GENERAL PRINCIPLES OF COLORIMETRY

M. M. Gurevich, Leningrad

At the present time there exist many books and articles devoted to an exposition of the foundations of colorimetry. The methods of exposition adopted in most of them do not seem to the author entirely satisfactory. The present article is an attempt at a more systematic exposition of the question, intended mainly for the physicist and especially for the photometrist. The article considers a model of the color-sensitive apparatus of the eye, consisting, in accordance with Young–Helmholtz ideas, of two or three receptors with different curves of spectral sensitivity. The color-sensitive properties of the model are analyzed. The possibility is investigated of studying such an apparatus without measuring the reactions of the receptors. On the basis of this consideration all the laws of colorimetry are established. The rationality and origin of the international colorimetric system of the I.C.I. of 1931 (the \(X, Y, Z\) system) are demonstrated.

I. INTRODUCTION

1. The ability to distinguish the forms of surrounding objects—vision—is one of the basic functions of every organism that has reached a certain stage of its development. Different representatives of the animal world solve this problem in different ways, attaining in its solution one or another degree of perfection. One may indicate a rather large number of quantities characterizing the degree of perfection of the visual apparatus. One of the most essential quantities of this kind is visual acuity, or the resolving power of the eye, inversely proportional to the minimum angle under which the eye can distinguish two objects, for example two points. The resolving power of the eye is connected with the subdivision of the visual apparatus into a greater or smaller number of separate receiving elements. We know that in this respect the human eye, as well as the eyes of many animals and insects, have attained a high degree of perfection. However, an object need not necessarily have vanishingly small dimensions in order to be invisible. Protective coloration matching the color of the surroundings makes it possible to mask its presence. This possibility is widely used by nature and by man.

Thus, a subdivision of the eye alone into a series of separate color-sensitive elements is insufficient for complete and confident orientation in space. Some sensitivity to the distinction of contrasts is absolutely necessary, and the greater this sensitivity—

The more strongly developed this ability is, the more perfect the eye will be. Therefore the minimum contrast that can be perceived by the eye is another, also very essential, sign of its perfection. However, the concept of contrast proves to be considerably more complex than the concept of resolving power.

The phenomenon of contrast may be divided into two types: quantitative contrast and qualitative contrast. By quantitative contrast is meant such a contrast in which the spectral composition of the light emitted by the object coincides with the spectral composition of the light emitted by the background, and the difference consists only in the magnitudes of their brightnesses. In qualitative contrast the spectral composition (quality) of the light emitted by the object differs from the spectral composition (quality) of the light emitted by the background. In the first case the compositions of the radiations which illuminate the images of the object and of the background formed in the eye will be identical, while in the second they will be different.

It is quite obvious that without the ability to distinguish quantitative contrast no eye can function. It is known that in this respect the human eye, and that of other highly developed animals, has also attained a high degree of development.

But in the overwhelming majority of practical cases the spectral compositions of the radiations sent by the object and by the background are different. The question arises as to how the eye will react to a difference in the spectral composition of the radiations entering it. One can imagine two fundamentally different answers to this question. First, the eye may fail to perceive differences in the spectral composition of the radiations emitted by the object and by the background surrounding it (a color-blind eye). This does not mean, of course, that such an eye will not see the object at all. Generally speaking, the object will be seen, but in this case the eye will not be able to distinguish qualitative contrast from quantitative contrast. The vision of such an eye will correspond to ordinary photography, capable of giving only a greater or lesser darkening of the photosensitive layer for any spectral composition of the light. And just as ordinary photography often does not notice boundaries between white and blue surfaces, so a color-blind eye in a number of cases will not notice the contours even of very large objects. Secondly, the eye may notice differences in the spectral composition of the radiations entering it and give qualitative evaluations of them, for example, green, yellow, etc. (a color-sensitive eye). As a rule, such an eye will always see an object against the background surrounding it, if the spectral compositions of the radiations coming from them are different.

1. Completely color-blind eyes are found in human beings extremely rarely. Nevertheless, we shall briefly dwell on the properties of such an eye, because this consideration will serve us as a starting point for the subsequent analysis of the properties of the color-sensitive eye.

We may imagine that a color-blind eye possesses a normal curve of spectral sensitivity (visibility), na-

beginning at about 400 \(m\mu\), reaching a maximum at about 555 \(m\mu\), and disappearing at about 750 \(m\mu\). This curve, which determines the differences in the eye’s luminous evaluation of equal radiant powers in different parts of the spectrum, in no way determines the color properties of the eye. In particular, such an eye may be completely color-blind. We may imagine a model of such an eye in the following way. Let each light-sensitive element of the eye be a system similar to a photocell with a spectral sensitivity that would coincide with the sensitivity of the eye. Greater or lesser illumination of such elements will produce a corresponding increase or decrease in the photocurrents and will enable the eye to distinguish quantitative contrast. Since radiation of any spectral composition will produce in the photocell a current of one direction, of greater or lesser strength, the system, responding only to the magnitude of the photocurrent, will not be able to distinguish when the photocurrent changes as a result of a quantitative change and when as a result of a qualitative change in the radiation. Responding only to the magnitude of the photocurrent, an eye corresponding to the scheme considered will be capable only of quantitatively distinguishing radiations and will therefore be completely color-blind.

The totality of the reactions of such an eye to all conceivable radiations can be represented graphically in the form of a single line, straight or curved indifferently.

We shall conventionally take one point of this line as zero, and another, likewise arbitrarily chosen point, as the representation of some unit stimulus. A stimulus \(n\) times stronger or weaker than the unit one will be represented by a point \(n\) times farther from or nearer to the zero point.

Thus the variety of stimuli, and with it the reactions of a color-blind eye, will be a one-dimensional variety.

1. The normal human eye distinguishes in the surrounding environment a multitude of colors and shades, and this fact poses for technology and science the question of a quantitative expression for color that would make it possible to record color with sufficient accuracy and to reproduce it. It is obvious that such a problem—the fundamental problem of colorimetry—can be solved only on the basis of a quantitative study of the laws of the eye’s color perception.

Before proceeding to the exposition of the basic facts and relations in the field of color, it is necessary to make several general remarks about what we shall include under the concept “color,” and also to agree on the use of certain terms.

It is customary to speak of the color of light sources, of diffusely scattering surfaces (plant and animal coverings, paper, paints and varnishes, etc.), or of the color of transparent objects (crystals, glass, etc.). If the first is usually in good agreement from the physical point of view, then the second and third do not withstand critical examination. Indeed, the radiation emitted by a source has a certain definite spectrum characteristic of it.

ral composition, to which a definite color corresponds. As for light-scattering surfaces or transparent bodies, they emit no radiation of their own, but reflect or transmit the luminous flux incident upon them from outside. If this luminous flux approximates daylight in its composition, then we see objects in their usual coloration. If, however, the incident luminous flux proves to be monochromatic (for example, green with wavelength 546 mμ), or approximates it in its composition, then whatever the properties of the surface (or transparent layer) on which it falls, its color will always coincide with the color of the radiation, i.e., in our case it will always be green, of greater or lesser brightness.

Thus we naturally arrive at the conclusion that the only object to which color can in all cases be ascribed is radiation, or luminous flux. Whether it comes directly from a source, has passed on its way through a light filter, or has been reflected from a surface, it always has some spectral composition, to which a definite color also corresponds. Having agreed as to the color of a luminous flux, it is not difficult to extend the concept of color to all the other photometric quantities as well. In doing so, it will immediately appear that the color of luminance will differ from the color of illuminance in practically all cases, except the case of ideally neutral reflection, in which the reflection coefficient does not depend on wavelength.

Having agreed to speak of the color of any photometric quantity, we obtain the possibility—apart from the absolute measure, i.e., apart from the number of lumens, phots, stilbs, etc.—of characterizing each photometric quantity also by parameters that determine its chromaticity and depend on the spectral composition of the radiation. Often, however, it is convenient to abstract from a definite photometric quantity and speak of color in general. In such cases we shall distinguish two aspects in color: quantitative and qualitative. The quantity (absolute magnitude) of color in such cases we shall call the “lightness” of the color. Thus the concept of “lightness” will be a concept common to the quantity of lumens, phots, stilbs, etc., of the corresponding photometric quantity. The qualitative aspect of color, determined by the spectral composition of the radiation and not connected with its quantity (lightness), we shall call “chromaticity.” Thus the concept of “chromaticity” will likewise be common to all photometric quantities, but will express only the qualitative aspect of color.

II. TWO-DIMENSIONAL COLOR PERCEPTION

4. One of the most probable hypotheses concerning the essence of color perception is, as is known, the Young–Helmholtz–Maxwell hypothesis, which consists in the fact that in the apparatus of the normal hu-

human eye, the existence is admitted of three receivers or three processes excited differently by monochromatic radiations. It is assumed that, for example, red light predominantly excites one receiver, green another, and blue a third. The different ratio of the excitations of these three receivers or processes determines the psychological difference between the perceptions excited by different radiations, i.e. color. Expressed in a few words, Young’s hypothesis contains great possibilities for development. First of all, a generalization suggests itself to the case when the number of receivers or processes determining color is not equal to three, but to any integer greater than one.

Having an arbitrary number \(n\) of receivers, excited differently, i.e. possessing different curves of spectral sensitivity, we shall have to introduce a generalized concept of color as a complex reaction of the aggregate of all \(n\) receivers, an aggregate consisting of \(n\) simple reactions of each of the receivers separately.

Thus it may be asserted that \(n\) receivers define color as a point in a space of \(n\) dimensions. All our subsequent arguments, which for practical reasons will be limited to the cases of two-dimensional and three-dimensional color spaces, may without fundamental difficulties be extended to the case of any number of dimensions.

Another direction in which Young’s theory may be generalized lies in the consideration of various curves of spectral sensitivity that the chosen number of receivers may possess. The form of these curves determines, as we shall see, all the properties of the color perception of the system.

Without entering into any physiological assumptions about the essence of the processes determining the form of the sensitivity curve, we shall try to consider, from a purely mathematical point of view, those relations that must arise in the joint operation of several (and first of all two and three) receivers with different spectral properties.

The basic question that will occupy us here will be the following. What possibilities for distinguishing radiations of unequal composition does an aggregate of several receivers with noncoincident sensitivity curves possess?

As receivers we shall consider photoelements with a blocking layer as the most illustrative case of a receiver that directly converts radiation of any composition into an electric current of one direction.

To begin with, let us consider the simplest system, consisting of two photoelements.

1. Let the curves \(V_1\) and \(V_2\) (Fig. 1) represent the spectral characteristics of two photoelements 1 and 2, each of which is connected to a galvanometer, as shown in Fig. 2. In what follows we shall assume that both our photoelements are—

are in completely identical illumination conditions, i.e., the radiations incident upon them are exactly the same. Let us consider how a change in the radiation incident upon the photocells is related to a change in the readings of the galvanometers, i.e., to the reaction of our system.

Fig. 1

Fig. 2

If the change in the incident radiation is only quantitative, i.e., if only the illumination of the photocells changes, while the composition of the radiation remains constant, then the readings of both galvanometers will change in proportion to the change in illumination, and the ratio of the readings will likewise remain constant. But if, when the radiation incident upon the photocells changes, the readings of the galvanometers change by different factors, or, for example, one galvanometer increases its deflection while the other decreases it, then we may confidently assert that the composition of the incident radiation has changed. Thus our system will be able to register not only quantitative, but also qualitative changes in the radiation incident upon it, i.e., it will prove to be a color-sensitive system. Let us examine in more detail the “color sensitivity” of our system.

