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Formulas for the cosines of the higher order phase invariants that arise in crystal structure analysis are derived as expected values from determinantal joint probability distributions. The values of the cosines of the invariants are expressed in terms of averages over simple functions of known structure factor magnitudes. The formulas are termed "general" to distinguish them from formulas that have more restricted averages and are termed "special." As with the special formulas, the best opportunity for obtaining useful information from these formulas is provided by embedded seminvariants formed from the invariants by use of relationships among the phases that arise from the space group symmetries.

General phase-invariant formulas of higher order: Expected values