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  1. Porter, Proc. Roy. Soc.113, p. 347, Proc. Roy. Soc.120, p. 313, Proc. Roy. Soc.136, p. 445.
  2. Kennelly and Whiting, National Electric Light Assn., 1907 convention.
  3. Luckiesh, Physical Review 4, 1, p. 1; 1914.
    [Crossref]
  4. Ives and Kingsbury, Phil. Mag.April1916, p. 290.
    [Crossref]
  5. Ives, Critical Frequency Relations in Scotopic Vision, J. Opt. Soc. Am.May, 1922, p. 254.
    [Crossref]
  6. Troland, Am. Jn. Physiology 32, [May] p. 8; 1913.
  7. See Drysdale, Proc. Optical Conv., p. 173, 1905.
  8. Hecht, Science, April 15, p. 347, 1921.
    [Crossref]
  9. Goldman, Ann. der Physik,  27, p. 449, 1908.
    [Crossref]
  10. The electrical behavior of these photoelectric cells under illumination is strikingly like that of the excised eye, as studied by Waller and other. Notable resemblances are shown in the preliminary negative response on commencement, and terminal positive twitch on cessation of illumination, and in the reversal of the reaction with age. Bose (“Response in the Living and non-Living,” p. 169) remarks “there is not a single phenomenon in the responses, normal or abnormal, exhibited by the retina, which has not its counterpart in the sensitive cell constructed of inorganic material.”
  11. I am very greatly indebted to Mr. T. C. Fry for assistance in the mathematical work which follows, and for helpful discussions of the general problem.
  12. It will be noted that this result follows, whatever the form of the re-composition function. The choice of beΘ for this function is in deference to the generally accepted idea of a logarithmic response to a steady stimulus. The farther the re-composition function departs from a simple direct proportionality to the reaction strength the higher must be the frequency to insure Talbot’s law holding. It is a fact of experiment that this law already holds at the critical frequency for flicker disappearance.
  13. See Preston’s “Heat,”2d Ed., p. 654.
  14. The mutually inclined ω-log I lines obtained by the writer previously for the case shown in Fig. 4 (Phil. Mag., April, 1917, p. 360) are apparently in error, due probably to the short range of intensities available for study in the apparatus then used.
  15. For the influence of diffusion on the response of a liquid photoelectric cell, see Samsonow, Zeits, f. Wiss. Phot. XI, 1912, p. 33.

1921 (1)

Hecht, Science, April 15, p. 347, 1921.
[Crossref]

1917 (1)

The mutually inclined ω-log I lines obtained by the writer previously for the case shown in Fig. 4 (Phil. Mag., April, 1917, p. 360) are apparently in error, due probably to the short range of intensities available for study in the apparatus then used.

1916 (1)

Ives and Kingsbury, Phil. Mag.April1916, p. 290.
[Crossref]

1914 (1)

Luckiesh, Physical Review 4, 1, p. 1; 1914.
[Crossref]

1913 (1)

Troland, Am. Jn. Physiology 32, [May] p. 8; 1913.

1912 (1)

For the influence of diffusion on the response of a liquid photoelectric cell, see Samsonow, Zeits, f. Wiss. Phot. XI, 1912, p. 33.

1908 (1)

Goldman, Ann. der Physik,  27, p. 449, 1908.
[Crossref]

1905 (1)

See Drysdale, Proc. Optical Conv., p. 173, 1905.

Drysdale,

See Drysdale, Proc. Optical Conv., p. 173, 1905.

Goldman,

Goldman, Ann. der Physik,  27, p. 449, 1908.
[Crossref]

Hecht,

Hecht, Science, April 15, p. 347, 1921.
[Crossref]

Ives,

Ives and Kingsbury, Phil. Mag.April1916, p. 290.
[Crossref]

Ives, Critical Frequency Relations in Scotopic Vision, J. Opt. Soc. Am.May, 1922, p. 254.
[Crossref]

Kennelly,

Kennelly and Whiting, National Electric Light Assn., 1907 convention.

