  The rigoruos mathematical theory of the coupling and response of a spherical
gravitational wave detector endowed with a set of resonant transducers is
presented and developed. A perturbative series in ascending powers of the
square root of the ratio of the resonator to the sphere mass is seen to be the
key to the solution of the problem. General layouts of arbitrary numbers of
transducers can be assessed, and a specific proposal (PHC), alternative to the
highly symmetric TIGA of Merkowitz and Johnson, is described in detail.
Frequency spectra of the coupled system are seen to be theoretically recovered
in full agreement with experimental determinations.
