A thorough investigation by means of X-rays has been carried out with the purpose to determine the nature of the ternary phase τ in Al-Cu-Ni alloys. In contrast with the conventional concept of alloy phase which is characterized by a definite type of crystal structure, systematic structure changes are found in the single phase field of τ which occupies quite an extensive area in the isothermal section of the phase diagram at room temperature. There are eight types of structures altogether, all derived from a basic rhombohedron with corners occupied by Al atoms and centres either occupied by the heavy atoms or remaining vacant. The basic rhombohedron is the building stone in the crystal architecture. By transforming the basic rhombohedron into a hexagonal prism in the usual way, all structures may be considered to be built up by stacking together a number of these hexagonal prisms along the triad. The transformation of one structure into another is quite systematic in the way that the number of the stacking stories in the unit cell increases according to the order 10, 11, 12, 13, 14, 15, 16, 17. The atomic arrangements in the different structures are closely related too, in the respect that they are all superstructures due to the presence of ordered vacancies in the rhombohedral centres.The principal factor controlling the formation of these structures has been fully considered. In view of the fact that the change of structure types follows closely with the content of Ni or Cu for alloys of constant Al content, the atomic size factor appears to be unimportant in the formation of these alloys. It has been shown that for alloy phases of the defect lattice type as the r-phase, the most fundamental factor is the average number of valency electrons per structural unit which is the basic rhombohedron in the present case. By assuming Hume-Rothery's valencies, the average number of valency electrons remains remarkably constant throughout the entire phase field, while the electron concentration varies with compositions. It has also been pointed out that for alloy phases where there is no unit cell change, the average number of electrons per structural unit is equivalent to the number of electrons per unit cell, and for alloy phase where there is no defect, this is in effect equivalent to the electron concentration.