  Recent inelastic neutron scattering (INS) experiments on
La$_{2-x}$Sr$_x$CuO$_4$ have established the existence of a {\it magnetic
coherence effect}, i.e., strong frequency and momentum dependent changes of the
spin susceptibility, $\chi''$, in the superconducting phase. We show, using the
spin-fermion model for incommensurate antiferromagnetic spin fluctuations, that
the magnetic coherence effect establishes the ability of INS experiments to
probe the electronic spectrum of the cuprates, in that the effect arises from
the interplay of an incommensurate magnetic response, the form of the
underlying Fermi surface, and the opening of the d-wave gap in the fermionic
spectrum. In particular, we find that the magnetic coherence effect observed in
INS experiments on La$_{2-x}$Sr$_x$CuO$_4$ requires that the Fermi surface be
closed around $(\pi,\pi)$ up to optimal doping. We present several predictions
for the form of the magnetic coherence effect in YBa$_2$Cu$_3$O$_{6+x}$ in
which an incommensurate magnetic response has been observed in the
superconducting state.
