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  1. Sections I and II of this paper are intended to present only a condensed formal synopsis of fundamental relations rather than explanations or proofs. The intelligent reading of this paper presumes an acquaintance with the following previous papers: Ives: “Transformation of Color-Mixture Equations,”.J.F.I. 180, pp. C73–701; December, 1915, and J.F.I. 195, pp. 23–44; January, 1923. Troland: “Report of the Colorimetry Committee of the O. S. A. 1920–21,” J.O.S.A. & R.S.I., 6, pp. 527–596; August, 1922. (Separate copies of this committee report may be had at fifty cents per copy of F. K. Richtmyer, Bus. Mgr., J.O.S.A. & R.S.I., Cornell University, Ithaca, New York.)
    [Crossref]
  2. Whether we call this color “white” or “gray,” is, for our present purpose, a matter of indifference. Usage on this point is not well established.
  3. Grassman: Pogg. Ann.,  89, p. 70; May, 1853, (or Grassmann’s “Math. & Phy. Werke”2, part 2, p. 162). Popular expositions may be found in Rood’s “Modern Chromatics,” Abney’s “Colour Measurement and Mixture,” and Luckiesh’s “Color and its Applications.” See also Nutting, B. S. Bulletin 9, pp. 1–5; 1913and Priest: J.O.S.A. & R.S.I. 8, pp. 173–200; January, 1923.
    [Crossref]
  4. Cf. “Report of Colorimetry Committee.” J.O.S.A. & R.S.I., 6, p. 564; August, 1922.
  5. Cf. Peddie: “Colour Vision” (London, 1922), pp. 16–17, 52–53, 108–109.
  6. Cf. Ives: Trans, I. E. S.,  5, pp. 196–198; April, 1910. Priest: B. S. Sci. Pap. 417, pp. 256–258, particularly Fig. 15; August, 1921.
  7. See J.O.S.A. & R.S.I.,  6, p. 534; 1922.
  8. Ives: Phil. Mag., (6),  24, pp. 845–853; Dec., 1912. Priest: J.O.S.A. & R.S.I.,  8, pp. 198–200; Jan., 1924.
    [Crossref]
  9. Maxwell: “On the Theory of Compound Colours …” Phil. Trans., 1860. (Or Maxwell’s “Scientific Papers,”I, pp. 410–444.) Koenig and Dieterici: Sitzb. der Akad. Berlin, pp. 805–829; July29, 1886. (Or Koenig’s “Gesam. Abhand. Physiol. Optik,” pp. 60–87.) Koenig and Dieterici: Zeit. für Phych. und Physiol. der Sinnesorgane 4, pp. 241–347; 1892. (Or Koenig’s “Gesam. Abhand,” pp. 214–321). Abney: “Researches in Colour Vision” (London, 1913) Chap. XV. (1)For general treatments see: Helmholtz’ “Physiol. Optik,”2nd ed. (Leipzig, 1896), pp. 311–384. (2)Köllner’s “Die Stoerungen des Farbensinnes” (Berlin, 1912). (3)Parsons’ “Introduction to Study of Colour Vision” (Cambridge, 1915). (4)Peddie’s “Colour Vision.” (London, 1922). For a brief elementary treatment, see Watson’s “Text-Book of Physics” (London, 1903) Chap. IX.
    [Crossref]
  10. This “absolute impurity” of course presupposes a given set of “elementary stimuli.” It is not “absolute” if a change of “elementary stimuli” is involved.
  11. J.F.I.,  180, pp. 673–701; Dec., 1915.
  12. J.F.I.,  195, pp. 23–44; Jan., 1923.
  13. J.O.S.A. & R.S.I.,  6, pp. 527–596; August, 1922. See particularly Fig. 9, p. 575, and Table 14A, pp. 586–587.
    [Crossref]
  14. J.F.I,  180, p. 684; Dec., 1915.
  15. J.F.I.,  195, Fig. 9, p. 35; Jan., 1923.
  16. J.O.S.A. & R.S.I.,  6, pp. 586–587; August, 1922.
  17. See Sec. IV-2 below for details.
  18. “This formula” [21, in the present paper] “appears to be valid, although in a form which differs from any heretofore proposed, in the use of ‘c’ and ‘C,’ the ‘least coordinates’ ”—From letter of Herbert E. Ives to the Director of the Bureau of Standards, October 1, 1923.
  19. J.F.I.,  180, p. 684; Dec., 1915.
  20. Letter, Ives to Director, Bureau of Standards, Oct. 1, 1923.
  21. See appendix, this paper.
  22. “It is gratifying that formulas arrived at through such entirely independent lines of thought and which look so different are identical in the value they give”—Letter, Ives to Priest, Feb. 27th, 1924, referring to the numerical agreement of values of p computed respectively by formulas 21 and 22.
  23. Casual inspection of these formulas (22) might suggest the mathematical possibility of negative values of p, for it is apparent that the numerator of the first factor and the denominator of the second factor may be negative. Negative values of p would have no physical meaning. However, these factors are always of the same sign whether positive or negative. Consequently the computed value of p is always positive.
  24. J.F.I.,  195, p. 40; Jan., 1923.
  25. Cf. Sec. IV-1, this paper.
  26. Letter, Ives to Priest, Feb. 27, 1924.
  27. See J.O.S.A. & R.S.I.,  6, pp. 582–585; August, 1922.
  28. See IV-2 above.
  29. J.O.S.A. & R.S.I.,  6, pp. 586–587; August, 1922.
  30. See Sec. IV-1 above.
  31. Note that Weaver’s tabulation is actually given in terms of (1−p) in per cent, not directly in terms of p.
  32. Or the formula derived above (equation 21) which is identical with it.
  33. J.O.S.A. & R.S.I.,  6, p. 575; August, 1922.
  34. In a letter of July 30, 1924, addressed to the author after reading this paper in manuscript Mr. Weaver expresses the opinion that these deviations are to be attributed to inconsistent data for the “excitation-curves” and the “visibility-curves.” (Cf. IV-2 above.) I.G.P. Aug. 4, 1924.
  35. See first paragraph, p. 585, J.O.S.A. & R.S.I., August, 1922.
  36. Table 14A, pp. 586–587, J.O.S.A. & R.S.I., August, 1922.
  37. J.O.S.A. & R.S.I.,  6, p. 527 and p. 548; August, 1922.
    [Crossref]
  38. Cf. last paragraph of Sec. IV above.
  39. Advance abstract already published. See proceedings Eighth Annual Meeting, Optical Society of America, J.O.S.A. & R.S.I.,  8, pp. 28–29; January, 1924. A further communication will be made at Ninth Annual Meeting, Boston, October, 1924. The complete paper will appear later as a B. S. Sci. Paper.