First of all, when we speak of the color sensitivity of a system of photocells, we introduce a certain new concept of color, which, in comparison with the usual concept, is deprived of its psychophysiological part and contains only the physico-mathematical part. In this new understanding, color will be called the set of reactions (photocurrents) of the system of receivers (photocells). Therefore, constant reactions of the system will correspond to a constant color, and conversely, a change in reactions is associated with a change in color.

In what follows we shall constantly use the concept of color in this mathematical sense.

The magnitudes \(\alpha\) and \(\beta\) of the photocurrents excited by the radiation in the first and second photocells may be represented as follows:

\[ \begin{aligned} \alpha &= \int E(\lambda)V_1(\lambda)d\lambda,\\ \beta &= \int E(\lambda)V_2(\lambda)d\lambda, \end{aligned} \tag{1} \]

where \(E(\lambda)\,d\lambda\) represents the monochromatic illumination of the photocells in the spectral interval \(\lambda, \lambda+d\lambda\), and \(V_1(\lambda)\) and \(V_2(\lambda)\) are the spectral characteristics of the photocells. Expressions (1) contain certain assumptions about the properties of the photocells we use, which we shall always consider to be satisfied. These assumptions consist in the fact that, first, the photocurrents excited by the action upon them of different parts of the spectrum are summed (additivity), and, second, that the photocurrents are proportional to the radiant power incident upon them.

As we have already noted, changes in the relation between the photocurrents \(\alpha\) and \(\beta\) indicate a change in the spectral composition of the \(E(\lambda)\)-radiation illuminating the photocells. However, not every change in spectral composition entails a change in the magnitudes of the currents \(\alpha\) and \(\beta\).

Indeed, the totality of pairs of values of the current magnitudes \(\alpha\) and \(\beta\) is a manifold of the second order, i.e. it is laid out on a plane. But the totality of different kinds of functions \(E(\lambda)\) is a manifold of infinitely high order. In fact, every finite spectral interval in which the function \(E(\lambda)\) is specified can be divided into an arbitrarily large number of narrower spectral intervals. The intensity of radiation in each such narrow interval may have an arbitrary positive value. For the characterization of the entire radiation, each such intensity has the value of an independent variable, the number of which can be made as large as desired. Therefore the total number of different possible functions \(E(\lambda)\) will be infinite of arbitrarily high order. It follows from this that one and the same pair of current values \(\alpha\) and \(\beta\) can be obtained by formulas (1) for an arbitrarily large number of functions \(E(\lambda)\), i.e. for an infinitely large number of radiation compositions1. All these compositions will be “perceived” by our system as completely identical, fully coinciding, and therefore indistinguishable from one another. Thus physically different radiations may be combined into groups according to the identity of their effect on our system, which proves unable to make any distinctions between them.

Let us agree to call these groups groups of “single-colored” radiations. Taking into account the additivity of the action of our receivers, we must conclude that for any combinations of radiations incident on the photocells, each radiation may be replaced by another radiation single-colored with it without any influence on the reactions of the system.

The belonging of physically different radiations to “single-color” radiations is determined, obviously, by the form of the functions \(V_1(\lambda)\) and \(V_2(\lambda)\). When these latter change, the groups of single-color radiations are rearranged. Those radiations which previously were single-color may cease to be so, and radiations which previously produced different effects on the system of photoelements may turn out to be single-color.

1. Let us consider one of the possible systems for graphically representing the “color perception” of a system of two photoelements. We shall plot along the abscissa axis the quantities \(\alpha\) of the currents excited by the radiation in one photoelement, and along the ordinate axis the quantities \(\beta\) of the currents of the other photoelement. To each radiation acting on the photoelements there will correspond a pair of values of the quantities \(\alpha\) and \(\beta\), which will determine some point \(C_1\) on the plane (Fig. 3). All single-color radiations will evoke the same currents in the photoelements and will therefore be represented by the very same point \(C_1\), which will thus represent a certain “color.” Every radiation producing a different action on our photoelements will be characterized by another point on the plane, i.e., will correspond to another “color.” If the change of the radiation incident on the photoelements consists in a change of illumination with an unchanged spectral composition, then the photocurrents \(\alpha\) and \(\beta\) will change proportionally by one and the same number of times, and the corresponding point on the plane will move along a straight line passing through the origin of coordinates, i.e., along a radius vector. An objective change in the illumination of the photoelements by \(K\) times will correspond to a change in each of the photocurrents \(\alpha\) and \(\beta\) also by \(K\) times, i.e., to a change in the distance from the point \(C\) to the origin of coordinates by \(K\) times.

Fig. 3

For example, let us plot on the graph of \(\alpha,\beta\) (Fig. 3) the points corresponding to monochromatic radiations of different wavelengths. We shall assume that the monochromatic radiations produce on the surfaces of the photoelements one and the same, and moreover unit, energy illumination. The corresponding photocurrents are plotted in Fig. 1 in the form of the curves \(V_1\) and \(V_2\), which will help us solve the problem posed.

If we plot the points \(\alpha,\beta\) in Fig. 3, then we obtain a series of points characterizing the colors of unit energy illuminations at different wavelengths. By joining these points with a smooth curve,

we shall obtain a graphical characteristic of the color sensitivity of our system. Against each point let us mark the wavelength of the corresponding radiation. Considering this curve, we see that it intersects every radius vector, including the one that passes through the point \(C_1\). Having determined the wavelength \(\lambda\) at which the line \(OC_1\) intersects the characteristic curve (in our case \(\lambda \simeq 520\,m\mu\)), we assert that it is possible to choose such an illumination of the photoelements by radiation of wavelength \(\lambda\) that this radiation will be of the same color as the radiation having color \(C_1\). In other words, we arrive at the conclusion that, for a two-dimensional system of color perception, in each group of same-color radiations there is at least one monochromatic radiation. The wavelength of this monochromatic radiation in this case determines the chromaticity, i.e. the qualitative aspect of the color.

As another example, let us plot on our color diagram (Fig. 3) the chromaticity \(\varepsilon\), corresponding to an equal-energy spectrum, i.e. to radiation characterized by the composition \(E(\lambda)=\mathrm{const}\).

Then expressions (1) will show us that

\[ \alpha_{\varepsilon}=\mathrm{const}\int V_1(\lambda)\,d\lambda \quad \text{and} \quad \beta_{\varepsilon}=\mathrm{const}\int V_2(\lambda)\,d\lambda . \]

An elementary calculation carried out for the curves \(V_1\) and \(V_2\) (Fig. 1) gives the value of the ratio \(\dfrac{\beta_{\varepsilon}}{\alpha_{\varepsilon}}\), which will determine the direction of the corresponding straight line. Plotting it on the diagram, we see that in our case the chromaticity of equal-energy radiation coincides with the chromaticity of monochromatic radiation with wavelength about \(558\,m\mu\).

1. Let us consider the laws of addition of colors, or the laws of mixing radiations, i.e. let us determine the action on our system of the sum of two radiations as a function of the action of each of them separately. Let one radiation with composition \(E_1(\lambda)\) be characterized by the photocurrent magnitudes \(\alpha_1\) and \(\beta_1\) (point \(C_1\) in Fig. 4), and the other—\(E_2(\lambda)\)—by the magnitudes \(\alpha_2\) and \(\beta_2\) (point \(C_2\)). Both radiations—\(E_1(\lambda)\) and \(E_2(\lambda)\)—acting simultaneously, constitute a third radiation

\[ E(\lambda)=E_1(\lambda)+E_2(\lambda), \]

which will be characterized by the photocurrent magnitudes \(\alpha\) and \(\beta\). As is readily seen from (1),

\[ \left. \begin{aligned} \alpha&=\alpha_1+\alpha_2\\ \beta&=\beta_1+\beta_2 . \end{aligned} \right\} \tag{2} \]

The point \(C\), representing the color of the mixture of two radiations, will be located according to the rule of the parallelogram constructed

Fig. 4

on the segments \(OC_1\) and \(OC_2\). The geometric summation of the segments \(OC_1\) and \(OC_2\), to which we resort in order to find the color of a mixture of two colors represented by the points \(C_1\) and \(C_2\), leads us to the conclusion that a vector representation of color is advisable. Instead of representing colors by points, we may represent colors by vectors having a common origin at the origin of coordinates \(O\) and an endpoint at the point serving as the representation of the color. In the vector representation one may write

\[ \mathbf{C}=\mathbf{C}_1+\mathbf{C}_2. \tag{3} \]

It should be noted that the possibility of a vector representation of colors is due to the additive properties of receivers. If the receivers did not possess additivity, then vector addition of colors would be impossible.

If we mix the colors \(C_1\) and \(C_2\) not in the initial quantities, but in arbitrary other quantities, for example \(xC_1\) and \(yC_2\), then, when \(x\) and \(y\) change, the point \(C\) will move in accordance with the change of the sum

\[ \begin{aligned} \alpha &= x\alpha_1 + y\alpha_2,\\ \beta &= x\beta_1 + y\beta_2. \end{aligned} \tag{4} \]

In the vector representation of color, for such a case we could write that

\[ \mathbf{C}=x\mathbf{C}_1+y\mathbf{C}_2, \tag{5} \]

where \(\mathbf{C}_1\), \(\mathbf{C}_2\), and \(\mathbf{C}\) would represent the vectors corresponding to the colors. In this case the quantities \(x\) and \(y\) acquire the character of coordinates of an arbitrary color.

In order that the quantities \(x\) and \(y\) may be coordinates of any color, we must also take into consideration their negative values.

Then expression (5) will correspond not only to the addition, but also to the subtraction of colors, i.e., to an operation that is physically impossible. However, by transferring the negative term (there can be only one) to the left-hand side of the equality, we again return to the feasible operation of adding colors.

In Fig. 5 there is presented a case in which the color \(\mathbf{C}\) cannot be obtained from \(\mathbf{C}_1\) and \(\mathbf{C}_2\) for any positive values of \(x\) and \(y\), but can be obtained as the sum \(\mathbf{C}=x\mathbf{C}_1+(-y\mathbf{C}_2)\). In other words, a mixture of the given color \(\mathbf{C}\) and the quantity \(y\) of the color \(\mathbf{C}_2\) will produce on the system of photoelements the same action as the quantity \(x\) of the color \(\mathbf{C}_1\).

\[ \mathbf{C}+y\mathbf{C}_2=x\mathbf{C}_1. \]

Generally speaking, every color equation of type (5) indicates that the colors or mixtures of colors standing in different parts of such an equation

…are of the same color and can completely replace one another with respect to their action on the color-perceiving system.

Returning to the system of equations (4), we may assert that, for any values of \(\alpha, \beta, \alpha_1, \beta_1\) and \(\alpha_2, \beta_2\) (i.e., for any colors \(C, C_1\), and \(C_2\)), this system has one definite solution with respect to \(x\) and \(y\), provided only that

\[ \left| \begin{matrix} \alpha_1 & \alpha_2\\ \beta_1 & \beta_2 \end{matrix} \right| \ne 0 \tag{6} \]

or, in other words, provided that

\[ \frac{\alpha_1}{\beta_1} \ne \frac{\alpha_2}{\beta_2}. \]

Fig. 5

The latter condition reduces to the requirement that the points \(C_1\) and \(C_2\) should not lie on one and the same radius-vector, i.e., that the vectors \(OC_1\) and \(OC_2\) should not coincide in direction, or, what is the same thing, that the colors \(C_1\) and \(C_2\) should not be qualitatively identical. Ultimately we arrive at the following assertion: by mixing two arbitrary, but qualitatively different colors \(C_1\) and \(C_2\), taken in appropriate quantities, we can obtain, and moreover in a unique way, a mixture of the same color as any third color \(C\). The quantity of one of the colors may be negative.

