  We investigate the relevance to the physics of underdoped
YBa$_2$Cu$_3$O$_{\rm 6+x}$ and YBa$_2$Cu$_4$O$_8$ of the quantum critical point
which occurs in a model of two antiferromagnetically coupled planes of
antiferromagnetically correlated spins. We use a Schwinger boson mean field
theory and a scaling analysis to obtain the phase diagram of the model and the
temperature and frequency dependence of various susceptibilities and relaxation
rates. We distinguish between a low $\omega ,T$ coupled-planes regime in which
the optic spin excitations are frozen out and a high $\omega ,T$
decoupled-planes regime in which the two planes fluctuate independently. In the
coupled-planes regime the yttrium nuclear relaxation rate at low temperatures
is larger relative to the copper and oxygen rates than would be naively
expected in a model of uncorrelated planes. Available data suggest that in
YBa$_2$Cu$_4$O$_8$ the crossover from the coupled to the decoupled planes
regime occurs at $T 700K$ or $T \sim 200K$. The predicted correlation length is
of order 6 lattice constants at $T=200K$. Experimental data related to the
antiferromagnetic susceptibility of YBa$_2$Cu$_4$O$_8$ may be made consistent
with the theory, but available data for the uniform susceptibility are
inconsistent with the theory.
