West and Brill (J Math Biol 1982;15:249-258) posed necessary and sufficient conditions on reflectance for Von Kries ratios to be illuminant-invariant. The statement of the conditions allowed the illuminant unlimited freedom in a function subspace. If the illuminant change is only a pairwise substitution, greater reflectance freedom at first seems available, but this appearance is illusory. Implications for sharpened-sensor theory are discussed, as well as application to an analogous formalism that emerges from Imura's (Color Res Appl 2007;32:195200) fluorescent estimation algorithm. Such studies may clarify how to use the West and Brill criterion of "perfect invariance" to quantify invariance that may be "good enough." (c) 2008 Wiley Periodicals, Inc.