III. Differential Coupling of Color-Sensitive Systems

§8. In considering the properties of a system of two photoelements possessing different spectral sensitivities, we have seen that:

a) The system possesses the ability to distinguish radiations of different spectral composition and therefore may be regarded as a color-sensitive system. The system distinguishes the qualitative and quantitative properties of color, which we called “chromaticity” and “lightness.”

b) There exist groups of same-colored radiations which have different spectral composition but produce completely identical actions on our system.

c) By mixing two colors that differ in quality in various proportions, we can obtain a mixture of the same color as any third color.

All these properties of a system of two photoelements either coincide with, or are entirely analogous to, the generally known color properties of the eye. This result, quite natural from the point of view of the Young–Helmholtz theory, compels us to examine more attentively the comparison

properties of the eye with the properties of a system of photoelements. In doing so, however, we note an essential methodological difference, consisting in the fact that a system of photoelements allows us to measure the currents excited in them by light, whereas the eye provides nothing equivalent to such a measurement. Galvanometers make it possible to measure the curve of spectral sensitivity of each of the photoelements composing the system, whereas the eye is an indivisible instrument, and the very existence of three receivers determining its color properties is only a hypothesis. Naturally the question arises whether, with sufficient grounds, it is possible to compare the color sensitivity of the eye with the color-discriminating properties of a system of photoelements. The further exposition must serve as justification for an affirmative answer to this question.

In connection with this, we shall in what follows have to abandon the measurement of the magnitudes of photocurrents $\alpha$ and $\beta$ and learn to characterize colors by means of such procedures as we use in visual observations. The only method of estimation that allows the eye to have a sufficiently accurate judgment about radiation consists in its assessment of the equality or inequality of two adjacent photometric fields. Under favorable conditions such estimates can be made with an accuracy down to fractions of a percent. In connection with this we shall rebuild the system of our photoelements so that it gives us the same possibility, and we shall try to use this possibility for the quantitative and qualitative estimation of color.

1. Let us imagine that, instead of one pair of photoelements with which we have dealt up to now, we have two exactly identical pairs. All the properties of these two pairs of photoelements and, first of all, their spectral sensitivities must be exactly identical. But instead of four simple galvanometers we shall connect our four photoelements to two differential galvanometers, as shown in Fig. 6. In this arrangement identical photoelements will act on one galvanometer, and the currents arising under the influence of light will tend to deflect the pointer in opposite directions. Let only one mark be applied on the scales of both galvanometers, indicating the position of the pointer when there are no currents in the instrument, or in the case when the opposite actions of two equal currents cancel one another.

Fig. 6. Differential connection of two color-sensitive systems $a$ and $b$, consisting of two photoelements

Such a system of connection of photoelements, which may be called a differential system, will allow us to establish the equality or inequality of the action of two radiations, from

which the first illuminates one pair of photoelements, and the second—the other pair. We shall have equal action of both radiations when the needles of both differential galvanometers are set at the zero marks. This position of the needles will testify to equality of the “colors.” If, however, one or both needles give readings different from zero, this means that the “colors” are not identical. In this case we shall act in exactly the same way as we would act with the aid of the eye, forcing the two radiations being compared to pass through adjacent parts of the photometric field.

With the aid of the differential system we shall also be able to judge whether the difference in colors is only quantitative or also qualitative. If the galvanometers have deflected from the zero position in different directions, then it is obvious that the difference in colors is qualitative. By a deflection “in one direction” we shall take those deflections which are observed if a system in equilibrium is brought out of it by darkening one pair of photoelements with the aid of a neutral filter, i.e. without changing the spectral composition of the light. If we observe the deflection of both galvanometers in one direction, then the solution of the question whether the colorities of the two radiations falling on the two pairs of photoelements are identical or different is somewhat complicated. Here, too, however, for the solution it is sufficient to have at our disposal a neutral wedge. By covering with its aid one of the pairs of photoelements, we can bring the reading of one galvanometer to zero. If, in doing so, the other galvanometer also sets itself at zero, then this means that we have dealt only with a quantitative difference in colors (the colorities are identical). If, however, the other galvanometer deflects from zero, then the difference is qualitative (the colorities are different). It should be noted that, in the case of the eye, small differences in the colority of the comparison fields can likewise be detected only on the condition that the brightnesses of both parts of the field are equal. A considerable difference in brightnesses makes it impossible to notice them.

10. Let us use the differential scheme just described for connecting two identical pairs of photoelements for a more detailed investigation of the laws of color mixture. In this investigation we shall use only the possibility of establishing equilibrium on two differential galvanometers. The method for investigating the laws of mixture suggests itself. For this we shall use two circumstances. First, the possibility established by us (see § 7) of reproducing any third color by mixing two differently colored colors, and, second, the obvious circumstance that every radiation is a mixture of a large number of weaker monochromatic radiations. We shall therefore proceed in the following way. We shall choose two colors different in quality—let us call them the fundamental colors \(C_1\) and \(C_2\)—and shall mix them in various proportions on the surface of the first pair of photoelements. The second pair of photoelements we shall illuminate with mono-

Fig. 7

chromatic radiations, creating energetically equal units of illuminance at different wavelengths. The colors corresponding to them will be denoted (see § 7) by the letter \(C_\lambda\). The quantities of colors \(C_1\) and \(C_2\) which must be mixed on the first pair in order to obtain equality will be denoted by \(\xi(\lambda)\) and \(\eta(\lambda)\), and we shall write the equation

\[ C_\lambda=\xi(\lambda)C_1+\eta(\lambda)C_2 . \tag{7} \]

With a change in the wavelength \(\lambda\), the quantities \(\xi\) and \(\eta\) will change. These changes are conveniently represented graphically in the form of curves (Fig. 7), which are often called mixing curves because they indicate in what quantities the primary colors should be mixed in order to obtain colors indistinguishable from pure monochromatic ones.

The form of the mixing curves is not difficult to establish from Fig. 3. For this it is sufficient to plot on it the primary colors \(C_1\) and \(C_2\), to draw through them and the origin two new axes, with respect to which the quantities \(\xi\) and \(\eta\) are to be determined. Thus, for example, for \(\lambda=510\,m\mu\), as is seen from the drawing, \(\xi(510)=1.1\); \(\eta(510)=-0.1\). The validity of such a construction is confirmed by the established parallelogram rule (§ 7). Having found the functions \(\xi(\lambda)\) and \(\eta(\lambda)\), we can plot them in the form of a single curve (Fig. 8), similar to the curve of Fig. 3, on rectangular axes.

From expressions (1) it is evident that, for the monochromatic color \(C_\lambda\),

\[ \begin{aligned} \alpha_\lambda&=V_1(\lambda),\\ \beta_\lambda&=V_2(\lambda), \end{aligned} \left\} \tag{8} \]

where \(\alpha_\lambda\) and \(\beta_\lambda\) are the coordinates of the color \(C_\lambda\).

Substitution of (8) into the coordinate expression corresponding to relation (7) gives

\[ \begin{aligned} V_1(\lambda)&=\xi(\lambda)\alpha_1+\eta(\lambda)\alpha_2,\\ V_2(\lambda)&=\xi(\lambda)\beta_1+\eta(\lambda)\beta_2. \end{aligned} \left\} \tag{9} \]

These latter equalities establish a linear relation between the curves of spectral sensitivity \(V_1\) and \(V_2\) of the photoelements and the mixing curves \(\xi\) and \(\eta\).

Having the values \(\xi(\lambda)\) and \(\eta(\lambda)\), we can determine the quantities \(x\) and \(y\) of the colors \(C_1\) and \(C_2\) in which they must be mixed in order that the mixture be indistinguishable from any

Fig. 8. Characteristic curve of two-color color perception

of the color \(C\), having a known spectral composition \(\varepsilon(\lambda)\), i.e., we can find the coefficients \(x\) and \(y\) in the equation

\[ C = xC_1 + yC_2 . \tag{5} \]

As is not difficult to see,

\[ \begin{aligned} x &= \int E(\lambda)\,\xi(\lambda)\,d\lambda,\\ y &= \int E(\lambda)\,\eta(\lambda)\,d\lambda, \end{aligned} \tag{10} \]

where \(E(\lambda)\,d\lambda\) represents the monochromatic illumination produced by the color \(C\) in the wavelength interval \(\lambda,\ \lambda + d\lambda\).

We see that formulas (10) can replace for us expressions (1), with \(x\) and \(y\) taking the place of \(\alpha\) and \(\beta\), while \(\xi(\lambda)\) and \(\eta(\lambda)\) play the role of the sensitivities of the photoelements \(V_1(\lambda)\) and \(V_2(\lambda)\). Nothing prevents us from considering the quantities \(x\) and \(y\) as coordinates of the color \(C\) with respect to the colors \(C_1\) and \(C_2\) chosen as primaries (see Fig. 8).

1. However, while the quantities \(\alpha\) and \(\beta\), representing currents from two photoelements, were, as it were, natural coordinates of color, the quantities \(x\) and \(y\), introduced by equations (5) and (10), contain a considerable element of arbitrariness, consisting in the uncertainty of the choice of the primary colors \(C_1\) and \(C_2\). In connection with this uncertainty, it is natural to think that an arbitrary choice may not turn out to be the best. In order subsequently to make the most rational choice, it is first of all necessary to examine the changes associated with transition to other primary colors. Therefore let us consider how the mixture functions \(\xi(\lambda)\) and \(\eta(\lambda)\), as well as the color coordinates \(x\) and \(y\), change if, instead of the colors \(C_1\) and \(C_2\), we choose as primaries two other, also qualitatively different colors \(C'\) and \(C''\).

According to § 7, we can always write

\[ \begin{aligned} C_1 &= a_1 C' + b_1 C'',\\ C_2 &= a_2 C' + b_2 C'', \end{aligned} \tag{11} \]

where \(a_1,\ b_1,\ a_2\) and \(b_2\) are the quantities in which the colors \(C'\) and \(C''\) must be mixed so that the mixture is equivalent to the colors \(C_1\) and \(C_2\).

Substituting (11) into (7), for which, according to the same § 7, we have full justification, we obtain

\[ \begin{aligned} C_\lambda &= \xi(\lambda)C_1 + \eta(\lambda)C_2 = \xi(\lambda)(a_1C' + b_1C'') +{}\\ &\quad + \eta(\lambda)(a_2C' + b_2C'') = [a_1\xi(\lambda) + a_2\eta(\lambda)]C' +{}\\ &\quad + [b_1\xi(\lambda) + b_2\eta(\lambda)]C'' . \end{aligned} \]

Denoting by \(\xi'(\lambda)\) and \(\eta'(\lambda)\) the quantities of the colors \(C'\) and \(C''\) in which they must be mixed in order to obtain the pure mono-

of chromatic color \(C_\lambda\) of constant radiant power, we can write, proceeding from the last equality,

\[ \left. \begin{aligned} \xi'(\lambda)&=a_1\xi(\lambda)+a_2\eta(\lambda),\\ \eta'(\lambda)&=b_1\xi(\lambda)+b_2\eta(\lambda), \end{aligned} \right\} \tag{12} \]

Since the coefficients \(a_1, b_1, a_2\), and \(b_2\) do not depend on wavelength, relations (12) indicate that the new mixing curves are linear functions of the old curves.

An arbitrary color \(C\) may be obtained by mixing \(C'\) and \(C''\) in quantities which we shall denote by \(x'\) and \(y'\). Substituting (11) into (5), we easily obtain the expressions

\[ \left. \begin{aligned} x'&=a_1x+a_2y,\\ y'&=b_1x+b_2y, \end{aligned} \right\} \tag{13} \]

which, just like equations (12), represent transformations for passing from one system of rectilinear coordinates in the plane to another having the same origin (the constant term is absent).

