a theory of intermittent vision

A THEORY OF INTERMITTENT VISIONBY

HERBERT E. IvEs

INTRODUCTION

Through the work of Porter,' Kennelly and Whiting,2 Luckiesh,3

Ives and Kingsbury,4 and others, there have been determinedrelationships between critical flicker frequency and the intensityand wave-form (intensity distribution with time) of the illumina-tion of the observing target. These relationships are compara-tively simple, the frequency occurring as a logarithmic functionof the other variables. They constitute probably the most com-plete and definite data we have on the time factor in visualresponse. This being the case, it would appear that from theserelationships it should be possible to form an idea of the kind ofmechanism and processes involved in vision. It is the purpose ofthis paper to describe a mechanism and processes, which, oncertain not improbable assumptions, would behave toward inter-mittent illumination substantially as does the eye.

REVIEW OF EXPERIMENTAL DATA

The theory to be developed is based on data already published,the only new data being certain sets of confirmatory observationswhich are used in the illustrations. The method employed toobtain the new data presents no points of striking novelty andneed not be described.

The principal high intensity critical frequency relations asobtained by the experimenters quoted may be summarized asfollows:

1. For all wave forms, critical frequencies plot against thelogarithm of illumination as parallel straight lines', 2, 3.

2. If the flicker range is varied, the range factor (amplitude)enters as a multiplier of the illumination.2

'Porter, Proc. Roy. Soc. 113, p. 347, 120, p. 313, 136, p. 445.2 Kennelly and Whiting, National Electric Light Assn., 1907 convention.3Luckiesh, Physical Review 4, 1, p. 1; 1914.Ives and Kingsbury, Phil. Mag. April 1916, p. 290.

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HERBERT E. IvEs [J.O.S.A. & R.S.I., VI

3. If the ratio of exposure to obscuration is varied, criticalspeeds are to a close approximation the same for complementaryopenings', .

These relationships may be summarized symbolically as below:where w is critical speed in cycles per second, I is illumination, or

more properly brightness;

for 1, =a+b[log I+1og F] where F is some constantcharacteristic of the waveform ........... (1)

for 2, co= a'+ b' log I a where a is the amplitude (aspecial case of (1) ) ........... (2)

for 3, w = a"+ b" log I+f[4' (1 - ')] where 4' is the fraction of theperiod during which exposureoccurs ............. (3)

At very low intensities, using short wave radiation (scotopicvision) critical speeds are independent of intensity and vary onlywith the waveform.5 The experimental results may be summar-ized in the empirical formula

2WX log - . ............... (4)

where W is the coefficient of the first periodic term of the Fourierexpansion representing the waveform, divided by the meanvalue of the stimulus; c and a are constants.

PREvious THEORETICAL TREATMENTS

The writer knows of but three previous attempts to correlatethese critical speed relations with specific theories of visual re-sponse. Their salient points may be briefly mentioned here, butthe original papers should be consulted for details.

Kennelly and Whiting2 state that their observations "conformsubstantially to the Weber-Fechner law of sensation and stimu-lus," that is that

AS=I

6 Ives, Critical Frequency Relations in Scotopic Vision, J. Opt. Soc. Am. May,

1922, p. 254.

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THEORY OF INTERMITTENT VISION

and are led from this to speak of the "sensation of visibly flickeringillumination" as following the same law with relation to stimulusas does the sensation due to steady illumination. While thispostulation of a "flicker sensation" introduces a conceptionexactly in accordance with the experimental facts (being but arestatement of them) it is questionable if it gives real aid in pictur-ing a visual mechanism.

L. T. Troland6 assumes a process of decomposition and recom-position which leads, in the case of equal light and dark intervals,to the experimentally obtained logarithmic relation. Uponapplying the same line of reasoning to the case of unequal exposureand obscuration it will be found that the critical speed should be,for any illumination, a linear function of the fraction obscuredinstead of the symmetrical function of opening required by experi-ment and symbolized in (3).

