334 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS same wavelength,* and this prediction may be made for a large number of wavelengths. It will be apparent, therefore, that by means of equa- tions 1 and 2 the spectrophotometric curve of any mixture of pigments may be predicted, providing the values for the individual pigments are known. An example will help to clarify the method. Table I lists the reflectances of 1% concentrations of red oxide, Mapico yellow, and ultramarine blue in titanium dioxide. These values were taken directly from the spectrophotometric curves of the pigments shown in Figs. 2 and 3 and are given for the wavelengths shown. In the next column to the right of that giving the reflectance, the cor- responding K/S values are given. These values are for the mixture of 1% colored pigment and 99% white and are equivalent to the K/S• value given by equation 2 for one colored pigment plus white. In order to get the K/S values for the pigment alone, the K/S values for white, which are also given in the table, are subtracted. This has been done to obtain the "corrected" K/S values in the next column to the right. It is these corrected values labeled K/Sy, K/SR, and K/SB which must be used in equation 2. Now let us try to predict the reflectance at 500 nm of a mixture of: Red oxide ............................................ 0.5% Mapico yellow ........................................ 0.5% Ultramarine blue ...................................... 0.2% White ............................................... 98.8% Using equation 2 and the values in Table I one finds: K/S•u -- 0.5 X 1.128 q- 0.5 X 0.389 + 0.2 X 0.62 + 0.012 = 0.782 The tables of K/S rs. R indicate that a K/S value of 0.782 corresponds to a reflectance of 30.7%. Thus one would predict the reflectance of this mixture to be 30.7% at 500 nm. The actual reflectance of the mixture, labeled "True" in Fig. 1, is 28.6%. Similar calculations can be made at each of the wavelengths for which data are tabulated. This has been done, and the results are plotted in Fig. 1 (labeled "Com- puted") so that the predicted curve may be compared with the actual curve. It will be noted that the predicted curve is 1•5 to 2% higher than the true curve. The two different colors represented by these two curves would not be a sufficiently good match for most cosmetic applica- * In practice, however, tables of K/S rs. R are used rather than calculations in accordance with equation 1. Such a table appears in Ref. 1.

INSTRUMENTATION IN COSMETIC COLOR CONTROL 335 tions, but they would be close enough so that an adjustment, as will be described below, would produce a color which would be an acceptable match to the standard. The error in prediction may be due to a num- ber of causes such as variations in pigment strength, grind, or errors in sample preparation. The manner in which this spectrophotometric theory is utilized will be described below. For a full understanding of the computer, how- ever, some colorimetric theory will also be required. The color of a sample may be described in terms of its tristimulus values, X, Y, and Z. If two samples have the same tristimulus values, they will match under the illuminant for which these values were com- puted even if the spectrophotometric curves are not identical. The tristimulus values may be computed from the spectrophotometric curve by means of equation 3. x = y ERdx Z = Y E is the relative distribution of energy in the light source used for view- ing the sample, oe, y, and • are values dependent on the characteristics of the human eye, R is the reflectance of the color, X is the wavelength of light, and the integration is carried out over the entire visible spectrum. All of these values vary with wavelen•h, and the values for •, •, •, and E have been standardized by the International Commission on Illumination. The computation is usually made either by an automatic computer attached to a recording spectrophotometer or by a combina- tion of filters and photocells in a colorimeter. Similarly, the difference in color between two samples having differences of •R in their spectro- photometric cu•es is given by equation 4. • = • E•RdX •z = ff E•RdX If one can predict the spectrophotometric cu•e of a sample having a given pigment formula by means of equations 1 and 2, then one can also predict the color by use of equation 3 and the predicted reflectance values. It will be apparent that one can also predict the color difference between two samples having known pigment composition by use of equations 1, 2, and •. Mthough this technique predicts color from known pigment formulas, the method gives, at least in theory, a means