Comparing expressions (1) with (9) and (10), we may write that

\[ \left. \begin{aligned} \alpha&=\int E(\lambda)V_1(\lambda)\,d\lambda=\alpha_1x+\alpha_2y\\ \text{and}\qquad \beta&=\int E(\lambda)V_2(\lambda)\,d\lambda=\beta_1x+\beta_2y, \end{aligned} \right\} \tag{14} \]

where \(\alpha\) and \(\beta\), \(\alpha_1\) and \(\beta_1\), \(\alpha_2\) and \(\beta_2\), as before, are the photocurrents arising in photoelements 1 and 2 under the influence of an arbitrary color \(C\) and the primary colors \(C_1\) and \(C_2\).

We see that expression (14) is completely identical in form with (13), and that, consequently, the quantities \(\alpha\) and \(\beta\), being linear functions of \(x\) and \(y\), may be regarded as a particular case of color coordinates determined by the corresponding choice of coordinate system, i.e. of the primary colors.

IV. QUANTITY OF COLOR

12. From the two preceding chapters we saw that color is determined by the magnitudes \(\alpha\) and \(\beta\) of the currents arising under its influence in the photoelements, or by the quantities \(x\) and \(y\) of the primary colors \(C_1\) and \(C_2\) that must be mixed on one pair of photoelements in order to bring the differential circuit into equilibrium. In both cases we had the possibility of distinguishing colors from one another and of establishing whether the difference is qualitative or only quantitative. In each color we distinguished these two aspects. The quality of a color was determined by the ratio of the coordinates \(\dfrac{\alpha}{\beta}\) or \(\dfrac{x}{y}\), which made it possible to compare the qualities of different colors with one another. With the quantitative aspect the matter is more complicated.

If two colors coincide in quality (chromaticity), then their quantitative comparison presents no difficulty. It is obtained as the ratio of the corresponding coordinates of these two colors, or as the ratio of the lengths of the radius-vectors corresponding to them. These ratios do not depend on the choice of the primary colors.

The difficulty lies in the quantitative comparison of colors that do not coincide in quality. Graphically, such colors are represented by points lying on different radius-vectors. The ratios of corresponding coordinates will not give identical values and, moreover, will depend on the choice of the directions of the coordinate axes. The ratio of the lengths of the radius-vectors is plainly unsuitable, since it depends on the scale in which the primary colors are laid off on the axes (obviously, we can be interested only in such a quantitative relation as would not depend on accidental circumstances of this kind). The relations established by us up to now do not provide a criterion for judging the quantitative relation between colors that differ in quality. In order to make possible such comparisons, independent of the choice of coordinates, we shall have to introduce some additional condition. This condition must enable us quantitatively to compare also our primary colors \(C_1\) and \(C_2\), which until now, for lack of other possibilities, we have taken as units, i.e. as quantities equivalent in certain ratios.

In order to satisfy the requirement that the quantitative characteristic be independent of the choice of coordinates, let us take as the measure of the quantity of a color a function of the photocurrents \(\alpha\) and \(\beta\) arising in the photoelements under the action of the color in question. Since the photocurrents \(\alpha\) and \(\beta\) are proportional to the illumination on the surfaces of the photoelements, we shall henceforth denote the quantity of color by \(E(\alpha,\beta)\).

At first sight it may seem that almost any function of the photocurrents can be taken as the measure of the quantity of a color. However, such assumptions would be in contradiction with our previous suppositions.

Indeed, first of all we must consider that the quantity of color changes in direct proportion to the quantitative change of radiation that is constant in composition. But in this case the quantities \(\alpha\) and \(\beta\) change proportionally to the radiation. Therefore we must suppose that the quantity of color changes in direct proportion to the simultaneous change of the photocurrents and must go to zero together with them.

Thus \(E(\alpha,\beta)\) must be a homogeneous function, and specifically of the first degree. Moreover, the photometrist and the lighting engineer cannot conceive that the quantities of mixed colors would not add algebraically, i.e. that the color equality

\[ C_1 + C_2 = C \]

should not correspond to the equality

$$ E(\alpha_1,\beta_1)+E(\alpha_2,\beta_2)=E(\alpha,\beta), $$

where \(E(\alpha_1,\beta_1)\) and \(E(\alpha_2,\beta_2)\) are the quantities of the mixed colors \(C_1\) and \(C_2\), and \(E(\alpha,\beta)\) is the quantity of the resulting color \(C\). But since, according to (2), \(\alpha=\alpha_1+\alpha_2\) and \(\beta=\beta_1+\beta_2\), we have

$$ E(\alpha_1,\beta_1)+E(\alpha_2,\beta_2)=E(\alpha_1+\alpha_2,\beta_1+\beta_2). $$

The latter expression can be satisfied only under the condition that

$$ E(\alpha,\beta)=p\alpha+q\beta, \tag{15} $$

where \(p\) and \(q\) are certain constant coefficients.

In order not to return to the necessity of measuring the photocurrents \(\alpha\) and \(\beta\), an operation having no visual counterpart, let us again turn to the differential scheme shown in Fig. 6. Here various cases are possible.

a) We may choose, as the measure of the quantity of color, the current \(\alpha\) from the first photocell, i.e. put \(q=0\) in (15). The differential scheme will give us the simple possibility of establishing quantitative relations between any colors illuminating the 1st and 2nd pairs of photocells. The colors will be quantitatively identical if the galvanometer \(g_1\) is in the zero position. Its deflection to the right or to the left will indicate which color is greater in quantity. In this case the readings of the other galvanometer \(g_2\) will have no significance for comparison of the quantity of the colors. If, when the galvanometer \(g_1\) is in the zero position, the galvanometer \(g_2\) also proves to be in the zero position, this will testify that the colors are identical not only quantitatively but also qualitatively, i.e. that we are dealing with monochromatic radiations.

b) If we take the current \(\beta\) from the second photocell as the measure of the quantity of color, i.e. put \(p=0\) in expression (15), then the galvanometer \(g_2\) will serve as the indicator of equality or inequality of the quantity of two colors. In all other respects this case will not differ from the first.

c) If, as the measure of the quantity of color, we take an arbitrary linear function of the photocurrents \(p\alpha+q\beta\), then the scheme shown in Fig. 6 will no longer be able to satisfy us. However, it is not difficult to imagine such a modification of this scheme as will allow the quantitative equality of colors to be established by this criterion as well. For the case, for example, when \(p=q=1\), the corresponding scheme is shown in Fig. 9, where both currents pass through the galvanometer \(g_{1+2}\)

Fig. 9

$\alpha$ and $\beta$, and which therefore makes it possible to compare quantities of colors, if the sum $\alpha+\beta$ is taken as the measure of this quantity.

In the general case the quantity of color is evaluated by expression (15) for arbitrary values of the coefficients $p$ and $q$. Expression (15) makes it possible to give a quantitative estimate to the primary colors $\mathbf{C}_1(\alpha_1,\beta_1)$ and $\mathbf{C}_2(\alpha_2,\beta_2)$.

From (15) it follows that

\[ \begin{aligned} E_1&=p\alpha_1+q\beta_1=L,\\ E_2&=p\alpha_2+q\beta_2=M, \end{aligned} \tag{16} \]

where $L$ and $M$ are the quantitative coefficients of the primary colors $\mathbf{C}_1$ and $\mathbf{C}_2$.

The quantity $E$ of an arbitrary color may be written as

\[ E=xL+yM, \tag{17} \]

where $x$ and $y$ are the color coordinates from equation (5).

If monochromatic fluxes with different wavelengths $\lambda$ fall on our system, producing on the photoelements energetic illuminances equal to unity, then the quantity $E_\lambda$ of the color $\mathbf{C}_\lambda(\alpha_\lambda,\beta_\lambda)$ will be determined by the following relations:

\[ E_\lambda=p\alpha_\lambda+q\beta_\lambda=pV_1(\lambda)+qV_2(\lambda)=V(\lambda), \tag{18} \]

where $V_1(\lambda)$ and $V_2(\lambda)$ are the spectral sensitivities of each of the two photoelements, and $V(\lambda)$ represents a quantitative estimate of monochromatic radiation whose power is equal to unity; i.e., it is the quantitative spectral sensitivity of the system of photoelements. If, instead of monochromatic light, we illuminate the photoelements with a color $\mathbf{C}(\alpha,\beta)$ of arbitrary composition $E(\lambda)$, so that $E(\lambda)d\lambda$ will represent the monochromatic illuminance produced by it in the interval of wavelengths $\lambda$, $\lambda+d\lambda$, then, in view of the additive properties of the receivers (photoelements), the quantitative estimate $E$ of the total radiation may be determined in the following manner [see equations (1)]:

\[ \begin{aligned} E&=p\alpha+q\beta =p\int E(\lambda)V_1(\lambda)\,d\lambda +q\int E(\lambda)V_2(\lambda)\,d\lambda \\ &=\int E(\lambda)\,[pV_1(\lambda)+qV_2(\lambda)]\,d\lambda =\int E(\lambda)V(\lambda)\,d\lambda . \end{aligned} \tag{19} \]

The last expression coincides with the generally accepted photometric estimate of the quantity of radiation, if it is assumed that $V(\lambda)$ coincides with the sensitivity of the average human eye.

Substituting (9) into (18) and assuming that the photocurrents corresponding to the primary colors $\mathbf{C}_1$ and $\mathbf{C}_2$ are denoted by $\alpha_1,\beta_1$ and $\alpha_2,\beta_2$, we find that

\[ V(\lambda)=(p\alpha_1+q\beta_1)\xi(\lambda)+(p\alpha_2+q\beta_2)\eta(\lambda) =L\xi(\lambda)+M\eta(\lambda), \tag{20} \]

where $L$ and $M$ are the quantitative coefficients of the primary colors $\mathbf{C}_1$ and $\mathbf{C}_2$.

On passing to another pair of primary colors $\mathbf{C}'$ and $\mathbf{C}''$, the color-matching functions change and, according to (12), pass into $\xi'(\lambda)$ and $\eta'(\lambda)$. The quantitative coefficients of the primary colors $L$ and $M$ also change and pass into $L'$ and $M'$, which may be determined from equation (11):

\[ L=a_1L'+b_1M' \quad \text{and} \quad M=a_2L'+b_2M'. \tag{21} \]

Substituting (21) into (17) and using (13), we easily obtain

\[ E = Lx + My = L'x' + M'y'. \tag{22} \]

We see that the quantity \(E\), as was to be expected, proves to be invariant with respect to a change of the primary colors.

Entirely similarly to formula (22), for monochromatic radiation we obtain

\[ V(\lambda)=L\xi(\lambda)+M\eta(\lambda)=L'\xi'(\lambda)+M'\eta'(\lambda). \tag{23} \]

Thus, starting from the linear estimate of the quantity of color

\[ E=p\alpha+q\beta \]

—the only one that can be reconciled with the assumption of additivity of color quantities—we arrive at the following results:

a) The spectral sensitivity of a system of photocells \(V(\lambda)\) is a linear function of the sensitivities of the photocells composing it, with the same coefficients that enter into the linear function (15).

b) The spectral sensitivity of a system of photocells \(V(\lambda)\) is at the same time a linear function of the mixing functions, and the coefficients multiplying the mixing functions prove to be the quantitative coefficients \(L\) and \(M\) of the corresponding primary colors \(C_1\) and \(C_2\).

c) Relation b) is preserved for any new primary colors \(C'\) and \(C''\).

d) The quantity of any color \(C\) can be represented in the form of a linear function of its color coordinates \(x\) and \(y\), the coefficients of these coordinates being the quantitative coefficients \(L\) and \(M\) of the primary colors \(C_1\) and \(C_2\).

