The method of spherical harmonic expansion of the hydromagnetic equations for incompressible media involves two well-known integral forms that are evaluated by standard numerical integration methods. The results are tabulated to 10 significant figure accuracy for all sets of principled integers inclusive of an 8th-degree harmonic analysis. Extension to compressible media introduces two further surface integral forms that are each simply evaluated in therms of a finite sum of products of the above tabulated integrals. A general development is presented of the known inter-relationships between these four basis integrals, and their selection rules are discussed. Their occurrence in the coupling elements is made explicit in the appendix.