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  1. “Illuminating engineering nomenclature and photometric standards,” Illum. Eng. 36, 820 (1941).
  2. Parry Moon, “A system of photometric concepts,” J. Opt. Soc. Am. 32, 348 (1942).
    [Crossref]
  3. Usually called “visibility” or “luminosity.” See Parry Moon, “Basic principles in illumination calculations,” J. Opt. Soc. Am. 29, 108 (1939). It has been customary to use two distinct symbols to represent the same set of values, v(λ) being employed in photometry and y¯(λ) in colorimetry. The two symbols represent identical relative values, on the basis of a maximum ordinate of approximately unity. There seems to be no merit in having two symbols for the same function, so in this paper y¯ (λ) will indicate relative values while v(λ) will be in lumens per watt.
    [Crossref]
  4. D. A. Goldhammer, “Die Farbenempfindlichkeit des Auges und die photometrische Helligkeit der leuchtenden Körper,” Ann. d. Physik 16, 621 (1905).
    [Crossref]
  5. P. G. Nutting, “The luminous equivalent of radiation,” Bull. Bur. Stand. 5, 261 (1908–9).
    [Crossref]
  6. E. F. Kingsbury, “A spectral luminosity curve equation and its use,” Phys. Rev. 7, 161 (1916).
    [Crossref]
  7. K. S. Gibson and E. P. T. Tyndall, “Visibility of radiant energy,” Sci. Pap. Bur. Stand. 19, 131 (1924); Trans. I. E. S. 19, 176 (1924).
    [Crossref]
  8. Commission Internationale de l’Eclairage, Geneva, Switzerland, 1924. See report by E. C. Crittenden, Trans. I. E. S. 19, 607 (1924).
  9. E. P. T. Tyndall and K. S. Gibson, “Visibility of radiant energy equation,” J. Opt. Soc. Am. 9, 403 (1924).
    [Crossref]
  10. J. W. T. Walsh, “Visibility of radiant energy equation,” J. Opt. Soc. Am. 11, 111 (1925).
    [Crossref]
  11. R. A. Houstoun, “The visibility of radiation and dark adaptation,” Phil. Mag. 10, 416 (1930). “Color vision,” Phil. Mag. 11, 565 (1931). Vision and Color Vision (Longmans, Green and Company, London, 1932), p. 81.
  12. C. V. L. Charlier, Die Grundzüge der mathematischen Statistik (Hamburg, 1920). A. Fisher, Mathematical Theory of Probabilities (Macmillan, New York, 1930). N. R. Jørgensen, Undersøgelser over Frequensflader og Korrelation (København, 1916).
  13. K. Pearson, “Supplement to a memoir on skew variation,” Roy. Soc., Phil. Frans. 197A, 446 (1901). W. P. Elderton, Frequency Curves and Correlation (Cambridge University Press, 1938).
  14. Tables of the Exponential Function ex (Nat. Bur. Stand., Washington, 1939).
  15. Tables of Probability Functions (Nat. Bur. Stand., Washington, 1941), Vol. I.
  16. J. Peters, Zehnstellige Logarithmen (Berlin, 1922), Vol. I, Table 6.
  17. T. C. Fry, Probability and Its Engineering Uses (Van Nostrand Company, New York, 1928), p. 429.
  18. Tables of Sine, Cosine, and Exponential Integrals (Nat. Bur. Stand., Washington, 1940), Vol. I.
  19. D. B. Judd, “Extension of the standard visibility function to intervals of 1 millimicron by third-difference osculatory interpolation,” J. Opt. Soc. Am. 21, 267 (1931). Z. Yamauti, “Interpolation of the standard visibility function with one millimicron steps,” Res. Elec. Lab.Tokyo, No. 388 (1935).
    [Crossref]
  20. Bowditch and Null, “Selected ordinates for computing trichromatic coefficients and candlepower of a light source,” J. Opt. Soc. Am. 28, 500 (1938).
    [Crossref]
  21. Parry Moon, “A table of Planck’s function from 3500 to 8000°K,” J. Math. Phys. 16, 33 (1937); M. I. T., E. E. Contr. No. 131 (1938).
  22. These suggestions have no official sanction and are merely the personal opinions of the authors. It may seem visionary in 1943 to mention international agreement; but a little thought will indicate that the present is an excellent time to initiate a program, for it will allow time to assess the merits of the proposal in the U. S. and to determine if it should be submitted to the C. I. E.
  23. A. Dresler, “Über eine jahreszeitliche Schwankung der Spektralen Hellempfindlichkeit,” Das Licht, 10 (1940).
  24. K. S. Gibson, “Spectral luminosity factors,” J. Opt. Soc. Am. 30, 51 (1940). W. Arndt, “Über neue Beobachtungen beim subjektiven Photometrieren,” Das Licht 6, 75 (1936). M. Jaggi, “Beitrag zur Kenntnis der spektralen Hellempfindlichkeit des menschlichen Auges,” Helv. Phys. Acta 12, 77 (1939).
    [Crossref]
  25. P. Jainski, “Die spektrale Hellempfindlichkeit des menschlichen Auges und ihre Bedeutung für die Lichtmesstechnik,” Diss. Tech. Hochschule, Berlin (1938).
  26. J. S. Preston, “The relative luminosity of radiation for the average observer,” Proc. Phys. Soc. London 51, 757 (1939).
    [Crossref]
  27. J. Guild, “The colorimetric properties of the spectrum,” Phil. Trans. Roy. Soc. London 230A, 149 (1932).
  28. The Illuminating Engineering Society (ref. 1) recommends the Judd interpolated values but still leaves the question of extrapolation undecided. The C. I. E. has not changed its original specification of v(λ) at intervals of 0.01μ.

