In a previous publication, the results of an investigation of the aluminium-rich portion of the aluminium-iron-nickel equilibrium diagram were reported and interpreted theoretically (Raynor & Pfeil 1946-7a). On the basis of this theoretical work, the forms of the previously unknown equilibrium diagrams for the systems aluminium-iron-cobalt and aluminium-cobalt-nickel were deduced, and subsequently verified quantitatively by experiment (Raynor & Pfeil 1946-76; Raynor & Waldron 1948). In the present paper, similar reasoning has been applied to the quaternary aluminium-iron-cobalt-nickel system, and the form of diagram expected compared with experiment. Previous work has suggested that Co2Al9and the isomorphous FeNiAl9are analogous to electron compounds. If the theory be accepted that transitional metal atoms may absorb electrons in aluminium-rich alloys to an extent governed by the vacancies in their atomic orbitals (or 3dshells), the two compounds have closely similar electron: atom ratios, and both dissolve nickel to the same limiting electron: atom ratio. Co2Al9will dissolve more iron than FeNiAl9; the reason for this has been discussed. In the quaternary system, it would be expected that a continuous series of solid solutions, of composition (Fe.Co.Ni)2Al9, would exist between the two compounds, and that the aluminium-rich boundary of the quaternary body in the tetrahedral equilibrium model would be a plane corresponding to a constant proportion of 2 solute atoms to 9 aluminium atoms. It would also be expected that the whole boundary corresponding to saturation of the quaternary body with nickel would occur at a constant electron-atom ratio. These considerations imply that no other phases, apart from FeAl3and NiAl3(which dissolve relatively small amounts of the other transitional solutes), enter into equilibrium with the aluminium-rich solid solution, and that the liquidus surfaces for FeNiAl9and Co2Al9merge continuously into each other without a break. Further predictions with regard to the form of the equilibrium diagram may also be made, and these are discussed in the paper. The results of the investigation, which are discussed, confirm previous suggestions that the inter-metallic compounds formed by aluminium and transitional metals may, if sufficient aluminium is present, be regarded as analogous to electron compounds, with absorption of electrons by the transitional metal atoms occurring. The theory may be used, in favourable cases, to predict quantitatively the forms of uninvestigated complex equilibrium diagrams. In the present case, equilibrium relationships in a quaternary system have been predicted from those in the three subsidiary ternary systems, two of which were themselves largely predicted as a result of the original theoretical in terpretation of the aluminium-iron-nickel alloys.