Abstract

In order to compute the dominant wave length and purity of a color stimulus by means of the O. S. A. “excitation” data, two values must be obtained by interpolation. The adoption of the osculatory formula for this interpolation permits the computations to be made with perfect reproducibility. Each of the O. S. A. curves by this method is represented as a series of parabolas of the fifth degree which join at the values specified at every 10 mμ so as to have a common slope and curvature at the junction point. Interpolated values have been computed according to this formula for every millimicron.

© 1931 Optical Society of America

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References

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  1. L. T. Troland, Report of the Committee on Colorimetry for 1920–21, J.O.S.A. and R.S.I.,  6, 547–553; 1922.
    [Crossref]
  2. Spectrophotometry; Report of O. S. A. Progress Committee for 1922 –3, J.O.S.A. and R.S.I.,  10, 230; 1925.
  3. I. G. Priest, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  9, 503–520; 1924.D. B. Judd, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  13, 133–152; 1926.
    [Crossref]
  4. D. B. Judd, Extension of the Standard Visibility Function to Intervals of 1 Millimicron by Third-difference Osculatory Interpolation, B. S. Jour. Research,  6, 465–471; 1931;J.O.S.A.,  21, 267–275; 1931.
    [Crossref]
  5. T. B. Sprague, Explanation of a New Formula for Interpolation, J. Inst. Actuaries,  22, 270–285; 1880.
  6. J. Karup, On a New Mechanical Method of Graduation, Trans. Second International Actuarial Congress, 82; 1899.
  7. George King, On the Construction of Mortality Tables from Census Returns and Records of Deaths, J. Inst. Actuaries,  42, 238–246; 1908.James Buchanan, Osculatory Interpolation by Central Differences; with an Application to Life Table Construction, J. Inst. Actuaries,  42, 369–394; 1908;see also an appendix by G. J. Lidstone, Alternative Demonstration of the Formula for Osculatory Interpolation, pp. 394–397.George King, On a New Method of Constructing and of Graduating Mortality and Other Tables, J. Inst. Actuaries,  43, 109–184; 1909.
  8. J. W. Glover, United States Life Tables, 1890, 1901, 1910, and1901–1910, 344–347, 372–388; 1921.
  9. J. W. Glover, Derivation of the United States Mortality Table by Osculatory Interpolation, Quarterly Publications of the American Statistical Association,  12, 87–93; 1910.
  10. The check was carried out by taking the differences in the ascending order rather than in the descending order as indicated in the formula. See Tables 2 and 4. It might naturally “be supposed that about twice the time to calculate nine values would be required to calculate and check them by an independent method, but this is not quite the case. The time actually required is considerably less than twice because the products found for checking the values in one interval may be used to calculate values in the four subsequent intervals.
  11. See footnote 3, p. 531. The values referred to here are included in Table 5 along with the values obtained from them by interpolation.
  12. The values for ρ0 from 451 to 459 and from 461 to 469 mμ result from substituting in the formula f(−20) equal to 4 and 1, respectively; that is, we have taken ρ0 for 430 and 440 mμ. equal to ρ0 for 470 and 460 mμ, respectively, instead of zero as shown in Table 5. This choice was made in order to bring the interpolated function to zero at 450 mμ with a zero slope; then, for λ less than 450 mμ, ρ0 is arbitrarily set at zero instead of at the values which would be obtained by mechanical application of the formula. Similarly for β0 between 590 and 610 mμ, we choose β0 in the formula as 1 and 2 for 620 and 630, respectively, although β0 is given in Table 5 as zero for wave lengths greater than 610 mμ.

1931 (1)

D. B. Judd, Extension of the Standard Visibility Function to Intervals of 1 Millimicron by Third-difference Osculatory Interpolation, B. S. Jour. Research,  6, 465–471; 1931;J.O.S.A.,  21, 267–275; 1931.
[Crossref]

1925 (1)

Spectrophotometry; Report of O. S. A. Progress Committee for 1922 –3, J.O.S.A. and R.S.I.,  10, 230; 1925.