1942 (1)

Parry Moon, “A system of photometric concepts,” J. Opt. Soc. Am. 32, 348 (1942).
[Crossref]

1941 (1)

“Illuminating engineering nomenclature and photometric standards,” Illum. Eng. 36, 820 (1941).

1940 (2)

1939 (2)

1938 (1)

1937 (1)

Parry Moon, “A table of Planck’s function from 3500 to 8000°K,” J. Math. Phys. 16, 33 (1937); M. I. T., E. E. Contr. No. 131 (1938).

1932 (1)

J. Guild, “The colorimetric properties of the spectrum,” Phil. Trans. Roy. Soc. London 230A, 149 (1932).

1931 (1)

1930 (1)

R. A. Houstoun, “The visibility of radiation and dark adaptation,” Phil. Mag. 10, 416 (1930). “Color vision,” Phil. Mag. 11, 565 (1931). Vision and Color Vision (Longmans, Green and Company, London, 1932), p. 81.

1925 (1)

1924 (2)

K. S. Gibson and E. P. T. Tyndall, “Visibility of radiant energy,” Sci. Pap. Bur. Stand. 19, 131 (1924); Trans. I. E. S. 19, 176 (1924).
[Crossref]

E. P. T. Tyndall and K. S. Gibson, “Visibility of radiant energy equation,” J. Opt. Soc. Am. 9, 403 (1924).
[Crossref]

1916 (1)

E. F. Kingsbury, “A spectral luminosity curve equation and its use,” Phys. Rev. 7, 161 (1916).
[Crossref]

1905 (1)

D. A. Goldhammer, “Die Farbenempfindlichkeit des Auges und die photometrische Helligkeit der leuchtenden Körper,” Ann. d. Physik 16, 621 (1905).
[Crossref]

1901 (1)

K. Pearson, “Supplement to a memoir on skew variation,” Roy. Soc., Phil. Frans. 197A, 446 (1901). W. P. Elderton, Frequency Curves and Correlation (Cambridge University Press, 1938).

Bowditch,

Charlier, C. V. L.

C. V. L. Charlier, Die Grundzüge der mathematischen Statistik (Hamburg, 1920). A. Fisher, Mathematical Theory of Probabilities (Macmillan, New York, 1930). N. R. Jørgensen, Undersøgelser over Frequensflader og Korrelation (København, 1916).

Dresler, A.

A. Dresler, “Über eine jahreszeitliche Schwankung der Spektralen Hellempfindlichkeit,” Das Licht, 10 (1940).

Fry, T. C.

