  We calculate spin Hall conductivities in the two dimensional electron systems
with Rashba spin-orbit interaction. The salient feature is that, apart from the
usual spin-Hall conductivity \sigma^z_{xy} which corresponds to the induction
of out-of-plane spin-current due to the application of transverse charge
current, there is a novel spin-Hall conductivity \sigma^\perp_{xy} which arises
due to the induction of transverse spin-polarized current in the transverse
direction by the application of in-plane spin-polarized current. This
phenomenon which we call as `spin-spin' Hall effect is a spin analog of
conventional Hall effect, but with no magnetic field. This contribution may be
understood through the spin-diffusive equation.