The third derivation to be noticed is that by the writer andMr. Kingsbury.4 In this the decrease of amplitude of a trans-mitted impression is ascribed to the process of conduction accord-ing to the Fourier diffusion law; change of illumination is supposedto cause a change in the diffusion constant. This theory indicatesthat the w-log I plots for different openings should be con-siderably inclined to each other, and that the small opening criticalspeeds should be higher than the complementary large openingspeeds. These predictions are qualitatively verified, but thedissymmetry is less than the theory calls for.

NEW THEORETICAL TREATMENT

In seeking for a theory of visual response which would lead tothe critical frequency relationships, the effort has been to makethe theory harmonize as closely as possible with the probablenature of the visual process, as indicated by lines of study otherthan that of critical frequency. Thus it is probable from severallines of evidence that the relation between stimulus and responseis approximately logarithmic; such a relationship should be in-cluded in a visual response theory. As another, and, indeed much

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

June, 1922] 345

HERBERT E. IES [J.O.S.A. & R.S.I., VI

more certainly established fact is the behavior summarized byTalbot's law, that the response (sensation) shall be the same forthe same mean illumination, no matter whether this is steady orintermittent, provided the speed of intermittence is so high thatflicker vanishes. The harmonizing of these two phenomena ofresponse presents considerable difficulty.7 More specific ideas ofvisual response lately current" indicate the initial reaction to bea photochemical one, followed by some relatively slow processof conduction.

The new theory here developed postulates three steps in theprocess of the perception of flicker. The first step consists of aphotochemical (photoelectric) reversible reaction of such a naturethat the equilibrium value under steady illumination is propor-tional to the logarithm of the stimulus. The second step consistsof a conduction process, according to the Fourier diffusion law,as developed to cover conduction accompanied by leakage, orre-composition of the diffused substance. The third step consistsin a perception process in which the criterion for perception is thatthe time rate of change of the transmitted reaction must exceed acertain constant critical value. These three steps will now betreated in detail.

The initial reaction is assumed to be of the kind occurring inphotoelectric cells with liquid electrolytes (Becquerel effect).. Inthese cells, as shown for instance by the work of Goldmann9 onmetal electrode cells containing dye solutions, the primary emis-sion of electrons is proportional to the intensity of the illumination.As the illumination continues there is an accumulation of chargedions which are continuously being neutralized, so that an equilibriumis reached under illumination which is, over a considerable range,proportional to the logarithm of the illumination.'0 On the removal

7 See Drysdale, Proc. Optical Conv., p. 173, 1905.8 Hecht, Science, April 15, p. 347, 1921.9 Goldman, Ann. der Physik, 27, p. 449, 1908.10 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 positivetwitch 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."

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of the illumination, the charge (potential) declines to its dark equili-brium. The potential variation of such a cell under intermittentillumination by a sector disc of open fraction 4' and period , isrepresented by a saw-tooth variation around the mean ( log I )rising during the time 4

'T and falling during the time (1-4)r. Thehigher the speed the smaller the amplitude of variation, and themore nearly the two slopes of the saw-tooth potential variationplot approximate to straight lines.

In order to study this kind of reaction quantitatively, we mayset down as representing the facts sufficiently closely, the follow-ing equation

c +be =f(I,t) ................ (5)dt

where c is the capacity of the "cell," e is the potential (concentra-tion of ions), b a constant, e the logarithmic base, f(I, t) themathematical expression of the time and intensity distribution ofthe illumination. The equation states that energy is being receivedby the system at a rate represented by f(I, ), is being stored(capacity factor), and is being lost or neutralized in such a way

dothat if a steady state obtains ( = o)dt

be0 =f(I, ), or ® = log f(I ) .............. (6)b

The general solution of (5) may be obtained" by multiplyingthrough by e-@; doing this, and substituting y for e-@ we get theequation

c -Y+yf(l,)=b................... (7)dt

of which geieral solution is

_ ff(I)dt f + f(It)ndt s -(Y=be e dt+const.

