the influence of the incommensurate spin
fluctuation peaks on the oxygen relaxation rates of LSCO. The inclusion of the
new $C'$ term also leads to a natural explanation, within the one-component
model, the different temperature dependence of the anisotropic oxygen
relaxation rates for different field orientations, recently observed by
Martindale $et~al$. The measured significant decrease with doping of the
anisotropy ratio, $R= ^{63}T_{1ab}/^{63}T_{1c}$ in LSCO system, from $R =3.9$
for ${\rm La_2CuO_4}$ to $R ~ 3.0$ for LSCO (x=15%) is made compatible with the
doping dependence of the shift in the incommensurate spin fluctuation peaks
measured in neutron experiments, by suitable choices of the direct and
transferred hyperfine coupling constants $A_{\beta}$ and B.