T. C. Fry, Probability and Its Engineering Uses (Van Nostrand Company, New York, 1928), p. 429.

Gibson, K. S.

Goldhammer, D. A.

D. A. Goldhammer, “Die Farbenempfindlichkeit des Auges und die photometrische Helligkeit der leuchtenden Körper,” Ann. d. Physik 16, 621 (1905).
[Crossref]

Guild, J.

J. Guild, “The colorimetric properties of the spectrum,” Phil. Trans. Roy. Soc. London 230A, 149 (1932).

Houstoun, R. A.

R. A. Houstoun, “The visibility of radiation and dark adaptation,” Phil. Mag. 10, 416 (1930). “Color vision,” Phil. Mag. 11, 565 (1931). Vision and Color Vision (Longmans, Green and Company, London, 1932), p. 81.

Jainski, P.

P. Jainski, “Die spektrale Hellempfindlichkeit des menschlichen Auges und ihre Bedeutung für die Lichtmesstechnik,” Diss. Tech. Hochschule, Berlin (1938).

Judd, D. B.

Kingsbury, E. F.

E. F. Kingsbury, “A spectral luminosity curve equation and its use,” Phys. Rev. 7, 161 (1916).
[Crossref]

Moon, Parry

Null,

Nutting, P. G.

P. G. Nutting, “The luminous equivalent of radiation,” Bull. Bur. Stand. 5, 261 (1908–9).
[Crossref]

Pearson, K.

K. Pearson, “Supplement to a memoir on skew variation,” Roy. Soc., Phil. Frans. 197A, 446 (1901). W. P. Elderton, Frequency Curves and Correlation (Cambridge University Press, 1938).

Peters, J.

J. Peters, Zehnstellige Logarithmen (Berlin, 1922), Vol. I, Table 6.

Preston, J. S.

J. S. Preston, “The relative luminosity of radiation for the average observer,” Proc. Phys. Soc. London 51, 757 (1939).
[Crossref]

Tyndall, E. P. T.

E. P. T. Tyndall and K. S. Gibson, “Visibility of radiant energy equation,” J. Opt. Soc. Am. 9, 403 (1924).
[Crossref]

K. S. Gibson and E. P. T. Tyndall, “Visibility of radiant energy,” Sci. Pap. Bur. Stand. 19, 131 (1924); Trans. I. E. S. 19, 176 (1924).
[Crossref]

Walsh, J. W. T.

Ann. d. Physik (1)

D. A. Goldhammer, “Die Farbenempfindlichkeit des Auges und die photometrische Helligkeit der leuchtenden Körper,” Ann. d. Physik 16, 621 (1905).
[Crossref]

Bull. Bur. Stand. (1)

P. G. Nutting, “The luminous equivalent of radiation,” Bull. Bur. Stand. 5, 261 (1908–9).
[Crossref]

Das Licht (1)

A. Dresler, “Über eine jahreszeitliche Schwankung der Spektralen Hellempfindlichkeit,” Das Licht, 10 (1940).

Illum. Eng. (1)

“Illuminating engineering nomenclature and photometric standards,” Illum. Eng. 36, 820 (1941).

J. Math. Phys. (1)

Parry Moon, “A table of Planck’s function from 3500 to 8000°K,” J. Math. Phys. 16, 33 (1937); M. I. T., E. E. Contr. No. 131 (1938).

J. Opt. Soc. Am. (7)

K. S. Gibson, “Spectral luminosity factors,” J. Opt. Soc. Am. 30, 51 (1940). W. Arndt, “Über neue Beobachtungen beim subjektiven Photometrieren,” Das Licht 6, 75 (1936). M. Jaggi, “Beitrag zur Kenntnis der spektralen Hellempfindlichkeit des menschlichen Auges,” Helv. Phys. Acta 12, 77 (1939).
[Crossref]

Parry Moon, “A system of photometric concepts,” J. Opt. Soc. Am. 32, 348 (1942).
[Crossref]