Kingsbury,

Ives and Kingsbury, Phil. Mag.April1916, p. 290.
[Crossref]

Luckiesh,

Luckiesh, Physical Review 4, 1, p. 1; 1914.
[Crossref]

Porter,

Porter, Proc. Roy. Soc.113, p. 347, Proc. Roy. Soc.120, p. 313, Proc. Roy. Soc.136, p. 445.

Samsonow,

For the influence of diffusion on the response of a liquid photoelectric cell, see Samsonow, Zeits, f. Wiss. Phot. XI, 1912, p. 33.

Troland,

Troland, Am. Jn. Physiology 32, [May] p. 8; 1913.

Whiting,

Kennelly and Whiting, National Electric Light Assn., 1907 convention.

Am. Jn. Physiology (1)

Troland, Am. Jn. Physiology 32, [May] p. 8; 1913.

Ann. der Physik (1)

Goldman, Ann. der Physik,  27, p. 449, 1908.
[Crossref]

Phil. Mag. (2)

Ives and Kingsbury, Phil. Mag.April1916, p. 290.
[Crossref]

The mutually inclined ω-log I lines obtained by the writer previously for the case shown in Fig. 4 (Phil. Mag., April, 1917, p. 360) are apparently in error, due probably to the short range of intensities available for study in the apparatus then used.

Physical Review (1)

Luckiesh, Physical Review 4, 1, p. 1; 1914.
[Crossref]

Proc. Optical Conv. (1)

See Drysdale, Proc. Optical Conv., p. 173, 1905.

Science (1)

Hecht, Science, April 15, p. 347, 1921.
[Crossref]

Zeits, f. Wiss. Phot. XI (1)

For the influence of diffusion on the response of a liquid photoelectric cell, see Samsonow, Zeits, f. Wiss. Phot. XI, 1912, p. 33.

Other (7)

Ives, Critical Frequency Relations in Scotopic Vision, J. Opt. Soc. Am.May, 1922, p. 254.
[Crossref]

Porter, Proc. Roy. Soc.113, p. 347, Proc. Roy. Soc.120, p. 313, Proc. Roy. Soc.136, p. 445.

Kennelly and Whiting, National Electric Light Assn., 1907 convention.

The electrical behavior of these photoelectric cells under illumination is strikingly like that of the excised eye, as studied by Waller and other. Notable resemblances are shown in the preliminary negative response on commencement, and terminal positive twitch on cessation of illumination, and in the reversal of the reaction with age. Bose (“Response in the Living and non-Living,” p. 169) remarks “there is not a single phenomenon in the responses, normal or abnormal, exhibited by the retina, which has not its counterpart in the sensitive cell constructed of inorganic material.”

I am very greatly indebted to Mr. T. C. Fry for assistance in the mathematical work which follows, and for helpful discussions of the general problem.

It will be noted that this result follows, whatever the form of the re-composition function. The choice of beΘ for this function is in deference to the generally accepted idea of a logarithmic response to a steady stimulus. The farther the re-composition function departs from a simple direct proportionality to the reaction strength the higher must be the frequency to insure Talbot’s law holding. It is a fact of experiment that this law already holds at the critical frequency for flicker disappearance.

See Preston’s “Heat,”2d Ed., p. 654.

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Figures (7)

Fig. 1
Fig. 1

Successive steps from stimulus to final reaction

Fig. 2
Fig. 2

The form factor for various ratios of rise and fall of a saw-tooth stimulus.

Fig. 3
Fig. 3

Values of f ( ω , μ ) = ( μ + μ 2 + ω 2 ) ½, in terms of ω.

Fig. 4
Fig. 4

Critical speeds (ω) versus log illumination for several flicker ranges, sine-curve stimuli.