Before using this solution to get the relations between I, 4' andX for particular values of f(I, t), it is of importance to determine

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

June, 1922] 347

HERBERT E. IVES [J.O.S.A. & R.S.I., VI

whether the kind of reaction represented will take care of Talbot'slaw. To do this we note that any periodic stimulus such as thatresulting from the interposition of a sector disc rotating at uniformvelocity before a light source, may be represented by the generalequation

f(I, t)=IP+A sin c+B sin 2 wt+C sin 3 wt+etc ..... (9)

where ID is the mean value of the stimulus. Introducing this into

(8), and making co = x, we get, since fA sin t= -A cos wt

fMrIdt.b clo i t + I d-d g| .............. (10)y=be J etJ dt+K . (10)

C~~~~~~~_I~t

.K'e .................. (11)

so that after such time as the second term becomes negligible

e = or log T . ................. (12)

But from (6), this is the value of E for a steady illumination ofvalue ID. Hence Talbot's law is obeyed. For speeds less thaninfinite, but still large, the fluctuations will be to either side of

=log I@, and Talbot's law will be more and more widely de-b

viated from, with decreasing speed; as the difference between themean position of the fluctuating potential corresponding to IMbecomes greater and greater.

We may now proceed with the solution of (8). For the presentwe shall consider only the case of the "square topped" stimulus, ofvalue I between t = 0 and t = (r, and of value zero between t =r

and t=r, (i.e. for time (1-)'r). For this (5) becomesAcl 21c- +be= I + I {sin vlcoswt+

dt

Y/ sin2vlIcos2t+½ sin3irccos 3wt+etc. .... (13)

where X is the frequency in cycles per second=-. The exact

348


solution of (13) is obtained by inserting the right hand memberin (8) for f(I, t), and performing the integration for a series ofvalues of I, ib and . From plots of the values obtained it isthen possible to find the desired factors, such as amplitudes, andslopes of the reaction. However, with the knowledge that theamplitude of the reaction must be kept small, (for Talbot's lawto hold), we may obtain an approximate solution without resortingto graphical methods, as follows:

Consider a high speed of alternation with the potential, ,varying by a small amount to either side of the value given by

e = . Now consider what happens immediately after a darkb

sector has covered the light. The potential will fall according tothe relation.

c.T+bee9=° ...................... (14)

Now since for small fluctuations bee differs but little from 1lbwe may write

L.= ............. (15)At

orAs .-at........................ (16)

C

Noting that A= (1- )T= ............. (17)

we get for the amplitude of the drop in potential before the light isagain thrown on

A®y -. I@(1-I) ............... (18)c Ct)

We have then, as the result of the intermittent illumination ofour photoelectric cell, a fluctuating potential in which, for largevalues of frequency, the amplitude is proportional to the intensity,and inversely as the frequency. 2 The shape of the potential-time

21 It will be noted that this result follows, whatever the form of the re-compositionfunction. The choice of bee for this function is in deference to the generally acceptedidea of a logarithmic response to a steady stimulus. The farther the re-compositionfunction departs from a simple direct proportionality to the reaction strength the highermust be the frequency to insure Talbot's law holding. It is a fact of experiment thatthis law already holds at the critical frequency for flicker disappearance.

June, 1922] 349


curve is that of a saw-tooth, varying from a long rise and shortdrop (> 1- l)) to a short rise and long drop ( <1-)). (Fig.1, b.)

