Abstract
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 "special" to distinguish them from formulas that have more general averages. The latter will be presented in a future publication. 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.