Given LMS cone quantum catches from a surface under a first illuminant what is the best method of predicting what the corresponding quantum catches will be for the same surface under a second illuminant given only the quantum catches of a white surface under both illuminants? The von Kries rule is one well known method. In this paper, two new prediction methods along with a variation on an existing third method are introduced and then compared experimentally. In contrast to the von Kries rule which is equivalent to a diagonal transformation, all three methods estimate a full 3-by-3 linear transformation mapping LMS values between illuminants. All the new methods perform better than the von Kries rule.