Usually called “visibility” or “luminosity.” See Parry Moon, “Basic principles in illumination calculations,” J. Opt. Soc. Am. 29, 108 (1939). It has been customary to use two distinct symbols to represent the same set of values, v(λ) being employed in photometry and y¯(λ) in colorimetry. The two symbols represent identical relative values, on the basis of a maximum ordinate of approximately unity. There seems to be no merit in having two symbols for the same function, so in this paper y¯ (λ) will indicate relative values while v(λ) will be in lumens per watt.
[Crossref]

D. B. Judd, “Extension of the standard visibility function to intervals of 1 millimicron by third-difference osculatory interpolation,” J. Opt. Soc. Am. 21, 267 (1931). Z. Yamauti, “Interpolation of the standard visibility function with one millimicron steps,” Res. Elec. Lab.Tokyo, No. 388 (1935).
[Crossref]

Bowditch and Null, “Selected ordinates for computing trichromatic coefficients and candlepower of a light source,” J. Opt. Soc. Am. 28, 500 (1938).
[Crossref]

E. P. T. Tyndall and K. S. Gibson, “Visibility of radiant energy equation,” J. Opt. Soc. Am. 9, 403 (1924).
[Crossref]

J. W. T. Walsh, “Visibility of radiant energy equation,” J. Opt. Soc. Am. 11, 111 (1925).
[Crossref]

Phil. Mag. (1)

R. A. Houstoun, “The visibility of radiation and dark adaptation,” Phil. Mag. 10, 416 (1930). “Color vision,” Phil. Mag. 11, 565 (1931). Vision and Color Vision (Longmans, Green and Company, London, 1932), p. 81.

Phil. Trans. Roy. Soc. London (1)

J. Guild, “The colorimetric properties of the spectrum,” Phil. Trans. Roy. Soc. London 230A, 149 (1932).

Phys. Rev. (1)

E. F. Kingsbury, “A spectral luminosity curve equation and its use,” Phys. Rev. 7, 161 (1916).
[Crossref]

Proc. Phys. Soc. London (1)

J. S. Preston, “The relative luminosity of radiation for the average observer,” Proc. Phys. Soc. London 51, 757 (1939).
[Crossref]

Roy. Soc., Phil. Frans. (1)

K. Pearson, “Supplement to a memoir on skew variation,” Roy. Soc., Phil. Frans. 197A, 446 (1901). W. P. Elderton, Frequency Curves and Correlation (Cambridge University Press, 1938).

Sci. Pap. Bur. Stand. (1)

K. S. Gibson and E. P. T. Tyndall, “Visibility of radiant energy,” Sci. Pap. Bur. Stand. 19, 131 (1924); Trans. I. E. S. 19, 176 (1924).
[Crossref]

Other (10)

Commission Internationale de l’Eclairage, Geneva, Switzerland, 1924. See report by E. C. Crittenden, Trans. I. E. S. 19, 607 (1924).

C. V. L. Charlier, Die Grundzüge der mathematischen Statistik (Hamburg, 1920). A. Fisher, Mathematical Theory of Probabilities (Macmillan, New York, 1930). N. R. Jørgensen, Undersøgelser over Frequensflader og Korrelation (København, 1916).

Tables of the Exponential Function ex (Nat. Bur. Stand., Washington, 1939).

Tables of Probability Functions (Nat. Bur. Stand., Washington, 1941), Vol. I.

J. Peters, Zehnstellige Logarithmen (Berlin, 1922), Vol. I, Table 6.

T. C. Fry, Probability and Its Engineering Uses (Van Nostrand Company, New York, 1928), p. 429.

Tables of Sine, Cosine, and Exponential Integrals (Nat. Bur. Stand., Washington, 1940), Vol. I.

The Illuminating Engineering Society (ref. 1) recommends the Judd interpolated values but still leaves the question of extrapolation undecided. The C. I. E. has not changed its original specification of v(λ) at intervals of 0.01μ.

P. Jainski, “Die spektrale Hellempfindlichkeit des menschlichen Auges und ihre Bedeutung für die Lichtmesstechnik,” Diss. Tech. Hochschule, Berlin (1938).

