A modified Helmholtz line-element in brightness-colour space

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Abstract
Helmholtz, Schrödinger and others have considered the relation between the trichromatic coordinates, x, y, z and x + δx y + δy, z + δz, of two juxtaposed light patches which can just be discriminated by the eye. The Helmholtz expression, (δx/(a + x))2 + (δy/(b + y))2 + (δz/(c + z))2 = const. = δs2 is the most readily interpretable in terms of the trichromatic theory, but to explain the observed variation of the hue limen through the spectrum it demands double-peak fundamental response curves which are at variance with other evidence. It also leads to a step-by-step visibility curve of wrong shape. The difficulties mentioned are removed by introducing different constant factors in the three terms. Such a modification is indicated independently by recent measurements of the liminal brightness increment, from which appropriate values of the factors have been derived. The factor in each term is simply related to the limiting Fechner fraction of the corresponding trichromatic mechanism, i.e., the Fechner fraction ΔB/B which would be observed at sufficiently high brightnesses B if visual discrimination depended on that mechanism only. The modified expression for δs2 is applied to the calculation of hue limens, the step-by-step visibility curve, the Fechner fraction curves, and the general colour limen at different points of the colour triangle. Certain main features of the experimental results are correctly reproduced, but discrepancies with Wright and MacAdam's measurements of general colour limens may indicate that the modified element ignores some factor which is operative in their experiments.

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