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  1. For a good summary of the previous literature and list of references, the reader is referred to S. Q. Duntley, J. Opt. Soc. Am. 32, 61 (1942).
    [CrossRef]
  2. P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931).
  3. L. Amy, Rev. d’optique 16, 81 (1938).
  4. Duntley’s equation reduces to the above Eq. (3) for P = 0, Q = 1 (see reference 1). See also D. R. Duncan, Proc. Phys. Soc. 52, 380 (1940).
    [CrossRef]
  5. J. L. Michaelson, J. Opt. Soc. Am. 28, 365 (1938).
    [CrossRef]
  6. See also A. E. Parker, Symposium on Color, p. 53, published by the American Society for Testing Materials (1941).
  7. See also J. W. Ryde, Proc. Roy. Soc. A131, 451 (1931). Ryde’s Eq. (59) reduces to the above Eq. (6) for (his) T=T′ = 0, R′ = R.
    [CrossRef]
  8. See also J. W. Ryde and B. S. Copper, Proc. Roy. Soc. A131, 464 (1931), especially page 467.
    [CrossRef]
  9. D. R. Duncan, reference 4, bases his calculations on θ ≡ e/μ. The use of Eq. (7) has been found by the author to be more convenient.
  10. U. S. Department of Commerce, National Bureau of Standards, Letter Circular, LC-547.

1942 (1)

1940 (1)

Duntley’s equation reduces to the above Eq. (3) for P = 0, Q = 1 (see reference 1). See also D. R. Duncan, Proc. Phys. Soc. 52, 380 (1940).
[CrossRef]

1938 (2)

L. Amy, Rev. d’optique 16, 81 (1938).

J. L. Michaelson, J. Opt. Soc. Am. 28, 365 (1938).
[CrossRef]

1931 (3)

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931).

See also J. W. Ryde, Proc. Roy. Soc. A131, 451 (1931). Ryde’s Eq. (59) reduces to the above Eq. (6) for (his) T=T′ = 0, R′ = R.
[CrossRef]

See also J. W. Ryde and B. S. Copper, Proc. Roy. Soc. A131, 464 (1931), especially page 467.
[CrossRef]

Amy, L.

L. Amy, Rev. d’optique 16, 81 (1938).

Copper, B. S.

See also J. W. Ryde and B. S. Copper, Proc. Roy. Soc. A131, 464 (1931), especially page 467.
[CrossRef]

Duncan, D. R.

Duntley’s equation reduces to the above Eq. (3) for P = 0, Q = 1 (see reference 1). See also D. R. Duncan, Proc. Phys. Soc. 52, 380 (1940).
[CrossRef]

Duntley, S. Q.

Kubelka, P.

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931).

Michaelson, J. L.

Munk, F.

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931).

Parker, A. E.

See also A. E. Parker, Symposium on Color, p. 53, published by the American Society for Testing Materials (1941).

Ryde, J. W.

See also J. W. Ryde, Proc. Roy. Soc. A131, 451 (1931). Ryde’s Eq. (59) reduces to the above Eq. (6) for (his) T=T′ = 0, R′ = R.
[CrossRef]

See also J. W. Ryde and B. S. Copper, Proc. Roy. Soc. A131, 464 (1931), especially page 467.
[CrossRef]

J. Opt. Soc. Am. (2)

Proc. Phys. Soc. (1)

Duntley’s equation reduces to the above Eq. (3) for P = 0, Q = 1 (see reference 1). See also D. R. Duncan, Proc. Phys. Soc. 52, 380 (1940).
[CrossRef]

Proc. Roy. Soc. (2)

See also J. W. Ryde, Proc. Roy. Soc. A131, 451 (1931). Ryde’s Eq. (59) reduces to the above Eq. (6) for (his) T=T′ = 0, R′ = R.
[CrossRef]

See also J. W. Ryde and B. S. Copper, Proc. Roy. Soc. A131, 464 (1931), especially page 467.
[CrossRef]

Rev. d’optique (1)

L. Amy, Rev. d’optique 16, 81 (1938).

Zeits. f. tech. Physik (1)

P. Kubelka and F. Munk, Zeits. f. tech. Physik 12, 593 (1931).

Other (3)

D. R. Duncan, reference 4, bases his calculations on θ ≡ e/μ. The use of Eq. (7) has been found by the author to be more convenient.

U. S. Department of Commerce, National Bureau of Standards, Letter Circular, LC-547.

See also A. E. Parker, Symposium on Color, p. 53, published by the American Society for Testing Materials (1941).

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Figures (10)

F. 1
F. 1

Graph relating the measured reflection R′ and the θ of Eq. (8).

F. 2
F. 2

Calculation of dye constants. Curve A: undyed molding; curve B: 0.01 percent dye. The absorption coefficients μ are calculated from Eq. (11).

F. 3
F. 3

Calculation of constants of white pigment. Curve A: 0.5 percent white pigment; curve B: 0.5 percent white pigment+0.01 percent dye. The scattering coefficients ρ are calculated from Eq. (15) and the absorption coefficients μ from Eq. (13).