It is obvious that this process alone does not yield the typical(logarithmic) critical frequency relationships. We proceed nowto the second step, the conduction process. We assume as ourstimulus, the potential or concentration of decomposition productsof the photochemical reaction varying in the manner just de-scribed. How will a typical conducting medium transmit-thisstimulus?

- ~~~~~~~~~~~~~~~C,FIG. I

Successive steps from stimulus to final reaction(a) Square-topped stimulus(h) Reaction of photoelectric cell(c) Reaction at far side of conducting medium.

The general expression for conduction according to the Fourierlaw is (19)2aw~~~~~~~ -= K- . ......................... ........ .(19)

at aX2

where ® is the potential (concentration) at a point distant x fromthe stimulated surface, and K is the diffusivity. This expressionassumes no loss or re-composition of the conducted material. Ifwe assume a leakage or process of neutralization we may modify(1 9) t 3 80 a2E®

-=K--u .(20)At Ox2

13 See Preston's "Heat," 2d Ed., p. 654.

350


placing the loss, as the simplest possible assumption, proportionalto the potential.

This equation is to be solved by introducing as the boundarycondition at x = 0, the proper expression for the saw-tooth stimu-lus, of which (17) is the fluctuating part. This expression consistsin general, of a constant (S), representing the minimum value ofthe stimulus, to which is added the Fourier expansion of thefluctuating portion. For the symmetrical saw-tooth ( =1 -4)the Fourier expansion of the fluctuating portion of amplitudeAO, is

A®A®\08 1 1-+ AO. - (sin cot- sin 6 cot+- sin 10 cwt+etc.) ...... (21)2 2 r 9 25

§LF1J1

I I I11 | O .~~1 . .$S .4 .5 . 7 -5 .9 10

FIG. 2The form factor for various ratios of rise and fall of a saw-tooth stimulus.

For the saw-tooth stimuli at the extremes of the series, where onearm of the tooth is vertical, the expansion is

A® A®( 2+-.-(sin cot+2 sin 2 wt+' 3 sin 3 wt+ .... ) .... (22)

2 2 7r

In general, the expression for the complete stimulus is,A A®

OH)=5+ +-. F(sin cot+al sin 2 cwt+a2 sin 3 co+....) . (23)2 2

where F is a form factor, depending on the ratio of the two armsof the saw-tooth, that is upon cP. Values of this form factor areplotted in Fig. 2.

I I I IX 1 '

z /l1I Ta v

l __ __ L_

June, 1922] 351


The solution of (20) for the boundary condition (23) is

L+ _X AJe 2+_ X (V2+ w2+I) {X@=+2 e a/2 +- F e \12K -Vikc~-

[Av/,2+,2_. J}

+a~ _ i_-2 (A, 4 2) in 2cot- V2-[P,\2 +4C,2_ -Al }+etc.]

(24)

Now if the amplitude of this function is small, the part contrib-uted by the periodic terms after the first, involving higher mul-tiples of c in the exponential term, may be neglected. Discardingthese, and putting in the value of A® from (18) we have

2c W L >K V2KL ]}]

where (25)

f(Cj, ).= (V\/y2+C.c2+,L)/ ............. (26)This expression states that we have, by the process of con-

duction, transformed our sharp saw-tooth stimulus, in general ofunsymmetrical shape, into a reaction, (at depth x) of symmetricalsine-curve contour, of much smaller amplitude, the magnitude ofthe fluctuations dependent both on the amplitude and shape ofthe saw-tooth stimulus. The three steps from the original flat-topped stimulus, through the photoelectric reaction of unsym-metrical saw-tooth contour, to the finally transmitted symmet-rical sine-curve reaction are shown diagrammatically in Fig. 1,a, b, and c.

At this point we must consider the third and last step. Whatcriterion shall we adopt for the visibility of flicker? Severalplausible ones suggest themselves, for instance the attainment of adefinite range of fluctuation in the transmitted impression; theattainment of a definite fractional range; the attainment of adefinite time rate of change of reaction; or any one of these, madeto vary in some manner with the magnitude of the illumination

352

June, 1922] THEORY OF INTERMITTENT VISION 353

or the sensation. These three criteria, all perhaps of equal dpriori probability, have been tried in the present case, with theresult that the attainment of a definite rate of change appears tobe the only one which, without introducing further complexity,will yield the desired final relations. It is therefore adopted onthis strictly pragmatic basis, for which however some additionalsupport is given later. From (25) we get

d= 12 (I -c)Fe

