  I derive a rigorous relation between the expectation value of the transverse
component of the angular momentum <J_T> of a quark in a transversely polarized
nucleon in terms of the Generalized Parton Distributions H and E, namely
<J_T(quark)> = 1/2M [P_0 \int dx x E_q(x,0,0) + M \int dx x H_q(x,0,0)] where
P_0 is the energy of the nucleon and where "quark" implies the sum of quark and
antiquark of a given flavor. A similar relation holds for gluons. The result is
remarkably similar to Ji's relation for the case of longitudinal polarization.
  In this revised version, I have rectified some misleading statements,
introduced a more precise notation and specified mpre clearly the meaning of
the terms in both the Ji and the new tranverse relation.