1924 (1)

Advance abstract already published. See proceedings Eighth Annual Meeting, Optical Society of America, J.O.S.A. & R.S.I.,  8, pp. 28–29; January, 1924. A further communication will be made at Ninth Annual Meeting, Boston, October, 1924. The complete paper will appear later as a B. S. Sci. Paper.

1923 (3)

J.F.I.,  195, Fig. 9, p. 35; Jan., 1923.

J.F.I.,  195, p. 40; Jan., 1923.

J.F.I.,  195, pp. 23–44; Jan., 1923.

1922 (9)

J.O.S.A. & R.S.I.,  6, pp. 527–596; August, 1922. See particularly Fig. 9, p. 575, and Table 14A, pp. 586–587.
[Crossref]

See J.O.S.A. & R.S.I.,  6, p. 534; 1922.

See J.O.S.A. & R.S.I.,  6, pp. 582–585; August, 1922.

J.O.S.A. & R.S.I.,  6, pp. 586–587; August, 1922.

J.O.S.A. & R.S.I.,  6, p. 575; August, 1922.

See first paragraph, p. 585, J.O.S.A. & R.S.I., August, 1922.

Table 14A, pp. 586–587, J.O.S.A. & R.S.I., August, 1922.

J.O.S.A. & R.S.I.,  6, p. 527 and p. 548; August, 1922.
[Crossref]

J.O.S.A. & R.S.I.,  6, pp. 586–587; August, 1922.