._~ ~ ( co s )(7)'cos (-A) ....... (27)

the maximum value of which is

(do )ma X-)Fe -V2K

2c......... (28)

40

° I > Z 4 S T t o tO

terms of w.

If our theory is correct we should be able to solve this expression

for w by. placing () =a constant, and obtain the criticaldt max

frequency-intensity relations. In order to do this it becomes

necessary to investigate the character of f(w, ,y) or (,u+ V,2+ w2)"4.

In Fig. 3 are shown calculated values of this function for

4I, )

FIG. 3Values of f(w, IA) = (,u+V/.A+w')2, in


various values of g. It will be seen that for all values of A above10, provided is greater than 20, f(w, ) is proportional to w, sothat this function is practically indistinguishable from (ma +n).Substituting this in (28) and solving for we get finally

_ -\/2K 1 log 14(1-)1 F.(29)x in c

(where c' is a constant involving (d) , c and n) as our generaldi max WA

expression for the case of alternating uniformly light and com-pletely dark intervals.

Z.

FIG. 4Critical speeds () versus log illumination for several flicker ranges, sine-curve stimuli.a=fractional excursion from mean position.Straight lines drawn to fit equation w =10 log Ia+26.4.

By similar reasoning to that adopted in deriving (29) we findthat if the original stimulus is of equal light and dark intervals,and fluctuates between 2+Ia, and 2-Ia, the amplitude amultiplies into I, giving finally

V\2KlI a Fw= - log '(0

x in 2c' . ....... .... (30)8

where F has the value corresponding to 4 = I - b, namely,-.

These expressions state that critical speeds plot as straight linesagainst the logarithms of the illuminations; that the speeds for

354

June, 1922] THEORY OF INTERMITTENT VISION

different amplitudes vary from each other by a factor! log a, thatm

the speeds for different openings are represented by a logarithmicfunction of the ratio of exposure to obscuration. Comparing thesefindings with the summary of critical frequency relationshipssymbolized in (1), (2) and (3) it appears that the findings of thetheory are in general agreement with the facts. How close thisagreement is will be seen from Figs. 4, 5, and 6. In Fig. 4 the

'A

…~~~~~~~~~~~~~~~~~~~4

FIG. 5Critical speeds () versus logarithm of amplitude (a); sine curve stimuli.

circles represent experimental points for a sine curve wave form,the full lines values for various amplitudes calculated from thea = 2 line in accordance with (30), the numerical equation beingX = 10 log Ia + 26. 4, where I is the arbitrary units. (Note that F,while different for a sine-curve stimulus than for a sharp transi-tion one is alike for all amplitudes and so permits the derivationof various amplitude values from a given amplitude irrespectiveof the wave form of the stimulus). In Fig. 5 where the circlesare again new experimental data, the straight line plot of log aagainst w is exactly what is called for by (30). We have also for this

355

HERBERT E. IVES [J.O.S.A. .& R.S.I., VI

case the data of Kennelly and Whiting, who derive empiricallythe same equation.14

As for the speeds at different openings, in Fig. 6 the circles arethe writer's 1916 data, the full lines the plot of (29) using thenumerical equation co = 10. 7 log IFJ(1 +4) +30 derived from the1916 data; the crosses are new 1921 data. It is evident thatthe general character of the relationship is well represented by theformula. The writer's experimental data, as already noted, indi-cate higher speeds for the small openings than for the large, while(29) is symmetrical about 4) = Y2. On the other hand T. C. Porterfinds these curves symmetrical. His empirical expression w =a(b +c log I) (log 4)) (1- ) is clearly very like (29). It is to benoted that in the derivation of (29) and (30) no variability with

11II N, I-N I

FIG. 6Critical speeds (w) versus sector opening ().Circles, 1916 data. Full lines, calculated curves from equation

a=10.7 log ID(1+()F+30.1.Crosses, 1921 data.

intensity has been ascribed to any of the factors. In all probabilitythe diffusivity (K), the rates of recomposition (b and tz) and eventhe critical rate of change used for the criterion of flicker visibilityare functions of the intensity. It is to be remembered as well thatthe process of derivation of our expressions is approximate only.There would therefore appear to be ample opportunity to accountfor deviations from the exact relations indicated; the importantthing is to account for the main characteristics of the critical fre-quency relations and this the theory appears to do for the highintensity conditions.

