A class of statistics suitable for testing against equality of multivariate distributions is described by Klebanov and co-workers in 2007. Referred to as N-statistics, their discriminating ability is based on various forms of distance kernels in R(d), the intention being to capture distinct forms of deviation from equality. This makes them particularly suitable for large-scale genomic screening applications, in which such variety of alternatives can be anticipated. One of these kernels, denoted as L(4), introduces weighting by directional densities, hence the evaluation of L(4) requires integration on the unit sphere in R(d). In this note we introduce a methodology for the evaluation of integrals related to L(4). It is shown that for a class of directional densities including, but not limited to, the uniform density L(4) reduces to Euclidean distance. For other cases, the methodology permits a direct interpretation of L(4) in terms of the directional weighting.