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An extension of the critical-phenomena universality hypothesis assumed previously to be valid for less singular terms is shown to be invalid. A reformulated extension of the universality hypothesis is found to be valid exactly for the spin-1/2 Ising model on the triangular, square, and honeycomb lattices, and, further support is supplied by highand low-temperature expansions for the susceptibility. The extension fails however for the Ising model on the kagome lattice and for the spherical model. At present there are insufficient data to test the extended principle for other models or for experimental systems.