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

356


Turning now to the low intensity phenomena, where criticalspeeds are independent of illumination, the obvious modificationdemanded if the same theory is to cover this region as well-a sim-pler one might be found adequate-is some assumption which willresult in the "I" term dropping out of (29). Perhaps the simplestassumption is that the process of dark adaptation, which isoperative in vision near the threshold, automatically increasesthe photoelectric sensitiveness (as by supplying more material, orexposing more surface), as the illumination is changed, so as tomaintain the mean value of the reaction constant. This adapta-tion process may be supposed to be altogether too slow to- followthe rapid fluctuations of the stimulus which constitute the periodic-ity to which flicker is due.

This assumption is introduced into the theory by multiplyingthe right hand side of (13) by the reciprocal of the mean intensity.It will be obvious, without going through the steps, that the var-ious expressions derived for the flicker-wave form relations therebybecome independent of the intensity. They reduce in fact to theempirical expression already quoted (4), with the exception thatthe expression for unequal exposure and obscuration becomes w=

-2K 1 log F, while the empirical expressions is co= c logx m' c

sin_ 7r -Upon plotting these two expressions however it is found7r

that they are quite indistinguishable in shape. Other expressions,included in the general empirical form, and applying to sine curveand other wave forms, are not handled by the present approximatetreatment, but it is highly probable that an exact solution of ourgeneral equation (as altered to cover the low intensity condition)would yield results agreeing equally well with the empirical expres-sion found to fit the experimental data. It may be pointed outthat the observation of a lower limit to flicker speed in the lowintensity investigation is exactly in accord with the criterion of adefinite rate of change of transmitted impression as the criticalcondition for perception of flicker. At low speeds the transmittedimpression rises and falls slowly because of the slowness of change

June, 1922] 357


of the stimulus; at high speeds it rises and falls slowly because ofthe smoothing out processes at work due to conduction. Thesetwo limits of flicker speed may be considered as support for theadoption of the rate of change criterion.

ELECTRICAL MODEL ILLUSTRATING THEORY

In the theory as stated there is nothing which absolutely bindsus to an electrical mechanism, probable although that may be.The initial reaction may be characterized simply as photo-chemical, the conduction process may be a diffusion of decomposi-tion products, the criterion of visibility of flicker may be rate ofchange of concentration of these products. The mathematicaltreatment is general and is the same for electrical as for chemicalprocesses and either may figure at some or all stages. It is, how-ever, of some interest, as contributing to concreteness, to illustratethe theory by an electrical model which is governed by the equa-tions used. In Fig. 7 let S be a photoelectric cell of the ordinaryvacuum type. At L let there be a leak, of high resistance which

AA~~-T

FIG. 7Electrical model illustrating theory.S photoelectric cellC condenserL variable resistance leakT " Cable" with distributed capacity and leakageN sensitive receiver.

decreases as the applied voltage increases, in such a manner thatthe attained potential is approximately as the logarithm of thestimulus (certain loose contacts approximate this property). AtC, close to the cell, let there be a capacity. The rest of the systemT consists of a "cable" consisting of a resistance path, withdistributed capacity and leakage,-the leakage in the cable beingcomposed of ordinary non-varying resistances. At the far end of

358


the line is to be placed a detecting instrument N which starts toindicate when the rate of change of potential across the two armsof the cable reaches a certain critical value. The amplitude of aninduced current would be a -criterion corresponding to thatpostulated.

