  The number and type of measurements needed to ascertain the amplitudes for
pseudoscalar meson photoproduction are analyzed in this paper. It is found that
8 carefully selected measurements can determine the four transversity
amplitudes without discrete ambiguities. That number of measurements is one
less than previously believed. We approach this problem in two distinct ways:
(1) solving for the amplitude magnitudes and phases directly; and (2) using a
bilinear helicity product formulation to map an algebra of measurements over to
the well-known algebra of the $4\times 4$ Gamma matrices. It is shown that the
latter method leads to an alternate proof that 8 carefully chosen experiments
suffice for determining the transversity amplitudes completely. In addition,
Fierz transformations of the Gamma matrices are used to develop useful linear
and nonlinear relationships between the spin observables. These relationships
not only help in finding complete sets of experiments, but also yield important
constraints between the 16 observables for this reaction.
