A. R. MacDonald and G. P. Bentley4 have shown that a physical instrument whose error is in percent of level rather than of full scale, e.g., ΔI/I resembling the logarithmic response of organisms, may “… be more precise than a spectrophotometer for many commercial measurements in either dark colors, or highly saturated colors, where reflectance on any one tristimulus filter may be low” (p. 366). However, the present analysis is not applicable to the logarithmic case, which is complicated by the fact that ΔTjk must be replaced, at every wavelength, by ΔTjk/Tjk. This, of course, would reduce the displacement quite markedly for dark samples.

This means that when the error operates to increase the X value, while it simultaneously decreases both the Y and Z values, the change in x will be at its maximum for a given magnitude of ∊. However, when this is so, the Δy and the Δz values are less than maximum, although definitely not minimum. How much less we cannot say except in a specific instance. And, mutatis mutandis, the same is true when Δx is at its minimum. Our analysis, in other words, entails the case that only one change, in either x, y, or z, may be maximum (or minimum) at any given time, for any given value of ∊. We do not consider those cases where the changes in two or in all three of the coordinates may be simultaneously maximum (or minimum). These issues, however, have little practical interest, for our empirical criterion necessarily involves changes in only one coordinate. Such changes are entirely sufficient for the production of chromaticness changes, and we do not need to be specifically concerned with simultaneous changes in either or both of the other coordinates.

The eye is not even approximately equally sensitive to changes in the z coordinate as it is to changes in the x and y coordinates, hence we do not take ∊ as the least of ∊x, ∊y, ∊z, but rather as the least of ∊x, ∊y. In fact, Δx=Δy is only a very rough approximation in the small; the standard deviations of chromaticity matches about a point are ellipses, not circles. Moreover, it may be, in any given case, that Δz, rather than Δx or Δy, is the most significant index of chromaticness change. Furthermore, it is entirely possible, in Eq. (5), to use δx≠δy≠δz, should all three contribute differently to the change in chromaticness.

Strictly, this is not true for the case where the chromaticity coordinates are very nearly of the same magnitude, for the values of the factors (C1, C2, C3) for CIE Source C then play a limiting role (compare the ∊x and ∊y values for filter CC-10B in Table IV).

In a series such as our No. 1, where the filters are of the same material but of varying thickness, excitation purity usually varies inversely with luminous transmittance. But the present error is related only to the transmittance and not at all to the purity.

The corresponding expression for Δxmax is:Δxmax=0.46024∊Y-0.40030∊.

Y cannot be 100%, as if no filter were used, for then we would not have the equal-energy point, but merely the proper x, y, z coordinates for Source C itself.

It is not important that this region be exactly specified (in fact the small size of the MacAdam ellipses in this area is not yet a universally acknowledged fact), but only the immediate neighborhood of a point.

It is possible for the method of tristimulus computation itself to be sufficiently crude so that ±0.002 changes will not be detected. Nickerson (reference 6), on the basis of disk colorimetry to a criterion of ±0.002, notes (p. 257) “… that a method of computation is required which gives x and y with an uncertainty no greater than 0.001.” Moreover, “… summation for intervals of 10 mμ gives x and y with an average uncertainty of 0.0004, … the 30-selected-ordinate method has an average uncertainty of 0.0010, and … the 10-selected-ordinate method has an average uncertainty … of 0.0035.” She goes on to note that the advantage of the 10 mμ weighted-ordinate method over the 30-selected-ordinate method may be spurious. It is on the basis of this work, that we choose the 30-selected-ordinate method to demonstrate our procedure.

Reflectance curves would have done as well; the principle is exactly the same in either case.

The following contains a full discussion of these various issues: R. von Mises, Vorlesungen aus dem Gebiete der angewandten Mathematik, Bd. I, Abschnitt III, No. 12.

If several samples are used, then the mean standard deviation can be taken to characterize the given instrument over all samples. The standard deviation for repeated runs on one sample would have a very limited meaning.