Direct methods: the identification of conditions which simplify the generation of inconsistent quadrupoles

An algorithm is implemented to determine the form and phase shift for inconsistent type II quadrupoles for any space group having glide or screw-axis translations which are not a consequence of lattice centering. Cumulatively there are only six different Miller index restrictions and nine different phase shift forms common to all space groups of orthorhombic or lower symmetry. A similar analysis has been performed for a newly discovered type III class of quadrupoles. The configuration of the phase connections among the four triples of the type III quadrupole is different from the common configuration previously described for both normal (type I) and inconsistent (type II) quadrupoles. A knowledge of these constraint conditions for type II and III quadrupoles greatly simplifies a procedure for generating these relationships.