In this article we explore the conditions under which a von Kries-like model of color constancy is exact. This investigation is motivated by the fact that in practice the model has been shown to work well for a wide range of imaging devices despite the fact that previous analysis of the necessary and sufficient conditions for the model to work predicts that it should perform poorly for real lights and surfaces. We present a modified theory that reconciles this apparent contradiction and that is based on the observation that von Kries-like adaptation treats sensor responses independently of one another. Starting from this point we show how to recover, for a single sensor, set of surfaces, and reference illuminant, the set of von Kries illuminants: all lights for which von Kries-like adaptation is a perfect model of illumination change. To help us in this task we use a linear model of surface reflectance, but importantly, we use a local model, that is, a model derived by looking at reflectance only in the region to which the sensor of interest is sensitive. Adopting such a model and treating sensors independently of one another, we show that our new theory accurately predicts the good practical performance of a von Kries-like model.