1915 (4)

J.F.I.,  180, p. 684; Dec., 1915.

Sections I and II of this paper are intended to present only a condensed formal synopsis of fundamental relations rather than explanations or proofs. The intelligent reading of this paper presumes an acquaintance with the following previous papers: Ives: “Transformation of Color-Mixture Equations,”.J.F.I. 180, pp. C73–701; December, 1915, and J.F.I. 195, pp. 23–44; January, 1923. Troland: “Report of the Colorimetry Committee of the O. S. A. 1920–21,” J.O.S.A. & R.S.I., 6, pp. 527–596; August, 1922. (Separate copies of this committee report may be had at fifty cents per copy of F. K. Richtmyer, Bus. Mgr., J.O.S.A. & R.S.I., Cornell University, Ithaca, New York.)
[Crossref]

J.F.I,  180, p. 684; Dec., 1915.

J.F.I.,  180, pp. 673–701; Dec., 1915.

1912 (1)

Ives: Phil. Mag., (6),  24, pp. 845–853; Dec., 1912. Priest: J.O.S.A. & R.S.I.,  8, pp. 198–200; Jan., 1924.
[Crossref]

1910 (1)

Cf. Ives: Trans, I. E. S.,  5, pp. 196–198; April, 1910. Priest: B. S. Sci. Pap. 417, pp. 256–258, particularly Fig. 15; August, 1921.

1860 (1)

Maxwell: “On the Theory of Compound Colours …” Phil. Trans., 1860. (Or Maxwell’s “Scientific Papers,”I, pp. 410–444.) Koenig and Dieterici: Sitzb. der Akad. Berlin, pp. 805–829; July29, 1886. (Or Koenig’s “Gesam. Abhand. Physiol. Optik,” pp. 60–87.) Koenig and Dieterici: Zeit. für Phych. und Physiol. der Sinnesorgane 4, pp. 241–347; 1892. (Or Koenig’s “Gesam. Abhand,” pp. 214–321). Abney: “Researches in Colour Vision” (London, 1913) Chap. XV. (1)For general treatments see: Helmholtz’ “Physiol. Optik,”2nd ed. (Leipzig, 1896), pp. 311–384. (2)Köllner’s “Die Stoerungen des Farbensinnes” (Berlin, 1912). (3)Parsons’ “Introduction to Study of Colour Vision” (Cambridge, 1915). (4)Peddie’s “Colour Vision.” (London, 1922). For a brief elementary treatment, see Watson’s “Text-Book of Physics” (London, 1903) Chap. IX.
[Crossref]

1853 (1)

Grassman: Pogg. Ann.,  89, p. 70; May, 1853, (or Grassmann’s “Math. & Phy. Werke”2, part 2, p. 162). Popular expositions may be found in Rood’s “Modern Chromatics,” Abney’s “Colour Measurement and Mixture,” and Luckiesh’s “Color and its Applications.” See also Nutting, B. S. Bulletin 9, pp. 1–5; 1913and Priest: J.O.S.A. & R.S.I. 8, pp. 173–200; January, 1923.
[Crossref]

Grassman,

Grassman: Pogg. Ann.,  89, p. 70; May, 1853, (or Grassmann’s “Math. & Phy. Werke”2, part 2, p. 162). Popular expositions may be found in Rood’s “Modern Chromatics,” Abney’s “Colour Measurement and Mixture,” and Luckiesh’s “Color and its Applications.” See also Nutting, B. S. Bulletin 9, pp. 1–5; 1913and Priest: J.O.S.A. & R.S.I. 8, pp. 173–200; January, 1923.
[Crossref]

Ives,

Sections I and II of this paper are intended to present only a condensed formal synopsis of fundamental relations rather than explanations or proofs. The intelligent reading of this paper presumes an acquaintance with the following previous papers: Ives: “Transformation of Color-Mixture Equations,”.J.F.I. 180, pp. C73–701; December, 1915, and J.F.I. 195, pp. 23–44; January, 1923. Troland: “Report of the Colorimetry Committee of the O. S. A. 1920–21,” J.O.S.A. & R.S.I., 6, pp. 527–596; August, 1922. (Separate copies of this committee report may be had at fifty cents per copy of F. K. Richtmyer, Bus. Mgr., J.O.S.A. & R.S.I., Cornell University, Ithaca, New York.)
[Crossref]

