Strengthening of the triplet relationship: a new probabilistic approach in P

A new probabilistic approach is described which can improve the reliability of the triplet relationships. The terms |E_K|, |E_{H1 + K}|, |E_{H1 + H2 + K}| are tested in order to give the probability of the sign of |E_H1E_H2E_{H1 + H2} rather than the 'exact' value of the invariant, as made in preceding approaches, e.g. B_3,0 formula [Karle & Hauptman, Acta Cryst. (1958), 11, 264-269] the modified triple product [Hauptman, Acta Cryst. (1964), 17, 1421-1433; Hauptman, Fisher, Hancock & Norton, Acta Cryst. (1969), B25, 811-814] and M-D-K-S formula [Hauptman, New Orleans Meet. Amer. Cryst. Assoc., 2 March. (1970), Abstract B8].