V. Rational Choice of Primary Colors

1. On the coordinate plane \(\alpha,\beta\) or \(x,y\), the expressions \(p\alpha+q\beta=\mathrm{const}\) and \(Lx+My=\mathrm{const}\) determine a family of parallel straight lines. These straight lines are the geometric loci of points representing colors of equal quantity, the measure of quantity being established by the value of the constant.

Fig. 10

The expressions (4) are formulas for transforming the coordinates \(\alpha,\beta\) into the coordinates \(x,y\). If they are substituted into (15), we obtain

\[ E=p\alpha+q\beta=(p\alpha_1+q\beta_1)x+(p\alpha_2+q\beta_2)y=Lx+My, \tag{24} \]

i.e. the equation of the straight line \(p\alpha+\)

\(+ q\beta = \mathrm{const}\) becomes the equation of a straight line \(Lx + My = \mathrm{const}\). In other words, both formulas belong to the same family with respect to different coordinate axes.

In Fig. 10 the axes \(\alpha, \beta\) and \(x, y\) are presented (passing through the primary colors \(C_1\) and \(C_2\)) and a family of equicolorous straight lines, denoted by the numerals \(0, 1, 2, 3\), etc. Thus points lying on the straight line denoted by the numeral \(5\), for example the point \(C'\) or \(C_z\), represent colors exceeding in quantity by 5 times the colors represented by the points of the straight line \(1\). If, moreover, the colors being compared are represented by points lying on one radius-vector, then the quality of these colors is the same; if on different ones, then their quality is different. That straight line from the family we have considered which passes through the origin of coordinates corresponds to colors with zero quantity and may be represented by the expression \(Lx + My = 0\). The only real color of this kind—the black color, represented by the origin of coordinates—corresponds to the complete absence of any light. All other points of this straight line have no corresponding real colors. Later we shall dwell in greater detail on these so-called imaginary colors.

Returning to the straight line of the equicolorous family corresponding to zero quantity, we note that it divides the color plane into two parts. In one part there will be colors with positive quantities, in the other—with negative ones. The latter will belong entirely to the class of imaginary colors.

1. Every real radiation consists of a greater or lesser number of more or less weak monochromatic component parts. Monochromatic colors are represented, in the case of constant radiation power, by a curved line similar to that which is shown in Fig. 10. In the case of two-dimensional color perception, the set of radius-vectors directed to all points of this curve exhausts the set of qualities of color of any composition. Indeed, the mixture of any two pure (monochromatic) colors gives, according to the parallelogram rule, a color whose quality is represented by a direction lying between the directions of the original vectors. Therefore, with any number of repetitions of such an operation, we shall have a resultant coinciding in quality with some pure monochromatic color.

Let us draw from the origin of coordinates two straight lines tangent to the curve of spectral colors. In Fig. 10 these will be the coordinate axes, i.e. the straight lines \(\beta = 0\) and \(\alpha = 0\), or, with respect to the axes \(x\) and \(y\),

\[ x\alpha_1 + y\alpha_2 = 0 \quad\text{and}\quad x\beta_1 + y\beta_2 = 0. \]

Every real color will be represented by a vector lying inside the angle between these straight lines. Any direction lying outside this angle will have no real correspondence. But for geometrical and mathematical constructions these directions may be very convenient. We can agree to extend

and color correspondences can also be established on them, but the colors corresponding to them will then have to be regarded as imaginary.

1. We have seen that all quantitative relations are invariant with respect to a transformation of coordinates, i.e., with respect to the choice of primary colors. This makes all pairs of primary colors, in principle, completely equivalent, and the choice of one or another pair is dictated by considerations of a purely practical, non-principled character. Among such practical circumstances is the greater or lesser difficulty of computing the quantities with which one has to operate in practice (practice deals with three-dimensional color perception, but the character of all the relations is preserved also in our case, which has the advantage of greater simplicity). It is quite evident that the computation of the color coordinates, i.e., of the integrals (10), will be substantially facilitated if the integrand does not change sign, i.e., if all elements of the integral are positive. This simplification will occur when the functions \(\xi'(\lambda)\) and \(\eta'(\lambda)\), taken with respect to the new primary colors \(C'\) and \(C''\), have only positive values. From Fig. 10 it is not difficult to conclude that such a situation will occur only when the coordinate axes \(x'\) and \(y'\) pass outside the region covered by the monochromatic colors. In other words, in order to attain the indicated convenience it is necessary that the colors \(C'\) and \(C''\) be imaginary. However, this condition still leaves a very large freedom in the choice of primary colors. Let us see whether it is possible to use this freedom to obtain additional simplifications in the quantities with which we shall have to operate.

Turning to formula (22), which determines the amount of color, we note that the quantities \(L\) and \(M\) are connected with the choice of the primary colors. Let us choose the new primary color \(C'\) so that, while remaining imaginary and therefore not violating the first condition, it gives us \(L'=0\). This is not difficult to do if \(C'\) is placed on the straight line \(p\alpha+q\beta=0\). Then expression (22) is transformed into the simpler

\[ E=M'y'=M'\int E(\lambda)\eta'(\lambda)\,d\lambda . \tag{25} \]

Comparing the last relation with expression (19), we see that

\[ V(\lambda)=M'\eta'(\lambda). \]

If we also take care that the other primary color \(C''\), while likewise remaining imaginary, lies on the straight line \(p\alpha+q\beta=1\), then the quantity \(M'\), obviously, will prove equal to unity, and we finally obtain

\[ V(\lambda)=\eta'(\lambda). \tag{26} \]

Thus, by an appropriate choice of primary colors, one can make one of the mixture functions coincide with the general spectral sensitivity of the photoelement system. Since \(L'=0\) and \(M'=1\), expression (22) gives us

\[ E=y', \tag{27} \]

i.e., the quantity of any color then coincides with one of its coordinates. The remaining undetermined length of the radius vector \(OC'\) we can determine from the additional requirement

\[ \int \xi'(\lambda)\,d\lambda=\int \eta'(\lambda)\,d\lambda, \tag{28} \]

which ensures a convenient position in the plane \(x',y'\) of the radius vector corresponding to the equal-energy (white) color \(E(\lambda)=\mathrm{const}\), which we shall still denote by the letter \(E\).

The values of the constants \(p\) and \(q\), which determine through (15) the quantity \(E\) of the color, will define the direction of the family of equal-quantity straight lines. A change in \(p\) and \(q\) will entail a change in the primary colors which we must choose as primaries if we wish to use the convenience of expressions (26) and (27).

If, in the particular case, \(p=0,\ q=1\), then the family of equal-quantity straight lines will be parallel to the \(\alpha\) axis, and the spectral sensitivity of the system will coincide with the sensitivity of the second photoelement.

VI. THREE-DIMENSIONAL COLOR PERCEPTION

1. A system of three photoelements having different spectral sensitivities also possesses possibilities for distinguishing spectral compositions, and here, obviously, possibilities greater than those of a system of two photoelements. However, the former method of analyzing these possibilities can be applied, which will considerably facilitate its execution.

Following the previous method of consideration, let us suppose that the curves of spectral sensitivity of the photoelements are known to us, and let these be the functions \(V_1(\lambda)\), \(V_2(\lambda)\), and \(V_3(\lambda)\).

It is quite natural that, having the possibility of measuring three photocurrents arising under the action of the incident radiation (Fig. 11), we can characterize each radiation by three quantities

\[ \left. \begin{aligned} \alpha&=\int E(\lambda)V_1(\lambda)\,d\lambda,\\ \beta&=\int E(\lambda)V_2(\lambda)\,d\lambda,\\ \text{and}\qquad \gamma&=\int E(\lambda)V_3(\lambda)\,d\lambda, \end{aligned} \right\} \tag{29} \]

where \(E(\lambda)\) characterizes the spectral composition of the radiation, and \(E(\lambda)d\lambda\) is the energetic illumination of the photoelements in the interval \(\lambda,\lambda+d\lambda\).

Fig. 11. Color-sensitive system of three photoelements

Having at our disposal the three quantities \(\alpha\), \(\beta\), and \(\gamma\), characterizing each radiation, we can regard them as coordinates of a point in any system of oblique-angled spatial coordinates. We shall call the space thus obtained the “color” spa-

space. To each radiation there corresponds a definite point of this space, i.e., a definite “color.”

The converse proposition is false. To each point of the color space, i.e., to each color, there does not correspond any particular radiation. As a rule, it may be stated that to each color there corresponds an infinite set of radiations forming a group of “same-colored” radiations, indistinguishable from one another by means of the system of photoelements under consideration. Groups of same-colored radiations are determined by the form of the functions \(V_1(\lambda)\), \(V_2(\lambda)\), and \(V_3(\lambda)\).

A change in radiation consisting in a proportional decrease or increase of the power of the radiations over the entire spectrum will, according to (29), produce proportional changes of all three coordinates and will be represented by a displacement of the point of the color space along a straight line passing through the origin of coordinates (along the radius vector).

The origin of coordinates \((\alpha=\beta=\gamma=0)\) corresponds to the absence of any radiation and represents black, into which every color passes under an unlimited weakening of the radiation.

It remains, however, to make sure that the totality of points corresponding to various radiations really fills some space, and is not located, for example, on a plane passing through the origin of coordinates or even on a straight line.

If the sensitivity curves of the photoelements \(V_1(\lambda)\), \(V_2(\lambda)\), and \(V_3(\lambda)\) differed from one another only by a constant multiplier, so that \(K_1V_1=K_2V_2=K_3V_3\) (\(K_1\), \(K_2\), and \(K_3\) do not depend on \(\lambda\)), then we would always have \(K_1\alpha=K_2\beta=K_3\gamma\), independently of the form of the function \(E(\lambda)\). In this case the points corresponding to any radiations would be arranged in the space \(\alpha,\beta,\gamma\) on one radius vector, expressed by the last equation. In this case we would have a color space of one dimension.

If the sensitivity curves of the photoelements were connected by one equation of the form \(K_1V_1+K_2V_2+K_3V_3=0\) (\(K_1\), \(K_2\), and \(K_3\) do not depend on \(\lambda\)), then for any function \(E(\lambda)\) we would always have the relation \(K_1\alpha+K_2\beta+K_3\gamma=0\). In this case the color space would degenerate into a plane passing through the origin of coordinates and represented by the last relation. This would be the case of a color space with two dimensions.

In all other cases, i.e., in those cases when the functions \(V_1\), \(V_2\), and \(V_3\) are linearly independent, the points representing various radiations do indeed occupy part of the space. In this case it is easy to select three colors represented by points not lying in a single plane with the origin of coordinates.