Ives: Phil. Mag., (6),  24, pp. 845–853; Dec., 1912. Priest: J.O.S.A. & R.S.I.,  8, pp. 198–200; Jan., 1924.
[Crossref]

Cf. Ives: Trans, I. E. S.,  5, pp. 196–198; April, 1910. Priest: B. S. Sci. Pap. 417, pp. 256–258, particularly Fig. 15; August, 1921.

Maxwell,

Maxwell: “On the Theory of Compound Colours …” Phil. Trans., 1860. (Or Maxwell’s “Scientific Papers,”I, pp. 410–444.) Koenig and Dieterici: Sitzb. der Akad. Berlin, pp. 805–829; July29, 1886. (Or Koenig’s “Gesam. Abhand. Physiol. Optik,” pp. 60–87.) Koenig and Dieterici: Zeit. für Phych. und Physiol. der Sinnesorgane 4, pp. 241–347; 1892. (Or Koenig’s “Gesam. Abhand,” pp. 214–321). Abney: “Researches in Colour Vision” (London, 1913) Chap. XV. (1)For general treatments see: Helmholtz’ “Physiol. Optik,”2nd ed. (Leipzig, 1896), pp. 311–384. (2)Köllner’s “Die Stoerungen des Farbensinnes” (Berlin, 1912). (3)Parsons’ “Introduction to Study of Colour Vision” (Cambridge, 1915). (4)Peddie’s “Colour Vision.” (London, 1922). For a brief elementary treatment, see Watson’s “Text-Book of Physics” (London, 1903) Chap. IX.
[Crossref]

Peddie,

Cf. Peddie: “Colour Vision” (London, 1922), pp. 16–17, 52–53, 108–109.

J.F.I (1)

J.F.I,  180, p. 684; Dec., 1915.

J.F.I. (6)

J.F.I.,  195, Fig. 9, p. 35; Jan., 1923.

J.F.I.,  180, p. 684; Dec., 1915.

Sections I and II of this paper are intended to present only a condensed formal synopsis of fundamental relations rather than explanations or proofs. The intelligent reading of this paper presumes an acquaintance with the following previous papers: Ives: “Transformation of Color-Mixture Equations,”.J.F.I. 180, pp. C73–701; December, 1915, and J.F.I. 195, pp. 23–44; January, 1923. Troland: “Report of the Colorimetry Committee of the O. S. A. 1920–21,” J.O.S.A. & R.S.I., 6, pp. 527–596; August, 1922. (Separate copies of this committee report may be had at fifty cents per copy of F. K. Richtmyer, Bus. Mgr., J.O.S.A. & R.S.I., Cornell University, Ithaca, New York.)
[Crossref]

J.F.I.,  180, pp. 673–701; Dec., 1915.

J.F.I.,  195, pp. 23–44; Jan., 1923.

J.F.I.,  195, p. 40; Jan., 1923.

J.O.S.A. & R.S.I. (9)

J.O.S.A. & R.S.I.,  6, pp. 527–596; August, 1922. See particularly Fig. 9, p. 575, and Table 14A, pp. 586–587.
[Crossref]

J.O.S.A. & R.S.I.,  6, pp. 586–587; August, 1922.

See J.O.S.A. & R.S.I.,  6, pp. 582–585; August, 1922.

J.O.S.A. & R.S.I.,  6, p. 575; August, 1922.

See first paragraph, p. 585, J.O.S.A. & R.S.I., August, 1922.

Table 14A, pp. 586–587, J.O.S.A. & R.S.I., August, 1922.

J.O.S.A. & R.S.I.,  6, p. 527 and p. 548; August, 1922.
[Crossref]

J.O.S.A. & R.S.I.,  6, pp. 586–587; August, 1922.

