D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

A review of empirical and inductive line elements can be found in G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 511–560.

G. Wyszecki, J. Opt. Soc. Am. 58, 290 (1968).

L. F. C. Friele, Die Farbe 10, 193 (1961).

L. F. C. Friele, J. Opt. Soc. Am. 55, 1314 (1965).

D. L. MacAdam, J. Opt. Soc. Am. 55, 91 (1965).

D. L. MacAdam, Acta Chromatica 1 (4), 1 (1965).

K. D. Chickering, J. Opt. Soc. Am. 57, 537 (1967).

G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).

C. L. Sanders and W. Gaw, Appl. Opt. 6, 1639 (1967).

See, for example, Ref. 3, pp. 536–541.

There are discrepancies between the coordinates (*x*_{0},y_{0},l_{0}) of the test color and the corresponding coordinates [Equation] of the mean color match calculated from the n observations. However, these discrepancies appear to follow the pattern we would expect from the statistical variations that characterize the observational data, superimposed on small unavoidable errors in the calibration of the instrumental scales of the two calorimetric fields. For approximately 90% of all of the color-matching ellipsoids [with (*ds*)^{2}=7.81] we have studied, the point (*x*¯_{0}, *y*¯_{0}, *l*_{0}) falls well within the ellipsoid centered at the point (*x*_{0}, *y*_{0}, *I*_{0}). For convenience of intercomparing different sets of color matches made by the same or by different observers on the same test color, the individual points (*x*_{i}, y_{i}, l_{i}) of each set of observations have been shifted so as to make their respective mean point (*x*¯_{0}, *y*¯_{0}, *l*_{0}) coincide exactly with (*x*_{0}, y_{0}, l_{0}). This small translation of the observed points is of no apparent consequence to the conclusions drawn from the experiment as a whole.

To illustrate the observer's precision of chromaticity matching, it might be more appropriate to use the (*x,y*) projection rather than the cross sections of the color-matching ellipsoid. However, in most cases we found only negligible differences between the two ellipses, that is, differences of the kind and magnitude shown in Fig. 3. This indicates that our color-matching ellipsoids are either only slightly tilted against the (*x,y*) plane, or their projections in the (*x,l*) and (*y,l*) planes are ellipses with little or only moderate eccentricities. The directions and magnitudes of the tilts are of interest in developing color-difference formulas or line elements, but we will not discuss this aspect of color-matching ellipsoids in the present paper.

W. R. J. Brown, J. Opt. Soc. Am. 47, 137 (1957).

Note that the g coefficients and parameters of the elliptical cross section of the mean ellipsoid are not the arithmetic means of the corresponding quantities of the ellipsoids derived from the data of the individual sessions.

W. S. Stiles, Proc. Phys. Soc. (London) 58, 41 (1946).

If the areas of two ellipses differ by a factor F, the chromaticity differences between the center color and a color on the ellipse differ by a factor √F.

The calculation of the FMC ellipses were made with the constant (*ds*)_{2} set equal to 7.81, that is, identical to the one used in calculating the GW ellipses [see Eq. (1)].

D. L. MacAdam, J. Opt. Soc. Am. 54, 1161 (1964).

To calculate Δ*x*_{i}, Δ*y*_{i} (see Fig. 19) (1) Calculate ε_{1} (in degrees) from tanε_{1} = - (a/b) tanθ. (2) Set up a series of angles ε_{i} at constant intervals Δε to give the desired number in of points on the semicircle ε_{i} = ε1+(i-1)Δε *i*=1, 2, … *m* with *m*=180/Δε. (3) Calculate the corresponding angles (ψ_{i}-θ) from tan(ψ_{i}-θ) = *(b/a)* tanε_{i} and record ψ_{i}. (4) Calculate Δ*x*_{i}=*r*_{i} cosψ_{i} Δ*y*_{i} =*r*_{i} sinψ_{i}, where *r*_{i}=*c*(*G*_{11} cos^{2}ψ*i*+*2G*_{12}sinψ*i* cosψ_{i}+*G22*sin^{2}ψ_{i})^{-½}. Note: For Δε=30°, that is *m*=6, MacAdam's^{20} Chickering’s ^{9} case is obtained.

When we square our computed value 〈*e*^{2}〉^{½}=0.175 for α=1.0, we obtain 0.03 (approx.) which agrees with the value Chickering^{9} reported.