With the aid of our system of three photoelements and three galvanometers we can trace the laws of color mixing, i.e., find that point \(C\) of the color space, with coordinates \(\alpha\), \(\beta\), and \(\gamma\), which corresponds to a mixture of three radiations. Let one of these radiations be represented by the point \(C_1(\alpha_1,\beta_1,\gamma_1)\), another by \(C_2(\alpha_2,\beta_2,\gamma_2)\)

and the third \(C_3(\alpha_3,\beta_3,\gamma_3)\). Since the general spectral characteristic \(E(\lambda)\) of the mixture of radiations is the sum of the characteristics of the mixed radiations \(E(\lambda)=E_1(\lambda)+E_2(\lambda)+E_3(\lambda)\), it is clear from (29) that

\[ \left. \begin{aligned} \alpha&=\alpha_1+\alpha_2+\alpha_3,\\ \beta&=\beta_1+\beta_2+\beta_3,\\ \gamma&=\gamma_1+\gamma_2+\gamma_3. \end{aligned} \right\} \tag{30} \]

Thus for the spatial mixing of colors we obtain the parallelepiped rule, corresponding to the parallelogram rule in the plane. If the radius-vectors of the colors \(C_1, C_2\), and \(C_3\) do not lie in one plane, then, by mixing them in definite proportions \(xC_1+yC_2+zC_3\), we can obtain any color with arbitrary coordinates \(\alpha,\beta\), and \(\gamma\). Indeed, the three equations

\[ \left. \begin{aligned} \alpha&=x\alpha_1+y\alpha_2+z\alpha_3,\\ \beta&=x\beta_1+y\beta_2+z\beta_3,\\ \gamma&=x\gamma_1+y\gamma_2+z\gamma_3 \end{aligned} \right\} \tag{31} \]

can be solved with respect to \(x,y\), and \(z\), and moreover in a unique manner, for any \(\alpha,\beta\), and \(\gamma\), provided only that

\[ \begin{vmatrix} \alpha_1 & \alpha_2 & \alpha_3\\ \beta_1 & \beta_2 & \beta_3\\ \gamma_1 & \gamma_2 & \gamma_3 \end{vmatrix} \ne 0 \tag{32} \]

Condition (32) is equivalent to the requirement that the radius-vectors \(C_1, C_2\), and \(C_3\) not lie in one plane. Having found the quantities \(x,y,z\), we can write

\[ C=xC_1+yC_2+zC_3. \tag{33} \]

One or two of the three numbers \(x,y,z\) may turn out to be negative.

Thus, if a system of three photoelements with linearly independent curves of spectral sensitivity is given, then such a system possesses color sensitivity; moreover this color sensitivity obeys Grassmann’s law, which is most fully stated in the following form:

1) there exist linearly independent aggregates of three colors, and

2) any four colors are always in a linear dependence, and uniquely so.

Grassmann’s law establishes the three-dimensionality of color space.

1. Let us construct a differential circuit for connecting two completely identical systems of three photoelements (Fig. 12), analogously to the way in which we connected two identical pairs of photoelements.

Fig. 12. Differential circuit for connecting two systems of three photoelements

Let the right-hand part of the circuit be illuminated by one radiation, and the left-hand part by another radiation. If, in this case, all three differential galvanometers \(G_1\), \(G_2\), and \(G_3\) give zero readings, this will mean that the two radiations being compared are of the same color. If the galvanometers are not in the zero position, but can be brought to it by increasing or decreasing the illumination of one of the halves of the differential circuit, without changing the spectral composition of the radiation, then this means that the colors of the original radiations differed only in lightness and coincided in chromaticity (in quality). If no increase or decrease in illumination, not disturbing the spectral composition of the radiation, can bring all three galvanometers at once into the zero position, then this means that the radiations illuminating the right and left sides of the differential circuit differ in chromaticity (i.e., in quality).

Having chosen three linearly independent colors \(C_1\), \(C_2\), and \(C_3\), we can, according to (33), compose from them and an arbitrary color \(C\) two one-color radiations. Using the three differential galvanometers as indicators, we can experimentally find the quantities \(x\), \(y\), and \(z\). Taking the colors \(C_1\), \(C_2\), and \(C_3\) as primary, we can characterize each color not by the photocurrents \(\alpha\), \(\beta\), and \(\gamma\), but by the quantities \(x\), \(y\), and \(z\), which are thus transformed into color coordinates.

Before proceeding to a more detailed analysis of the color sensitivity of the human eye, let us note some general properties of a color-sensitive system of three photoelements. Simi-

color equilibria between monochromatic (pure) colors \(C_\lambda\), producing energy illuminances equal to unity at different wavelengths, and the primary colors \(C_1, C_2\), and \(C_3\), we can find three color-matching functions \(\xi(\lambda)\), \(\eta(\lambda)\), and \(\zeta(\lambda)\) such that

\[ C_\lambda=\xi(\lambda)C_1+\eta(\lambda)C_2+\zeta(\lambda)C_3. \tag{34} \]

When plotted on a separate graph (Fig. 13), these functions give curves called color-matching curves. The functions \(\xi(\lambda)\), \(\eta(\lambda)\), and \(\zeta(\lambda)\) may at the same time be regarded as a parametric expression of a certain characteristic curve in the space \(x,y,z\), expressing the color properties of our system of three photoelements. A conical surface having its vertex at the origin and the characteristic curve as directrix for the lateral surface embraces a part of color space. It is not difficult to understand that all real colors will be represented by points lying in that part of the space which is embraced by this surface.

Fig. 13

The coordinates \(x,y,z\) of the color of radiation having composition \(E(\lambda)\) will be expressed, with respect to the primary colors \(C_1, C_2\), and \(C_3\), by the integrals

\[ \begin{aligned} x&=\int E(\lambda)\xi(\lambda)\,d\lambda,\\ y&=\int E(\lambda)\eta(\lambda)\,d\lambda,\\ z&=\int E(\lambda)\zeta(\lambda)\,d\lambda. \end{aligned} \tag{35} \]

If, instead of the colors \(C_1, C_2\), and \(C_3\), we take as primaries new colors \(C'\), \(C''\), and \(C'''\), and if the old primaries are related to the new ones by the relations

\[ \begin{aligned} C_1&=a_1C'+b_1C''+c_1C''',\\ C_2&=a_2C'+b_2C''+c_2C''',\\ C_3&=a_3C'+b_3C''+c_3C''', \end{aligned} \tag{36} \]

then instead of (33) and (34) we shall write

\[ C=x'C'+y'C''+z'C''' \tag{37} \]

and

\[ C_\lambda=\xi'(\lambda)C'+\eta'(\lambda)C''+\zeta'(\lambda)C''', \tag{38} \]

where

\[ \begin{aligned} x'&=a_1x+a_2y+a_3z,\\ y'&=b_1x+b_2y+b_3z,\\ z'&=c_1x+c_2y+c_3z. \end{aligned} \tag{39} \]

and

\[ \left. \begin{aligned} \xi'(\lambda)&=a_1\xi(\lambda)+a_2\eta(\lambda)+a_3\zeta(\lambda),\\ \eta'(\lambda)&=b_1\xi(\lambda)+b_2\eta(\lambda)+b_3\zeta(\lambda),\\ \zeta'(\lambda)&=c_1\xi(\lambda)+c_2\eta(\lambda)+c_3\zeta(\lambda). \end{aligned} \right\} \tag{40} \]

The new mixing functions thus represent linear combinations of the old mixing functions.

The new coordinates \(x'\), \(y'\), and \(z'\), relative to the new primary colors \(C'\), \(C''\), and \(C'''\), can also be expressed in the form of integrals

\[ \left. \begin{aligned} x'&=\int E(\lambda)\xi'(\lambda)\,d\lambda,\\ y'&=\int E(\lambda)\eta'(\lambda)\,d\lambda,\\ z'&=\int E(\lambda)\zeta'(\lambda)\,d\lambda. \end{aligned} \right\} \tag{41} \]

The latter expressions, which are easily obtained by substituting (35) and (40) into (39), coincide in form with relations (35). This is quite natural, since (35) and (41) represent the coordinates of a color referred to two entirely equivalent systems of primary colors \(C_1, C_2, C_3\) and \(C', C'', C'''\).

1. Passing to the analysis of quantitative relations between colors, we may repeat that when colors identical in quality are to be compared, i.e., those represented by points of the same radius vector, the question is resolved simply. The ratio of the coordinates of these colors gives the answer to the question posed.

But in the general case there is no such simple solution. We shall seek a measure of the quantity of color in the form of a function of the three photostimuli \(\alpha, \beta, \gamma\), and denote it by \(E(\alpha,\beta,\gamma)\). This assumption ensures the independence of the quantity of color from the choice of the coordinate system. In order that there be no contradictions with the preceding assumptions, we must suppose that \(E(\alpha,\beta,\gamma)\) changes directly proportionally to changes in \(\alpha, \beta, \gamma\), and that, moreover,

\[ E(\alpha_1,\beta_1,\gamma_1)+E(\alpha_2,\beta_2,\gamma_2) = E(\alpha_1+\alpha_2,\beta_1+\beta_2,\gamma_1+\gamma_2). \tag{42} \]

The latter relation is a consequence of the additivity of the quantities of mixed colors postulated by us. It can be satisfied only under the condition that \(E(\alpha,\beta,\gamma)\) is a linear homogeneous function, i.e.,

\[ E(\alpha,\beta,\gamma)=p\alpha+q\beta+r\gamma. \tag{43} \]

The primary colors \(C_1, C_2\), and \(C_3\) then receive definite quantitative values

\[ \left. \begin{aligned} \text{for } C_1\quad E(\alpha_1,\beta_1,\gamma_1)&=L=p\alpha_1+q\beta_1+r\gamma_1,\\ \text{for } C_2\quad E(\alpha_2,\beta_2,\gamma_2)&=M=p\alpha_2+q\beta_2+r\gamma_2,\\ \text{for } C_3\quad E(\alpha_3,\beta_3,\gamma_3)&=N=p\alpha_3+q\beta_3+r\gamma_3. \end{aligned} \right\} \tag{44} \]

The quantity \(E\) of each color having, relative to \(\mathbf{C}_1, \mathbf{C}_2\), and \(\mathbf{C}_3\), the coordinates \(x, y\), and \(z\), is found at once as

\[ E = xL + yM + zN . \tag{45} \]

Pure monochromatic colors \(\mathrm{C}_\lambda\), which according to (34) have coordinates \(\xi(\lambda)\), \(\eta(\lambda)\), and \(\zeta(\lambda)\), will be quantitatively evaluated as follows:

\[ E_\lambda = \xi(\lambda)L + \eta(\lambda)M + \zeta(\lambda)N = V(\lambda), \tag{46} \]

where \(V(\lambda)\) is the quantitative measure of equal-energy monochromatic radiations, i.e. determines the quantitative spectral sensitivity of the entire system.

On the other hand, according to (43),

\[ V(\lambda) = E_\lambda = p\alpha_\lambda + q\beta_\lambda + r\gamma_\lambda = pV_1(\lambda) + qV_2(\lambda) + rV_3(\lambda) \tag{47} \]

and for a color of any composition \(E(\lambda)\)

\[ E = \int E(\lambda)v(\lambda)d\lambda . \tag{48} \]

If we have a large number of sets of three photoelements similar in their properties, connected in a circuit with close values of the quantities \(p, q\), and \(r\), then we can observe different functions \(V(\lambda)\). In this case the difference may arise both from a mismatch of \(V_1, V_2\), and \(V_3\), and from a mismatch of the coefficients \(p, q\), and \(r\). If the difference is based on a mismatch of the curves \(V_1, V_2\), and \(V_3\), then such systems will differ in establishing the same color for radiations of different composition. If, however, the differences arise only because of a mismatch of the coefficients \(p, q\), and \(r\), then this will entail differences in the quantitative evaluation of colors of different quality (disagreements between observers in heterochromatic photometry).