1924 (1)

I. G. Priest, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  9, 503–520; 1924.D. B. Judd, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  13, 133–152; 1926.
[Crossref]

1922 (1)

L. T. Troland, Report of the Committee on Colorimetry for 1920–21, J.O.S.A. and R.S.I.,  6, 547–553; 1922.
[Crossref]

1921 (1)

J. W. Glover, United States Life Tables, 1890, 1901, 1910, and1901–1910, 344–347, 372–388; 1921.

1910 (1)

J. W. Glover, Derivation of the United States Mortality Table by Osculatory Interpolation, Quarterly Publications of the American Statistical Association,  12, 87–93; 1910.

1908 (1)

George King, On the Construction of Mortality Tables from Census Returns and Records of Deaths, J. Inst. Actuaries,  42, 238–246; 1908.James Buchanan, Osculatory Interpolation by Central Differences; with an Application to Life Table Construction, J. Inst. Actuaries,  42, 369–394; 1908;see also an appendix by G. J. Lidstone, Alternative Demonstration of the Formula for Osculatory Interpolation, pp. 394–397.George King, On a New Method of Constructing and of Graduating Mortality and Other Tables, J. Inst. Actuaries,  43, 109–184; 1909.

1899 (1)

J. Karup, On a New Mechanical Method of Graduation, Trans. Second International Actuarial Congress, 82; 1899.

1880 (1)

T. B. Sprague, Explanation of a New Formula for Interpolation, J. Inst. Actuaries,  22, 270–285; 1880.

Glover, J. W.

J. W. Glover, United States Life Tables, 1890, 1901, 1910, and1901–1910, 344–347, 372–388; 1921.

J. W. Glover, Derivation of the United States Mortality Table by Osculatory Interpolation, Quarterly Publications of the American Statistical Association,  12, 87–93; 1910.

Judd, D. B.

D. B. Judd, Extension of the Standard Visibility Function to Intervals of 1 Millimicron by Third-difference Osculatory Interpolation, B. S. Jour. Research,  6, 465–471; 1931;J.O.S.A.,  21, 267–275; 1931.
[Crossref]

Karup, J.

J. Karup, On a New Mechanical Method of Graduation, Trans. Second International Actuarial Congress, 82; 1899.

King, George

George King, On the Construction of Mortality Tables from Census Returns and Records of Deaths, J. Inst. Actuaries,  42, 238–246; 1908.James Buchanan, Osculatory Interpolation by Central Differences; with an Application to Life Table Construction, J. Inst. Actuaries,  42, 369–394; 1908;see also an appendix by G. J. Lidstone, Alternative Demonstration of the Formula for Osculatory Interpolation, pp. 394–397.George King, On a New Method of Constructing and of Graduating Mortality and Other Tables, J. Inst. Actuaries,  43, 109–184; 1909.

Priest, I. G.

I. G. Priest, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  9, 503–520; 1924.D. B. Judd, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  13, 133–152; 1926.
[Crossref]

Sprague, T. B.

T. B. Sprague, Explanation of a New Formula for Interpolation, J. Inst. Actuaries,  22, 270–285; 1880.

Troland, L. T.

L. T. Troland, Report of the Committee on Colorimetry for 1920–21, J.O.S.A. and R.S.I.,  6, 547–553; 1922.
[Crossref]

B. S. Jour. Research (1)

D. B. Judd, Extension of the Standard Visibility Function to Intervals of 1 Millimicron by Third-difference Osculatory Interpolation, B. S. Jour. Research,  6, 465–471; 1931;J.O.S.A.,  21, 267–275; 1931.
[Crossref]

J. Inst. Actuaries (2)

T. B. Sprague, Explanation of a New Formula for Interpolation, J. Inst. Actuaries,  22, 270–285; 1880.

George King, On the Construction of Mortality Tables from Census Returns and Records of Deaths, J. Inst. Actuaries,  42, 238–246; 1908.James Buchanan, Osculatory Interpolation by Central Differences; with an Application to Life Table Construction, J. Inst. Actuaries,  42, 369–394; 1908;see also an appendix by G. J. Lidstone, Alternative Demonstration of the Formula for Osculatory Interpolation, pp. 394–397.George King, On a New Method of Constructing and of Graduating Mortality and Other Tables, J. Inst. Actuaries,  43, 109–184; 1909.