These suggestions have no official sanction and are merely the personal opinions of the authors. It may seem visionary in 1943 to mention international agreement; but a little thought will indicate that the present is an excellent time to initiate a program, for it will allow time to assess the merits of the proposal in the U. S. and to determine if it should be submitted to the C. I. E.

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Figures (1)

Fig. 1
Fig. 1

Standard lamprosity curve and various analytic approximations. — standard C. I. E. curve; ○ computed by Walsh equation; ● Houstoun;×probability; □ Pearson; δ composite.

Tables (9)

Tables Icon

Table I The standard lamprosity function.

Tables Icon

Table II The composite curve.

Tables Icon

Table III The equal-energy spectrum. L(λ)=1.00000.

Tables Icon

Table IV Three-term energy distribution. L(λ)=8.1802–36.512λ+41.285λ2.

Tables Icon

Table V Planckian distribution. (C2=14320).

Tables Icon

Table VI Calculated values of lamprosity.

Tables Icon

Table VII Examples of calculations for Planckian radiators. C1=36970, C2=14320, p=182.19050, q=100.9370, log [621C1A·Γ(p+4)]=380.81916.

Tables Icon

Table VIII Planckian radiation calculated by means of Eq. (16a).

Tables Icon

Table IX λm for radiation from incandescent sources.

Equations (75)