Advance abstract already published. See proceedings Eighth Annual Meeting, Optical Society of America, J.O.S.A. & R.S.I.,  8, pp. 28–29; January, 1924. A further communication will be made at Ninth Annual Meeting, Boston, October, 1924. The complete paper will appear later as a B. S. Sci. Paper.

Phil. Mag., (6) (1)

Ives: Phil. Mag., (6),  24, pp. 845–853; Dec., 1912. Priest: J.O.S.A. & R.S.I.,  8, pp. 198–200; Jan., 1924.
[Crossref]

Phil. Trans. (1)

Maxwell: “On the Theory of Compound Colours …” Phil. Trans., 1860. (Or Maxwell’s “Scientific Papers,”I, pp. 410–444.) Koenig and Dieterici: Sitzb. der Akad. Berlin, pp. 805–829; July29, 1886. (Or Koenig’s “Gesam. Abhand. Physiol. Optik,” pp. 60–87.) Koenig and Dieterici: Zeit. für Phych. und Physiol. der Sinnesorgane 4, pp. 241–347; 1892. (Or Koenig’s “Gesam. Abhand,” pp. 214–321). Abney: “Researches in Colour Vision” (London, 1913) Chap. XV. (1)For general treatments see: Helmholtz’ “Physiol. Optik,”2nd ed. (Leipzig, 1896), pp. 311–384. (2)Köllner’s “Die Stoerungen des Farbensinnes” (Berlin, 1912). (3)Parsons’ “Introduction to Study of Colour Vision” (Cambridge, 1915). (4)Peddie’s “Colour Vision.” (London, 1922). For a brief elementary treatment, see Watson’s “Text-Book of Physics” (London, 1903) Chap. IX.
[Crossref]

Pogg. Ann. (1)

Grassman: Pogg. Ann.,  89, p. 70; May, 1853, (or Grassmann’s “Math. & Phy. Werke”2, part 2, p. 162). Popular expositions may be found in Rood’s “Modern Chromatics,” Abney’s “Colour Measurement and Mixture,” and Luckiesh’s “Color and its Applications.” See also Nutting, B. S. Bulletin 9, pp. 1–5; 1913and Priest: J.O.S.A. & R.S.I. 8, pp. 173–200; January, 1923.
[Crossref]

See J.O.S.A. & R.S.I. (1)

See J.O.S.A. & R.S.I.,  6, p. 534; 1922.

Trans, I. E. S. (1)

Cf. Ives: Trans, I. E. S.,  5, pp. 196–198; April, 1910. Priest: B. S. Sci. Pap. 417, pp. 256–258, particularly Fig. 15; August, 1921.

Other (18)

This “absolute impurity” of course presupposes a given set of “elementary stimuli.” It is not “absolute” if a change of “elementary stimuli” is involved.

Cf. “Report of Colorimetry Committee.” J.O.S.A. & R.S.I., 6, p. 564; August, 1922.

Cf. Peddie: “Colour Vision” (London, 1922), pp. 16–17, 52–53, 108–109.

Whether we call this color “white” or “gray,” is, for our present purpose, a matter of indifference. Usage on this point is not well established.

See Sec. IV-2 below for details.

“This formula” [21, in the present paper] “appears to be valid, although in a form which differs from any heretofore proposed, in the use of ‘c’ and ‘C,’ the ‘least coordinates’ ”—From letter of Herbert E. Ives to the Director of the Bureau of Standards, October 1, 1923.

Letter, Ives to Director, Bureau of Standards, Oct. 1, 1923.

See appendix, this paper.

“It is gratifying that formulas arrived at through such entirely independent lines of thought and which look so different are identical in the value they give”—Letter, Ives to Priest, Feb. 27th, 1924, referring to the numerical agreement of values of p computed respectively by formulas 21 and 22.

Casual inspection of these formulas (22) might suggest the mathematical possibility of negative values of p, for it is apparent that the numerator of the first factor and the denominator of the second factor may be negative. Negative values of p would have no physical meaning. However, these factors are always of the same sign whether positive or negative. Consequently the computed value of p is always positive.

Cf. last paragraph of Sec. IV above.

