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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 21, Iss. 9 — Sep. 1, 1931
  • pp: 531–536

INTERPOLATION OF THE O. S. A “EXCITATION” DATA BY THE FIFTH-DIFFERENCE OSCULATORY FORMULA1

DEANE B. JUDD  »View Author Affiliations


JOSA, Vol. 21, Issue 9, pp. 531-536 (1931)
http://dx.doi.org/10.1364/JOSA.21.000531


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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.

Citation
DEANE B. JUDD, "INTERPOLATION OF THE O. S. A “EXCITATION” DATA BY THE FIFTH-DIFFERENCE OSCULATORY FORMULA1," J. Opt. Soc. Am. 21, 531-536 (1931)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-21-9-531


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References

  1. Publication Approved by the Director of the Bureau of Standards of the U. S. Department of Commerce.
  2. L. T. Troland, Report of the Committee on Colorimetry for 1920–21, J.O.S.A. and R.S.I., 6, 547–553; 1922.
  3. Spectrophotometry; Report of O. S. A. Progress Committee for 1922–3, J.O.S.A. and R.S.I., 10, 230; 1925.
  4. 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.
  5. 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.
  6. T. B. Sprague, Explanation of a New Formula for Interpolation, J. Inst. Actuaries, 22, 270–285; 1880.
  7. J. Karup, On a New Mechanical Method of Graduation, Trans. Second International Actuarial Congress, 82; 1899.
  8. 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.
  9. J. W. Glover, United States Life Tables, 1890, 1901, 1910, and 1901–1910, 344–347, 372–388; 1921.
  10. J. W. Glover, Derivation of the United States Mortality Table by Osculatory Interpolation, Quarterly Publications of the American Statistical Association, 12, 87–93; 1910.
  11. 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.
  12. See footnote 3, p. 531. The values referred to here are included in Table 5 along with the values obtained from them by interpolation.
  13. The values for ρ0 from 451 to 459 and from 461 to 469 mµ result from substituting in the formula ƒ(-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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