This present work treats on the hydromagnetic phenomena associated with astrophysical bodies, with particular emphasis being given to the earth's core. The transformation, by the method of spherical harmonic expansion, for compressible media, of the amplification equation, the Navier-Stokes equation, and the thermal energy transport equation are carried out, and an analysis is made of the resulting partial differential equations that define the scaloidal, poloidal, and toroidal sets of radial functions. General mathematical methods of transforming arbitrary differential equations are discussed.