With new primary colors \(\mathrm{C}', \mathrm{C}''\), and \(\mathrm{C}'''\), the coordinates \(x, y, z\) pass into \(x', y', z'\) according to (39), and the quantitative coefficients \(L, M, N\) pass into \(L', M', N'\), where (36) gives

\[ \left. \begin{aligned} L &= a_1L' + b_1M' + c_1N',\\ M &= a_2L' + b_2M' + c_2N',\\ N &= a_3L' + b_3M' + c_3N'. \end{aligned} \right\} \tag{49} \]

The quantity \(E\) of the color \(\mathbf{C}\), independent of the choice of primary colors, can be represented through \(x', y', z'\) and \(L', M', N'\) in the following way:

\[ E = x'L' + y'M' + zN' . \tag{50} \]

Substituting (44) into (46) and comparing with (47), we find

\[ \left. \begin{aligned} V_1(\lambda) &= \alpha_1\xi(\lambda) + \alpha_2\eta(\lambda) + \alpha_3\zeta(\lambda),\\ V_2(\lambda) &= \beta_1\xi(\lambda) + \beta_2\eta(\lambda) + \beta_3\zeta(\lambda),\\ V_3(\lambda) &= \gamma_1\xi(\lambda) + \gamma_2\eta(\lambda) + \gamma_3\zeta(\lambda), \end{aligned} \right\} \tag{51} \]

i.e. the mixture curves \(\xi(\lambda)\), \(\eta(\lambda)\), and \(\zeta(\lambda)\) are linear combinations of the sensitivity curves of the three photoelements \(V_1(\lambda), V_2(\lambda)\), and \(V_3(\lambda)\).

1. In three-dimensional space \(\alpha, \beta, \gamma\) (or \(x, y, z\)) the expression
   \(p\alpha+q\beta+r\gamma=\mathrm{const}\) (or \(Lx+My+Nz=\mathrm{const}\)) represents a family of parallel planes, which are the geometric loci of equiquantitative colors. That plane of this family which passes through the origin corresponds to colors with zero quantity. Except for the origin, all colors of this plane are imaginary. If we choose two new primary colors \(C'\) and \(C'''\) so that they lie in this plane, then it is obvious that the corresponding quantitative coefficients \(L'\) and \(N'\) become zero.

If, in addition, we place the third primary color \(C''\) in the plane with quantity equal to unity (\(M'=1\)), then (50) becomes

\[ E=y'. \tag{52} \]

Comparing (48) with (41), we see that, with this choice of primary colors,

\[ V(\lambda)=\eta'(\lambda). \tag{53} \]

In other words, one of the mixture functions coincides with the general spectral sensitivity of the system.

Having chosen any two directions in the plane \(p\alpha+q\beta+r\gamma=0\) as the directions of the primary chromaticities, we can always place on them two of \(C'\) and \(C'''\) so that their mixture with a unit quantity of the color \(C''\) would give any color. In particular, one may require that the mixture of the primary colors give an “achromatic” or “white” color. It is most convenient to take as the white color that of equal-energy radiation \((E(\lambda)=\mathrm{const})\). It is not difficult to see that, according to (41), its coordinates are

\[ x'_E=\int \xi'(\lambda)\,d\lambda;\quad y'_E=\int \eta'(\lambda)\,d\lambda;\quad z'_E=\int \zeta'(\lambda)\,d\lambda . \]

In order that the mixture of the primary colors should give a color indistinguishable from the color \(E\), i.e. in order that

\[ C'+C''+C'''=E, \]

it is necessary and sufficient that \(x'_E=y'_E=z'_E\), i.e. that

\[ \int \xi'\lambda\,d=\int \eta'\,d\lambda=\int \zeta'\,d\lambda . \tag{54} \]

This condition establishes the scale along the axes \((OC'\) and \(OC''')\), i.e. determines the position of the primary colors \(C'\) and \(C'''\).

VII. COLOR SENSITIVITY OF THE NORMAL EYE

1. In proceeding to analyze the color-sensitive properties of the normal human eye, we must make several preliminary remarks.

It is known that the eye is a very complex organ, enabling a person to distinguish the contours of objects, their colors,

depth of their position under extremely wide variations in the external conditions of illumination—from a dark autumn night, when illumination is measured in thousandths and ten-thousandths of a lux, to a bright summer day with illumination on the order of tens of thousands of lux. It is also known that such a breadth of range is determined by the presence in the eye of two kinds of photosensitive elements—rods and cones.

The number of rods in the eye is much greater than the number of cones; they are situated around the yellow spot, which itself contains comparatively very few of them. The main mass of the sensitive elements of the yellow spot consists of cones, especially in its central part, which forms the so-called central depression, where rods are practically absent. The rods provide vision at low illuminations (so-called twilight vision), while the cones make it possible to see by day, when the very sensitive rods are in a state of complete dazzling.

Color sensitivity is one of the characteristic properties of the apparatus of daytime vision—the cones. Therefore all color measurements and all investigations of the color properties of the eye are carried out: 1) at a sufficiently high level of brightness of the visual field, excluding the rod apparatus, and 2) with a sufficiently small size of the visual field, ensuring the operation of only the central depression.

Moreover, it should be remembered that our discussion concerns only the physico-mathematical side of color perception and does not touch upon the very essential psychophysiological domain that determines a whole series of phenomena of an entirely different order. This includes, for example, the question of why a yellow color having low relative brightness appears to our eye, under ordinary conditions, as qualitatively different from bright yellow. This difference is so substantial that it has found its reflection in language, which has a special name for this case—brown. A whole series of questions of this kind (for example, the very widespread division of colors into “cold” and “warm,” all questions connected with the phenomenon of adaptation or fatigue) will remain entirely untouched by us. We shall confine ourselves only to considering the laws of color mixing and their qualitative and quantitative characterization under the specialized conditions indicated above—sufficiently high brightness and a small visual field.

For the enormous number of problems of scientific and technical colorimetry this will be sufficient.

1. As the basis of our discussion we shall take Grassmann’s law, established experimentally with respect to the color perception of the eye and having, as we have seen, a place for a three-dimensional color-perceiving system. Conversely, Grassmann’s law permits us to assert that the color-perception system of the normal eye is a three-dimensional system and makes it possible to characterize every color-

three coordinates, which define it as a mixture of three basic linearly independent colors.

With qualifications regarding the size of the visual field and the level of brightness, all the conclusions we have drawn concerning a three-dimensional system can be applied entirely to the eye.

The basis for constructing a system of color perception for the human eye should be the mixture curves obtained as averages from the investigation of a sufficiently large number of observers normal with respect to color.

The mixture curves are obtained, in general terms, as follows. We choose three linearly independent colors as the primaries. One half of a sufficiently small visual field is illuminated with monochromatic light while maintaining a sufficiently high level of brightness. The energy measure of this monochromatic radiation must be known. On the other half of the visual field we seek to mix the three primary colors in such a proportion that both parts of the visual field would become indistinguishable from one another, and we determine these proportions (coefficients) in certain arbitrary units. For some wavelengths this cannot be done. Then one (or two) of the primary colors must be directed onto the field illuminated by the monochromatic light under study, and in this way complete photometric and color equality is obtained. In this case one (or two) of the three mixture coefficients is assigned a negative sign. By dividing the magnitudes of the coefficients obtained by the energy quantity of the monochromatic radiation, we refer them to a unit of radiation.

The three coefficients, determined over the entire visible spectrum, constitute three mixture curves $\xi(\lambda)$, $\eta(\lambda)$, and $\zeta(\lambda)$, which make it possible to construct in color space a curve of double curvature characterizing the color properties of the eye.

As coordinate axes we shall take any three straight lines not lying in one plane; on them, on an arbitrary scale, we shall lay off the primary colors chosen by us. For the average eye we shall then obtain the following picture. The ends of our characteristic curve will abut the origin of coordinates. The segments adjoining the ends are rectilinear. At the short-wavelength end of the characteristic curve the rectilinear segment extends approximately to $\lambda = 420\,\mathrm{m}\mu$, and at the long-wavelength end—to $700\,\mathrm{m}\mu$.

The rectilinear segments of the curve mean that monochromatic radiations with wavelengths of $420\,\mathrm{m}\mu$ and shorter all have the same chromaticity, just as do the radiations corresponding to wavelengths of $700\,\mathrm{m}\mu$ and longer.

The portion of the characteristic curve adjoining the latter rectilinear segment and extending approximately to $570$–$580\,\mathrm{m}\mu$ lies in a plane passing through the origin of coordinates. Thus all spectral colors intermediate between $700$ and $570\,\mathrm{m}\mu$ can be obtained by mixing these two in some definite proportion. Only the middle part of the characteristic curve-

between wavelengths of 420 and 570 mμ has a double curvature, i.e., it is essentially a space curve.

The form of the characteristic curve can be traced in more detail in Fig. 14, where projections of this curve onto two planes are presented.

The curve itself is obtained in the following way. Let us imagine a rectangular system of spatial coordinates and plot along each of the axes the value of one of the coefficients of the three mixture curves $\bar r$, $\bar g$, $\bar b$, obtained for the primary colors corresponding to monochromatic radiations with wavelengths 700, 546.1, and 435.8 mμ. To each wavelength there correspond three coefficients of the mixture curves, which determine the position of one point of the characteristic curve. By varying the wavelength, we obtain a series of points in space that determine the entire characteristic curve.

The fifth, sixth, and seventh columns of the first table appended to the resolutions on colorimetry adopted by the Congress of 1931 give the mixture curves relative to the indicated primary colors. These numbers $\bar r$, $\bar g$, $\bar b$, established on the basis of measurements by Wright and Guild, completely determine the color-sensitive properties of the average human eye (the so-called standard observer of 1931). Figure 14 presents two projections of the characteristic curve onto the planes $\bar b\bar g$ and $\bar g\bar r$. The first of them is given by a solid line, the second by a dashed line. The numbers placed along these curves denote the corresponding wavelengths in mμ. The units in which the quantities of the monochromatic radiations 435.8, 546.1, and 700 mμ are measured are chosen so that a mixture of them, taken in equal quantities, would give a color coinciding with the color of equienergetic radiation.

Fig. 14

A conical surface having its vertex at the origin of coordinates and the characteristic curve as its generator represents that surface of color space on which the monochromatic (as they are sometimes called, “pure”) colors are situated. If this unclosed surface is completed by a plane passing through the origin of coordinates and the two rectilinear ends of the characteristic curve, then the cavity of our cone will isolate that part of color space in which all real ...

colors. To each color with a complex spectral composition there will correspond a point inside this strip. If the coordinate axes are cut by a plane at points equidistant from the origin of coordinates, then the section of this plane with the coordinate planes and with the cone of real colors will have the form shown in Fig. 15. The points \(r, g, b\), which are the vertices of an equilateral triangle formed by the intersection of the coordinate planes, correspond to monochromatic radiations of \(700\), \(546.1\), and \(435.8\) mμ.

This establishes their connection with the geometric locus of monochromatic radiations. The center of the equilateral triangle corresponds to the white color of equal-energy radiation.

1. The question of the quantitative estimate that the eye gives to various colors must be examined quite separately. In the case of a system consisting of several receivers, we could act as we pleased. We could connect them into arbitrary circuits, or simply compose some linear function of several variables—the reactions of the receivers—and take it, conventionally, as a quantitative measure of color. In the case of the eye, we must take account of experimentally established facts. Grassmann’s law, indicating that our system is three-dimensional, gives no indications whatever as to quantitative dependencies. Let us turn to the corresponding experiment. Suppose that on a photometric bench two sources \(A\) and \(B\), differing in color, are placed on two sides of the photometric head. Let us provide a suitable size of the field of view and its brightness.