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L = 0 w ( λ ) L ( λ ) d λ ,
y ¯ ( λ ) A λ p e p ( 1 - a / λ ) .
y ¯ ( λ ) = A 1 λ 200 e - 111.20 / λ + A 2 λ 400 e - 186.00 / λ + A 3 λ 1000 e - 610.00 / λ ,
y ¯ ( λ ) = A 1 λ 200 e - 111.00 / λ + A 2 λ 550 e - 333.85 / λ + A 3 λ 2000 e - 1046.0 / λ + A 4 λ 630 e - 294.21 / λ ,
y ¯ ( λ ) = A 1 λ 220 e - 122.32 / λ + A 2 λ 700 e - 427.00 / λ + A 3 λ 2000 e - 1046.0 / λ + A 4 λ 350 e - 166.25 / λ ,
y ¯ ( λ ) = A exp [ ( log λ - log λ 0 ) 2 2 σ 2 ] .
y ¯ ( λ ) = 1.024 exp [ - ( log λ - 1 ¯ .7451 ) 2 2 ( 0.0330 ) 2 ] = 1.024 exp [ - ( ln λ + 0.586929 ) 2 2 ( 0.075986 ) 2 ] .
f ( z ) = c 0 φ 0 ( z ) + c 1 φ 1 ( z ) + c 2 φ 2 ( z ) ,
φ 0 ( z ) = H 0 ( z ) exp [ - z 2 / 2 ] , φ 1 ( z ) = - H 1 ( z ) exp [ - z 2 / 2 ] , φ 2 ( z ) = H 2 ( z ) exp [ - z 2 / 2 ] , .         .         .         .         .         .         .         .
y ¯ ( λ ) = c 0 φ 0 ( λ ) + c 3 φ 3 ( λ ) + c 4 φ 4 ( λ )
φ 0 ( λ ) = ( 1 / 2 π σ ) exp [ - 1 2 ( λ - λ 0 / σ ) 2 ] ,
λ 0 = 0.560168 μ , σ = 0.0418466 , φ ( λ ) = 1 0.0418466 ( 2 π ) 1 2 exp [ - 1 2 ( λ - 0.560168 0.0418466 ) 2 ] , c 0 = 0.106832 , c 3 = - 5.14385 × 10 - 4 , c 4 = 6.21915 × 10 - 5 .
y ¯ ( λ ) = 1.0185 exp [ - 1 2 ( λ - 0.560168 0.0418466 ) 2 ] .
y ¯ ( λ ) = A 1 λ p e - q / λ .
p = 182.19050 , q = 100.93700 , log A 1 = 32.41070.
log y ¯ = 32.41070 - 182.19050 log λ - 43.83638 / λ .
y ¯ ( λ ) = 268 ( λ - 0.400 ) 3 + 5.7 × 10 4 ( λ - 0.400 ) 6 ,
y ¯ ( λ ) = 1.000 - 215.24 ( λ - 0.556 ) 2 ,
y ¯ ( λ ) = 1.05 × 10 4 ( 0.750 - λ ) 5 - 5.0906 × 10 6 ( 0.750 - λ ) 9 .
L l = a b y ¯ ( λ ) L ( λ ) d λ .
L ( λ ) = m = 0 K m λ m
L ( λ ) = n = 1 K n x 5 e - n x ,
L ( λ ) = C 1 λ 5 exp [ - C 2 λ T ] ,
L ( λ ) = C 1 λ 5 1 [ C 2 / λ T ] - 1 .
L ( λ ) = C 1 ( C 2 T ) - 5 x 5 [ e - x + e - 2 x + e - 3 x + ] ,
y ¯ ( λ ) = A exp [ - ( λ - a ) 2 / 2 σ 2 ] ,
y ¯ ( λ ) = A exp [ - ( ln λ - c ) 2 / 2 σ 2 ] ,
y ¯ ( λ ) = ( A / λ p ) e - q / λ ,
y ¯ ( λ ) = B n ( λ - b ) n .
L l = A 1 K m 0 λ m exp [ - ( λ - a ) 2 / 2 σ 2 ] d λ .
L l = A 1 K m σ m + 1 - ( x + a / σ ) m exp [ - x 2 / 2 ] d x .
L l = A 1 K m σ m + 1 [ - x m exp [ - x 2 / 2 ] d x + m ( a / σ ) - x m - 1 exp [ - x 2 / 2 ] d x + m ( m - 1 ) 2 ! ( a / σ ) 2 - x m - 2 exp [ - x 2 / 2 ] d x + + ( a / σ ) m - exp [ - x 2 / 2 ] d x ] .
- exp [ - x 2 / 2 ] d x = ( 2 π ) 1 2 , - x r exp [ - x 2 / 2 ] d x = 1 · 3 · 5 ( r - 1 ) ( 2 π ) 1 2 ,
L l = A 1 K m 0 λ m exp [ - ( ln λ - c ) 2 / 2 σ 2 ] d λ .
L l = A 1 K m σ e c ( m + 1 ) - e σ ( m + 1 ) x exp [ - x 2 / 2 ] d x = A 1 K m σ e c ( m + 1 ) exp [ σ 2 ( m + 1 ) 2 2 ] × - exp { - 1 2 [ x - σ ( m + 1 ) ] 2 } d x = ( 2 π ) 1 2 A 1 K m σ exp [ c ( m + 1 ) + σ 2 ( m + 1 ) 2 2 ] .
L l = A 1 K m 0 λ m - p e - q / λ d λ .
L l = A 1 K m q m - p + 1 0 z p - m - 2 e - z d z .
0 x r e - a x d x = Γ ( r + 1 ) a r + 1 ,
L l = A 1 K m Γ ( p - m - 1 ) q p - m - 1 .
