A probability theory of the cosine invariant cos(ϕ_h + ϕ_k + ϕ_l – ϕ_h+k+l)

A conditional joint probability distribution is derived in order to estimate the values of the cosine invariant cos (φ_h + φ_k + φ_l - φ_{h + k + l}) in terms of the magnitudes of E_h, E_k, E_l, E_{h + k}, E_{h + l}, E_{k + l}, E_{h + k + l}. The theory leads to values for the cosines which lie anywhere between - 1 and + 1. Some applications of the quartets in procedures for crystal structure determination are described.