Gorman's theory of demand is extended comprehensively to incomplete systems. The incomplete systems approach dramatically increases this class of models. The separate roles of symmetry and adding up are identified in the rank and the functional form of this class of models. We show that symmetry determines rank and the maximum rank is three. We show that adding up and 0o homogeneity determines the functional form and there is no functional form restriction for an incomplete system. We prove that every full rank system and reduced rank systems with a minimal level of degeneracy can be written as a polynomial in a single function of income. A complete set of closed form solutions for the indirect objective functions of this class of models is derived. A simple method to nest rank and functional form for incomplete systems is presented.