L l = A 1 C 1 0 λ - ( p + 5 ) exp [ - ( q + C 2 / T ) 1 λ ] d λ .
L l = A 1 C 1 ( q + C 2 / T ) p + 4 0 z p + 3 e - z d z = A 1 C 1 Γ ( p + 4 ) ( q + C 2 / T ) p + 4 .
L l = A 1 C 1 ( C 2 / T ) - ( p + 4 ) 0 x p + 3 × exp [ - ( n + q T / C 2 ) x ] d x = A 1 C 1 Γ ( p + 4 ) ( q + C 2 n / T ) p + 4 .
ρ = m = 0 a m λ m .
L l = A 1 C 1 ( C 2 T ) - ( p + 4 ) m = 0 n = 1 x = 0 x = a m λ m x p + 3 × exp [ - ( n + q T / C 2 ) x ] d x = A 1 C 1 ( C 2 T ) - ( p + 4 - m ) m = 0 n = 1 0 a m x p + 3 - m × exp [ - ( n + q T / C 2 ) x ] d x .
L l = A 1 C 1 a m ( C 2 T ) - ( p + 4 - m ) 0 x p + 3 - m × exp [ - ( n + q T / C 2 ) x ] d x = A 1 C 1 a m Γ ( p + 4 - m ) ( q + C 2 n / T ) p + 4 - m .
L l = K m B n a b λ m + n d λ = K m B n m + n + 1 × ( b m + n + 1 - a m + n + 1 ) .
L l = C 1 B n a b λ n - 5 e - C 2 / λ T d λ = C 1 B n ( C 2 T ) n - 4 C 2 / T b C 2 / T a e - x d x x n - 3 .
x m e - x d x = - x m e - x + m x m - 1 e - x d x ,
L l = - C 1 B n ( C 2 T ) n - 4 E i ( - x ) ,             n = 4 , L l = C 1 B n ( C 2 T ) n - 4 [ - x 3 - n e - x + ( 3 - n ) x 2 - n e - x d x ] ,             n 3 , L l = - C 1 B n ( C 2 T ) n - 4 [ e - x x n - 4 + e - x d x x n - 4 ] ,             n 5.
Γ ( m ) = ( m - 1 ) !
Γ ( x ) = ( 2 π ) 1 2 ( x - 1 ) x - 1 2 e - ( x - 1 ) × [ 1 + 1 12 ( x - 1 ) + 1 288 ( x - 1 ) 2 + ] ,
log Γ ( x ) = 0.39908993 + ( x - 1 2 ) log ( x - 1 ) - 0.4342945 ( x - 1 ) + log [ 1 + 1 12 ( x - 1 ) + ] .
ρ = 1.630 λ - 0.620.
v ( λ ) = A λ p e - q / λ ( lumens / watt ) ,
L l = 0 v ( λ ) L ( λ ) d λ .
L ( λ ) = m = 0 K m λ m ,
L l = A K m Γ ( p - m - 1 ) q p - m - 1 .
log L l = 35.203 7916 - 2.004 0505 ( 181.19050 - m ) + log Γ ( 181.19050 - m ) + log K m .
L ( λ ) = C 1 λ 5 1 exp [ C 2 / λ T ] - 1 .
L l = A C 1 n = 1 Γ ( p + 4 ) ( q + C 2 n / T ) p + 4 .
log L l = 380.819 158 - 186.19050 × log ( 100.9370 + 14320 n / T ) .
ρ = m = 0 a m λ m ,
L l = A C 1 a m Γ ( p + 4 - m ) ( q + C 2 n / T ) p + 4 - m .
log L l = 39.771 654 + log Γ ( 186.19050 - m ) + log a m - ( 186.19050 - m ) × log ( 100.93700 + 14320 n / T ) .
L ( λ ) = K m λ m
d ( v L ) d λ = [ - ( p - m ) λ p - m + 1 + q λ p - m + 2 ] A K m e - q / λ = A K m λ p - m + 2 [ - λ ( p - m ) + q ] e - q / λ .
λ m = q p - m = 100.9370 182.1905 - m .
L ( λ ) = ( C 1 / λ 5 ) exp [ - C 2 / λ T ] , v ( λ ) L ( λ ) = A C 1 λ p + 5 exp [ - 1 λ ( C 2 T + q ) ] ,
d d λ ( v L ) ] = A C 1 λ p + 7 [ ( C 2 T + q ) - λ ( p + 5 ) ] × exp [ - 1 λ ( C 2 T + q ) ] .
λ m = C 2 / T + q p + 5 = 100.9370 + 14320 / T 187.1905 = 0.5392207 + 76.49961 / T .
v ( λ ) L ( λ ) = A C 1 λ p + 5 e - q / λ exp [ C 2 / λ T ] - 1
d d λ ( v L ) = A C 1 e - q / λ λ p + 7 ( exp [ C 2 / λ T ] - 1 ) 2 × { [ q - λ ( p + 5 ) ] ( exp [ C 2 λ T ] - 1 ) + C 2 T exp [ C 2 λ T ] } .
q - λ m ( p + 5 ) = [ q - λ m ( p + 5 ) + C 2 / T ] × exp [ C 2 / λ m T ]
exp [ 14320 / λ m ] = 100.9370 - 187.1905 λ m 100.9370 - 187.1905 λ m + 14320 / T .
v ( λ ) = A λ p e - q / λ ,