The two varieties of resistance leaks assumed for ease ofdescription, can be reduced to one, the variable resistance kind,-since it would only be the resistances close to the cell whichwould be subjected to voltages high enough to utilize the depart-ture from Ohmic character. Also the capacity pictured near thecell may be merely the normal capacity of the cable near the cell.The whole system may, therefore, be physically somewhat simplerthan the coupling of photoelectric cell, capacity, leak, and specialtransmitting channel which must be considered as separate enti-ties for purposes of mathematical treatment. It is, in fact, quitepossible that all the recombination and diffusion propertiesrequired may be localized in the liquid photoelectric cell itself.'5

NUMERICAL FORMULAS

The values of the constants to be used with the formulas abovederived depend upon the illumination unit, the size of the observ-ing field, the particular observer. Porter's formula for equal darkand light intervals has been rather widely copied; it appearedtherefore worth while to calculate the constants for the newformulae to agree with his.

His formula for high intensities isco=12.4log +29.4 ................... (31)

where co is in cycles per second and I in meter candles. [His slope,(12.4) is higher than Kennelly and Whitings (11) and that fittingFig. 4, (10)]. Observing that according to our notation Ia mustbe substituted for I, a being 2, we get for equal light and darkexposures, of amplitude a

co=12.4 log I a+33.1 .................. (32)

For various openings (), for a = 2, we get similarlyX =12.4log I (1-4))F+38 ............. (33)

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

June, 1922] 359


where F is the fraction shown in Fig. 2. Making use of the observa-tion that b(1 -P)F is practically equivalent to

sin r const.

,r4we get a handier working formula

X w12.4log I Sin b +35.6 .............. (34)7r

For low intensities (blue light), using the writer's own observa-tions,5 for various amplitudes

w =13.3log a+18.6 ................... (35)for various openings

w=13.3 log (1-1D)F+21 ............... (36)sin 7zr 4,or noting that (1 -4)F is practically equivalent to si Xconst.

'7r4

we get this working formula:sin 7r 4,

co=13.3 log S +17.2 .............. (37)7rb,

With these formulas a complete family of low and high intensitycritical frequency curves, for abrupt transitions of illumination,may be plotted, which represent the experimental data closely incharacter and position.

DISCUSSION

The most serious objection to the theory of intermittent visionhere presented appears to the writer to be the fact that thelogarithmic relation between illumination and critical frequency isdue to the special type of conduction assumed for the products ofthe light action. It would appear priori much more likely thatthis is a more or less direct consequence of the logarithmic relationbetween stimulus and response. In that respect the lines ofthought in the attempted theoretical derivations of Kennelly andTroland are preferable.

It may also be felt, owing to the somewhat lengthy mathemati-cal development, that the processes assumed are unduly complex,and that some other simpler method of handling the variables atour command based on different assumptions might give the sameresults. It should be emphasized that the complexity is due to

360

June, 1922] THEORY OF INTERMITTENT VISION 361

the mathematical processes themselves, and that the assumedphysical processes-an initial photoelectric or photochemicalreaction, and a subsequent conduction-are simple and plausible.On the basis of these assumed physical processes the mathematicaltreatment cannot be much simpler, whatever direction it takes.Such ultimate explanation of the critical frequency phenomena asmay be developed will unquestionably involve processes of reac-tion to light and transmission of the results of the reaction. Thetheory here presented should therefore be at least a guide to a morecomplete (and probably even more complex) theory which can bebuilt up on a better knowledge than we now possess of photo-chemical reactions and of physiological conduction processes.

RESEARCH LABORATORIES OF THEAMERICAN TELEPHONE AND TELEGRAPH COMPANY AND THE

WESTERN ELECTRIC COMPANY, INCORPORATED.JULY, 1921.

a theory of intermittent vision

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