Cross sections for several processes involving electromagnetic fields in a nonlinear manner are derived from the electrodynamic scattering matrix and are expressed in terms of the fourth-order nonlinear vacuum polarization tensor. The differential cross section for the scattering of light by light is calculated as a function of energy and angle. Numerical values are given for scattering at zero and at ninety degrees in the center-of-mass system. Near 1.75 Mev the forward scattering cross section has its largest value of 4.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}31}$ ${\mathrm{cm}}^{2}$/sterad, while the maximum right-angle scattering takes place near 0.7 Mev with a cross section of 2.8\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}31}$ ${\mathrm{cm}}^{2}$/sterad, all for unpolarized radiation. Numerical results are also given for scattering at the above angles between circularly polarized states. The conclusions in this paper agree with all the results previously calculated for special cases.