In a letter of July 30, 1924, addressed to the author after reading this paper in manuscript Mr. Weaver expresses the opinion that these deviations are to be attributed to inconsistent data for the “excitation-curves” and the “visibility-curves.” (Cf. IV-2 above.) I.G.P. Aug. 4, 1924.

See IV-2 above.

See Sec. IV-1 above.

Note that Weaver’s tabulation is actually given in terms of (1−p) in per cent, not directly in terms of p.

Or the formula derived above (equation 21) which is identical with it.

Cf. Sec. IV-1, this paper.

Letter, Ives to Priest, Feb. 27, 1924.

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Figures (4)

Fig. 2
Fig. 2

(Reproduction of Fig. 9 from Ives’ Paper, p. 35, J.F.I., Jan. 1923)

Equations (44)

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Purity p B Λ B
Impurity i B w B
B = B Λ + B w
p + i = 1.00 )
ρ = γ = β
0 ρ λ E λ d λ = 0 γ λ E λ d λ = 0 β λ E λ d λ .
L r + L g + L b = 1.000
L r ρ λ + L g γ λ + L b β λ = V λ
L r 0 ρ λ E λ d λ + L g 0 γ λ E λ d λ + L b 0 β λ E λ d λ = 0 V λ E λ d λ
r ρ ρ + γ + β
g γ ρ + γ + β
b β ρ + γ + β
r + g + b = 1.000
r = g = b
i = { The component of brightness due to the joint action of the three stimuli each taken in the amount , c The brightness due to the joint action of the three stimuli taken respectively in the amounts r , g , and b . }
K c ( L r + L g + L b )
K ( r L r + g L g + b L b )
i = K c ( L r + L g + L b ) K ( r L r + g L g + b L b )
i = c ( r L r + g L g + b L b )
p c = 1 - c ( r L r + g L g + b L b )
p C = 1 - C ( R L r + G L g + B L b )
p = p c p C
p = 1 - c r L r + g L g + b L b 1 - C R L r + G L g + B L b
or or             p = ( r - g r L r + g L g + b L b ) ( R L r + G L g + B L b R - G ) p = ( b - r r L r + g L g + b L b ) ( R L r + G L g + B L b B - R ) p = ( b - g r L r + g L g + b L b ) ( R L r + G L g + B L b B - G ) }
p = 1 - r r L r + g L g + b L b 1 - R R L r + G L g + B L b
p = 1 - g r L r + g L g + b L b 1 - G R L r + G L g + B L b
p = 1 - b r L r + g L g + b L b 1 - B R L r + G L g + B L b
p = 3 - 1 r L r + g L g + b L b 3 - 1 R L r + G L g + B L b
L r = 0.568 L g = .426 L b = .006 _ L r + L g + L b = 1.000
L r = 0.370 L g = 0.617 L b = 0.012 _ L r + L g + L b = 0.999 }             ( See J. O. S. A. & R. S. I. , 6 , p. 551 and p. 582 ; August , 1922. )
ρ λ , γ λ , β λ , L r , L g , L b and V λ
p = q R L r + G L g + B L b r L r + g L g + b L b where q = r - g R - G = g - b G - B = b - r B - R
p = r L r + g L g + b L b - c R L r + G L g + B L b - C × R L r + G L g + B L b r L r + g L g + b L b
q = r L r + g L g + b L b - c R L r + G L g + B L b - C
q = x - r x - R
q = x - r x - R = x - g x - G = x - b x - B
q = L r ( x - r ) + L g ( x - g ) + L b ( x - b ) - ( x - c ) L r ( x - R ) + L g ( x - G ) + L b ( x - B ) - ( x - C )
q = x - c x - C
q = r L r + g L g + b L b - c R L r + G L g + B L b - C ,             Q. E. D.
r + g + b = 1 and R + G + B = 1
q = x - r + x - g + x - b x - R + x - G + x - B = 3 x - 1 3 x - 1
x = 1 / 3
q = 1 / 3 - b 1 / 3 - B = 1 / 3 - g 1 / 3 - G = 1 / 3 - r 1 / 3 - R
p = 3 - 1 r L r + g L g + b L b 3 - 1 R L r + G L g + B L b