A development in convenient matrix representation is given for the first four moments of the neutron scattering kernel K(ν,ν’), for a monatomic Maxwellian gas with arbitrary dependence of the scattering cross section on relative speed. The availability of explicit forms for (x,x’), components of the moments in terms of dimensionless speeds, forn = 0, I, 2, and 3, as single-integral expressions facilitates the spherical harmonics solution of neutron thermalization problems through the P3-approxima-tion.