We derive a new Galerkin quadrature (GQ) method for S calculations that differs from the two methods preceding it in that a matrix inverse for an matrix, where is the number of directions in the quadrature set, is no longer required. Galerkin quadrature methods are designed for calculations with highly anisotropic scattering. Such methods are not simply special angular quadratures but also are methods for representing the S scattering source that offers several advantages relative to the standard scattering source representation when highly truncated Legendre cross-section expansions must be used. Galerkin quadrature methods are also useful when the scattering is moderately anisotropic, but the quadrature being used is not sufficiently accurate for the order of the scattering source expansion that is required. We derive the new method and present computational results showing that its performance for two challenging problems is comparable to those of the two GQ methods that preceded it.