Observing through the eyepiece, we shall begin to move one of the sources, say \(B\), leaving the other source and the head itself fixed. Then the color of one part of the field of view—from source \(A\)—will remain unchanged, while the color of the other part—from source \(B\)—will change. At the same time, the change of color from \(B\) will be purely quantitative, since its spectral composition remains constant. In color space our experiment will be represented as follows: some color \(A\) remains unchanged. Another color, \(B\), moves along its radius-vector, now approaching the origin of coordinates (the field darkens), now moving away from it (the field becomes lighter). It is not at all difficult, by moving source \(B\), to arrive at such positions of it in which no one will doubt that the field from \(B\) is in one case darker and in the other lighter than the field from \(A\). And if this is so, then undoubtedly somewhere between these extreme positions of source \(B\) there exists a position at which both fields should appear equally bright. And indeed, after some practice the observer begins to establish some, not very definite, position of photometric equilibrium with much greater accuracy than could have been expected at first. The greater the difference in colors between sources \(A\) and \(B\), the more difficult it is to find the equilibrium position, the greater

GENERAL FOUNDATIONS OF COLORIMETRY

random error of comparison. With a small difference in the color of the fields, the accuracy of comparison is only slightly inferior to the accuracy of measurements performed when the color of both parts of the field of view is completely identical.

However, the magnitude of the error of comparison is not of fundamental importance, since random errors accompany every measurement. What is essential is that the eye can quantitatively compare differently colored radiations.

The next fundamental question will be whether this possibility of quantitative comparison is suitable for physical constructions or not; above all, whether the quantities obtained by the method of heterochromatic photometry satisfy the axiom: two quantities separately equal to a third are equal to one another. Direct experiments set up to resolve this question give an affirmative answer to it1. Thus the experiment described above, with comparison of differently colored radiations, becomes equivalent to some conventional criterion of quantitative equality of qualitatively different colors introduced by us above.

The only difference is that, in the case of the eye, the condition of quantitative equality is not assigned by us arbitrarily, but is established by the eye by virtue of one or another internal structure, and that the accuracy of comparison decreases with an increase in the difference between the colors being compared.

The experiment just analyzed corresponds to what is commonly called comparison of colors by their brightness. In other words, the so-called brightness of a color becomes a measure of its quantity.

If we now compare the rules that we derived for the quantity of color with the properties of color brightness established experimentally, we at once notice their complete correspondence.

Indeed, we have:

1. Direct proportionality between the quantity of color and the radiation, with unchanged spectral composition.

The same is true for the brightness of a color.

1. Summation of the quantity of color when colors are mixed.

The law of additivity of brightness has been established experimentally.

1. The quantity of each color can be determined from the color equation by replacing the colors with their quantitative coefficients.

In the same way, the brightness of a color can also be established through the brightness coefficients of the primary colors.

Experimentally this was done by the same authors (Exner, Kohlrausch). It follows from this that, within known limits of variation of the external conditions of illumination that provide sufficient brightness of the field of view and for sufficiently small central parts of the retina,

the impression of brightness in the eye is determined by a linear and homogeneous function of the reactions of the three receptors entering into it.

1. If we compare by quantity (by brightness) the three primary colors adopted by us for the color space shown in Fig. 14, it will turn out that they are very far from being equal to one another. It turns out that the brightness (quantitative) coefficients of red (700 mμ), green (546.1 mμ), and blue (435.8 mμ) are in the ratio \(1 : 4.5907 : 0.0601\). In other words, if we wish to obtain the “white” color \(e\) of an equal-energy spectrum, then the brightnesses of the monochromatic radiations 700, 546.1, and 435.8 mμ of the primary colors, which we shall have to mix for this purpose (in Fig. 14 these quantities are represented by equal segments on the axes \(\bar r\), \(\bar g\), \(\bar b\)), will be measured by the numbers \(1 : 4.5907\) and \(0.0601\). We could, of course, change the scale and represent the primary colors not by equal segments, but by such segments that their lengths would be proportional to their quantitative measure. Then quantitatively equal colors would be determined by equal segments on the axes. But such a scale would be extremely inconvenient, because the vector of the white color would then lie right up against the plane \(\bar r \bar g\), occupying in the plane of Fig. 15 (this plane cuts the coordinate axes at equal distances from the origin, i.e., in the case under consideration it is the plane of quantitatively equal colors) the position \(E'\). In this case the entire color space turns out to be occupied by purples and blues, while all reds, yellows, and greens turn out to be compressed toward the plane \(\bar r \bar g\), which is very inconvenient from the standpoint of the possibility of using graphical methods. It should be noted that the curve of spectral colors will acquire on this plane an entirely different form, and only the points \(r\), \(g\), and \(b\) will retain their positions. Taking these inconveniences into account, one has to adopt as primary colors, and represent by equal segments, not colors of equal quantity, but those which in mixture form white. Then this white color is located at the center of the triangle, and the space is distributed more uniformly among the various color tones.

If we wish to mark on the axes \(r\), \(g\), \(b\) of Fig. 14 colors equal in their quantity, then we shall have to lay off from the origin of coordinates segments inversely proportional to the quantitative coefficients. These segments must have lengths related as \(1 : 0.21783 : 166.4\). The three marked points determine the plane of equal-quantity colors. Every plane parallel to it will also be an equal-quantity plane, but corresponding to a different quantity of color. The plane of this family passing through the origin of coordinates intersects the planes \(\bar r \bar g\) and \(\bar g \bar b\) along straight lines (traces), shown by dashed lines in the drawing. The indicated plane will contain all colors with zero quantitative (brightness) coefficients.

It is also not difficult to draw in Fig. 15 the trace of the intersection of the zero plane with the plane of the equilateral triangle. The traces

Fig. 14 intersect the plane of Fig. 15 at the points \(A\) and \(B\). The dashed straight line connecting them is the intersection of the plane of the triangle with the “zero” plane (this straight line is sometimes called the “alychna”).

Two coordinate axes of the international system \(X, Y, Z\), namely \(X\) and \(Z\), lie in the “zero” plane and therefore must intersect the straight line \(AB\) (the alychna). In principle the direction of these axes may be arbitrary. Practically, however, it is convenient to arrange the other coordinate plane, namely the plane \(XY\), so that it coincides with the plane of the long-wavelength portion of the characteristic curve (580–760 \(m\mu\)). Such an arrangement ensures zero values of the coordinate \(z\) over this entire fairly considerable portion of the spectrum and entails a noticeable simplification of the computations. In Fig. 15 this latter requirement determines the position of the line \(XY\) and, together with the already marked line \(XZ\), fixes the position of the \(X\) axis. The coordinate plane \(ZY\) is chosen more or less arbitrarily. Its position has no special advantages. It almost touches the cone of spectral colors along the line corresponding to \(\lambda = 504\,m\mu\). By choosing its position relative to the cone of monochromatic colors, we thereby determine the still undetermined positions of the \(Y\) and \(Z\) axes.

Fig. 15

Having as a skeleton the coordinate trihedron \(\bar r, \bar g, \bar b\) and the characteristic curve established experimentally (Fig. 14), we can compute, in relation to them, the position of the axes \(X, Y\), and \(Z\), chosen on the basis of the considerations set forth above. Since all three axes lie outside the cone of spectral colors, they do not correspond to any real colors. Nevertheless, we can use these imaginary colors for constructing color equations. In particular, we can establish what lengths of vectors should be taken along the axes \(X, Y\), and \(Z\) so that their sums have directions coinciding with the directions of the axes \(\bar r, \bar g, \bar b\), and equal length.

Adding these three vectors, we obtain, on the one hand, the white color \(\varepsilon\), and on the other—the lengths of the vectors directed along \(X, Y, Z\), which must be added in order to obtain the white color. After this, nothing prevents us from taking these latter summed vector lengths along each of the axes as the primary colors and expressing all colors through them.

The next step may be as follows.

We take a new rectangular system of spatial coordinates and begin to construct on it a new form of the same color space. For this purpose we lay off on the axes, in equal scales, the new primary colors found for the axes \(X, Y, Z\). This will allow us to transfer the old color space into the new coordinate system, since for each color its new coordinates will be known. The characteristic curve will take on a new form without changing its color properties, which, obviously, cannot depend on the choice of one or another set of primary colors. The plane that cuts the new coordinate axes at equal distances from the origin forms, in its intersection with the coordinate planes, an equilateral triangle \(xyz\) (Fig. 16). The white color \(\varepsilon\) of equal-energy radiation will again be at the “center” of the triangle. The geometrical locus of the spectral colors will lie inside the triangle, and the side \(zy\) will still be almost tangent to it at the wavelength \(504\,m\mu\), while the side \(xy\) will coincide with the rectilinear portion of the long-wave end of the color curve.

Fig. 16. Section of the color space by a plane cutting off equal segments on the axes \(X, Y, Z\) (the scales on the axes \(X, Y, Z\) are chosen so that the color \(\varepsilon\) lies at the “center” of the triangle)

This same triangle, transformed into a rectangular one (Fig. 17), proves considerably more convenient for practical use. On the plane of this triangle, besides the point \(\varepsilon\), there are plotted the positions of the chromaticities corresponding to the standard sources \(A, B\), and \(C\).

Fig. 17. Orthogonal projection of the triangle of Fig. 16 onto the \(XOY\) plane.

References

1. I. Newton, Optics, State Publishing House, Moscow—Leningrad, 1927, translated by S. I. Vavilov. First book, Part II, Propositions IV, VII, pp. 107—128.

2. Th. Young, Lectures on natural philosophy, London, 1807, Vol. I, p. 439.

3. H. v. Helmholtz, Handb. d. physiolog. Optic, 3. Aufl., B. II, 1911, 52, 172.

4. J. C. Maxwell, Scientif. Pap., Vol. I, p. 126—154.

5. J. C. Maxwell, Scientif. Pap., Vol. I, p. 119—125.

6. J. C. Maxwell, Scientif. Pap., Vol. I, p. 410—444.

7. J. C. Maxwell, Scientif. Pap., Vol. II, p. 267—279.

8. H. Grassmann, Zur Theorie der Farbenmischung, Gesam. Abh. Leipzig, 1902, B. II, 161—173.

9. H. Grassmann, Bemerkungen zur Theorie der Farbenmischung, Gesam. Abh. Leipzig, 1902, B. II, 213—221.

10. E. Schrödinger, Ann. d. Phys., 63, 397—456, 481—620, 1920.

11. E. Schrödinger, Müller-Pouillet, Lehrb. d. Physik, 1926, B. II, 456—560.

12. Deane B. Judd, Bur. Stand. Journ. Res., 4, 515—548, 1930.

13. Deane B. Judd, J. O. S. A., 23, Nr. 10, 359—374, 1933.

14. J. Guild, Proc. Optic. Convention, V. 1, 61—147, 1926.

15. J. Guild, Phil. Trans., A 230, 149—187, 1931.

16. W. D. Wright, Trans. Optic. Soc., 30, 141—164, 1928—1929.

17. W. D. Wright, Trans. Optic. Soc., 31, 201—218, 1929—1930.

18. N. D. Nyuberg, “Mathematical foundations of the problem of color measurement,” article in N. T. Fedorov’s book The Present State of Colorimetry, GTTI, 1933, pp. 141—169.

More detailed bibliographic guidance on questions of colorimetry may be found in N. T. Fedorov’s book The Present State of Colorimetry, GTTI, 1933, as well as in R. Rösch’s article, Darstellung der Farbenlehre für die Zwecke des Mineralogen, Fortschr. e. Mineralog., Kristallogr. u. Petrogr., Bd. 13, 74—234, 1929, which contains about a thousand titles of works on colorimetry.

1. F. Exner, Sitz. Akad. Wiss. Wien, IIa, 129, 27, 1920; K. W. F. Kohlrausch, Physik. Z., 21, 423, 1920; Cl. Schaefer, Physik. Z., 26, 58, 908, 1925. ↩↩