F. 4
F. 4

Calculation of constants of red pigment. Curve A: 0.5 percent red pigment; curve B: 0.1 percent red+0.4 percent white; curve C: 0.4 percent red+0.1 percent white. The constants μ and ρ for the red pigment are determined by means of Eqs. (16) and (18). The constants are used to calculate the position of curve C.

F. 5
F. 5

Calculation of a pigment mixture, as carried out in Table I.

F. 6
F. 6

Use of the relative constants. Curve A: 0.5 percent white pigment; curve B: 0.5 percent white pigment +0.05 percent yellow dye; curve C: 0.5 percent white pigment+0.05 percent blue dye; curve D: 0.5 percent white pigment+0.025 percent yellow dye+0.025 percent blue dye. The relative constants for the white pigments and the yellow and blue dyes are calculated in Table II. The constants can be used to predict the curve D, as illustrated by Table III.

F. 7
F. 7

Calculation of a color match. Curve B is the attempted duplication of curve A of the original.

F. 8
F. 8

Curves showing the accumulative effect of the various pigments added to white.

F. 9
F. 9

The effect.of opacity. Curve A: 0.5 percent concentration, reflection; curve B: 0.1 percent concentration, reflection; curve C: 0.5 percent concentration, transmission; curve D: 0.1 percent concentration, transmission. Curves A and B are essentially the same where the transmission (curve D) is small. An opacity correction has been applied in calculating the points along curve B.

F. 10
F. 10

Curves of the reflection and transmission of the white pigment, used for making the opacity correction. Curves A to E are reflection curves of 1.0 percent, 0.5 percent, 0.2 percent, 0.1 percent, 0.05 percent concentration of white pigment, respectively. Curves F to K are transmission curves of the same samples in the order given.

Tables (3)

Tables Icon

Table I Calculation of the color of a Pigment mixture. The calculations of the reflection value of a sample containing 0.5 percent white pigment, 0.015 percent red pigment shown in Fig. 4, and 0.025 percent yellow pigment are carried out in this table. The calculations are made by means of Eq. (8), and the results are compared with the actual curve obtained for the sample in Fig. 5.

Tables Icon

Table II Calculation of “relative” constants. These calculations are given as an illustration of the calculation of the calculation of the relative constants of a white and two dayes by use of Eqs. (22) and (23).The date are taken from curves A, B, and C of Fig.6.

Tables Icon

Table III Calculation of the color of a mixture. The constants calculated in Table II are used to predict the curve D of Fig. 6 by means of Eq. (8).

Equations (29)

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T = 2 a e a t μ + ρ + a ( μ + ρ a ) e 2 a t ,
R = ρ ( 1 e 2 a t ) μ + ρ + a ( μ + ρ a ) e 2 a t ,
a = ( μ 2 + 2 μ ρ ) 1 2 .
T = 0 , R = ρ μ + ρ + a = 1 μ ρ + 1 + [ μ ρ ( μ ρ + 2 ) ] 1 2 .
μ = C 1 μ 1 + C 2 μ 2 + C 3 μ 3 + = C i μ i ,
ρ = C 1 ρ 1 + C 2 ρ 2 + C 3 ρ 3 + = C i ρ i ,
R = k 1 2 + ( 1 k 1 ) ( 1 k 2 ) R 1 k 2 R .
k 1 = [ n 1 n + 1 ] 2 .
θ μ ρ = C i μ i C i ρ i ,
T = ( 1 k 1 ) 2 T 1 k 1 2 T 2
T = exp ( μ 0 μ D C D ) .
T = T 0 exp ( μ D C D ) ,
T 0 = ( 1 k 1 ) 2 exp ( μ 0 ) .
θ W = C W μ W C W ρ W = μ W ρ W ,
θ W , D = C W μ W + C D μ D C W ρ W .
ρ W = D D μ D C W ( θ W , D θ W ) ,
θ R = C R μ R C R ρ R = μ R ρ R ,
θ W , R = C W μ W + C R μ R C W ρ W + C R ρ R .
ρ R = C W C R θ W , R ρ W μ W θ R θ W , R ,
μ R = ρ R θ R = C W C R θ W , R ρ W μ W θ R θ W , R θ R = C W C R θ W , R ρ W μ W 1 ( θ W , R / θ R ) ,
ρ W = 1 .
μ W = θ W .
μ D = ( θ W , D θ W ) C W C D .
ρ R = C W C R θ W , R θ W θ R θ W , R
θ = C W μ W + C B μ B C W ρ W + C B ρ B ,
θ = C W μ W + C B μ B + C R μ R C W ρ W + C B ρ B + C R ρ R ,
θ = C W μ W + C 1 μ 1 + C 2 μ 2 C W ρ W + C 1 ρ 1 + C 2 ρ 2 ,
θ = C W μ W + C 1 μ 1 + C 2 μ 2 C W ρ W + C 1 ρ 1 + C 2 ρ 2
θ W = μ 0 + C W μ W C W ρ W ,