J.O.S.A. and R.S.I. (3)

L. T. Troland, Report of the Committee on Colorimetry for 1920–21, J.O.S.A. and R.S.I.,  6, 547–553; 1922.
[Crossref]

Spectrophotometry; Report of O. S. A. Progress Committee for 1922 –3, J.O.S.A. and R.S.I.,  10, 230; 1925.

I. G. Priest, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  9, 503–520; 1924.D. B. Judd, The Computation of Colorimetric Purity, J.O.S.A. and R.S.I.,  13, 133–152; 1926.
[Crossref]

Quarterly Publications of the American Statistical Association (1)

J. W. Glover, Derivation of the United States Mortality Table by Osculatory Interpolation, Quarterly Publications of the American Statistical Association,  12, 87–93; 1910.

Trans. Second International Actuarial Congress (1)

J. Karup, On a New Mechanical Method of Graduation, Trans. Second International Actuarial Congress, 82; 1899.

United States Life Tables (1)

J. W. Glover, United States Life Tables, 1890, 1901, 1910, and1901–1910, 344–347, 372–388; 1921.

Other (3)

The check was carried out by taking the differences in the ascending order rather than in the descending order as indicated in the formula. See Tables 2 and 4. It might naturally “be supposed that about twice the time to calculate nine values would be required to calculate and check them by an independent method, but this is not quite the case. The time actually required is considerably less than twice because the products found for checking the values in one interval may be used to calculate values in the four subsequent intervals.

See footnote 3, p. 531. The values referred to here are included in Table 5 along with the values obtained from them by interpolation.

The values for ρ0 from 451 to 459 and from 461 to 469 mμ result from substituting in the formula f(−20) equal to 4 and 1, respectively; that is, we have taken ρ0 for 430 and 440 mμ. equal to ρ0 for 470 and 460 mμ, respectively, instead of zero as shown in Table 5. This choice was made in order to bring the interpolated function to zero at 450 mμ with a zero slope; then, for λ less than 450 mμ, ρ0 is arbitrarily set at zero instead of at the values which would be obtained by mechanical application of the formula. Similarly for β0 between 590 and 610 mμ, we choose β0 in the formula as 1 and 2 for 620 and 630, respectively, although β0 is given in Table 5 as zero for wave lengths greater than 610 mμ.

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Figures (1)

F. 1
F. 1

An example of interpolation of the O. S. A. “excitation” data by the fifth-difference, oscillatory formula.

Tables (5)

Tables Icon

Table 1 Computation of the leading descending major differences, λ0 = 580 mμ.

Tables Icon

Table 2 Compulation of the leading ascending major differences, λ0 = 580 mμ.

Tables Icon

Table 3 Coefficients, k1 to k5, for interpolation to tenths by fifth-difference osculatory interpolation.

Tables Icon

Table 4 Example of interpolation to tenths by the fifth-difference oscillatory formula, descending differences; check by ascending differences.

Tables Icon

Table 5 The O.S.A. “Excitation” Curves extended to values for every millimicron by fifth-difference osculatory interpolation.

Equations (4)

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f ( λ λ 0 ) = f ( 20 ) + k 1 Δ 1 f ( 20 ) + k 2 Δ 2 f ( 20 ) + k 3 Δ 3 f ( 20 ) + k 4 Δ 4 f ( 20 ) + k 5 Δ 5 f ( 20 ) , λ 0 < λ < λ 0 + 10
k 1 ( λ λ 0 + 20 ) / 10 k 2 ( λ λ 0 + 20 ) ( λ λ 0 + 10 ) / 200 k 3 ( λ λ 0 + 20 ) ( λ λ 0 + 10 ) ( λ λ 0 ) / 6 , 000 k 4 ( λ λ 0 + 20 ) ( λ λ 0 + 10 ) ( λ λ 0 ) ( λ λ 0 10 ) / 240 , 000 k 5 ( λ λ 0 ) 3 ( λ λ 0 10 ) ( 5 λ 5 λ 0 70 ) / 2 , 400 , 000
f ( λ λ 0 ) = 466 + 39 k 1 24 k 2 + 24 k 3 64 k 4 + 121 k 5
f ( λ λ 0 ) = 462 + 48 k 1 23 k 2 17 k 3 57 k 4 121 k 5