We derive the conditions for which the illuminant-independent behavior of von Kries quotients is good. They are in qualitative agreement with those obtained in the experiment. The key concepts are as follows: narrow sensitivities, their approximation by a delta-function at an appropriate wavelength, and the nature of the reflectance and spectral distribution functions fir which the delta-function approximation is good. Color constancy theory aims to construct strictly or almost illuminant-independent perceptual. achromatic and chromatic variables from illuminant-dependent tristimulus values. Deficiencies due to ignoring metamerism and, in constructions, the constraints that spectral distribution,functions are nonnegative and reflectance functions assume values between 0 and I. are identified and corrected, if remediable. Variables are constructed with a more favorable illuminant-independent behavior than von Kries quotients. Helmholtz's (conjectured) interpretation of color constancy is proved to be correct. If color and lightness are illuminant-independent functions of these variables, equality of appearance under test and reference sources occurs if (and under weak conditions only if) they possess the same values under these sources. The resulting chromatic adaptation transform specifies an earlier open proposal (Color Science. New York: Wiley; 1982. 434 p). We prove a result concerning the role of color in discrimination. (C) 2010 Wiley Periodicals, Inc. Col Res Appl. 35. 164-183. 2010; Published online 8 March 2010 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/col.20526