Sextets in P1: The probabilistic theory of the six-phase structure invariant ϕ = ϕ_h + ϕ_k + ϕ_l + ϕ_m + ϕ_n + ϕ_p

It is assumed that a crystal structure P1 is fixed and that the 31 non-negative numbers R₁, R₂, . . ., R₆; R₁₂, R₁₃, . . ., R₅₆; R₁₂₃, R₁₂₄, . . ., R₁₅₆ are also specified. The random variables (vectors) h, k, 1, m, n, p are assumed to be uniformly and independently distributed in the regions of reciprocal space defined by and |E_h| = R₁, |E_k| = R₂, . . . , |E_p| = R₆; (1) |E_{h + k}| = R₁₂, |E_{h + l}| = R₁₃, . . . , |E_{n + p}| = R₅₆; (2) |E_{h + k + l}| = R₁₂₃, |E_{h + k + m}| = R₁₂₄, . . . , |E_{h + n + p}| = R₁₅₆; (3) and h + k + l + m + n + p = 0. (4) Then the structure invariant φ = φ_h + φ_k + φ_l + φ_m + φ_n + φ_p, as a function of the primitive random variables h, k, l, m, n, p, is itself a random variable, and its conditional probability distribution, given (1), (2) and (3), is derived from the main result of the previous paper [equation (3.1); Fortier & Hauptman (1977). Acta Cryst. A33, 694-696.] and compared with the distribution when only (1) is given. The distribution leads to estimates for cos φ in terms of the 31 magnitudes (1), (2) and (3).