Fig. 5
Fig. 5

Critical speeds (ω) versus logarithm of amplitude (α); sine curve stimuli.

Fig. 6
Fig. 6

Critical speeds (ω) versus sector opening (Φ).

Fig. 7
Fig. 7

Electrical model illustrating theory.

Equations (39)

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for 1 , ω = a + b [ log I + log F ] where F is some constant characteristic of the wave form
for 2 , ω = a + b log I α where α is the amplitude ( a special case of ( 1 ) )
for 3 , ω = a + b log I + f [ Φ ( 1 - Φ ) ] where Φ is the fraction of the period during which exposure occurs
ω = c log 2 W δ
Δ S = Δ I I
c d Θ d t + b e Θ = f ( I , t )
b e Θ = f ( I , t ) , or Θ = log f ( I , t ) b
c d y d t + y f ( I , t ) = b
y = b e c - 1 c f ( I , t ) d t { e + 1 c f ( I , t ) d t d t + const. }
f ( I , t ) = I Φ + A sin ω t + B sin 2 ω t + C sin 3 ω t + etc
y = b c e - 1 c o t I Φ . d t { e + 1 c I Φ d t . d t + K }
= b I Φ + K e - I Φ t c
e Θ = I Φ b or Θ = log I Φ b
c d Φ d t + b e Θ = I Φ + 2 I π { sin π Φ cos ω t + ½ sin 2 π Φ cos 2 ω t + sin 3 π Φ cos 3 ω t + etc. }
c d Θ d t + b e Θ = 0
c Δ Θ Δ t = - I Φ
Δ Θ = - I Φ c Δ t
Noting that Δ t = ( 1 - Φ ) τ = 1 - Φ ω
Δ Θ = 1 c I Φ ( 1 - Φ ) ω
Θ t = K 2 Θ x 2
Θ t = K 2 Θ x 2 - μ Θ
Δ Θ 2 + Δ Θ 2 · 8 π 2 ( sin ω t - 1 9 sin 6 ω t + 1 25 sin 10 ω t + etc. )
Δ Θ 2 ± Δ Θ 2 · 2 π ( sin ω t + ½ sin 2 ω t + sin 3 ω t + . )
Θ = S + Δ Θ 2 + Δ Θ 2 · F ( sin ω t + a 1 sin 2 ω t + a 2 sin 3 ω t + . )
Θ = [ S + Δ Θ 2 ] e - x 2 μ + Δ Θ 2 · F [ e - x 2 K ( μ 2 + ω 2 + μ ) ½ { sin ω t - x 2 K [ μ 2 + ω 2 - μ ] ½ } + a 1 e - x 2 K ( μ 2 + 4 ω 2 + μ ) ½ { sin 2 ω t - x 2 K [ μ 2 + 4 ω 2 - μ ] ½ } + etc. ]
Θ = S + 1 2 c I Φ ( 1 - Φ ) ω F [ e - x K f ( ω , μ ) { sin ω t - x 2 K [ f ( ω , μ ) ] } ]
f ( ω , μ ) = ( μ 2 + ω 2 + μ ) ½
d Θ d t = 1 2 c I Φ ( 1 - Φ ) F e - x 2 K f ( ω , μ ) cos ( ω t - Ψ )
( d Θ d t ) m a x = I Φ ( 1 - Φ ) F e - x 2 K f ( ω , μ ) 2 c
ω = 2 K x 1 m log I Φ ( 1 - Φ ) F c
ω = 2 K x 1 m log I α F 2 c
ω = 12.4 log I + 29.4
ω = 12.4 log I α + 33.1
ω = 12.4 log I Φ ( 1 - Φ ) F + 38
sin π Φ π Φ × const.
ω = 12.4 log I sin π Φ π Φ + 35.6
ω = 13.3 log α + 18.6
ω = 13.3 log ( 1 - Φ ) F + 21
ω = 13.3 log sin π Φ π Φ + 17.2