Chapter IV

Radiation of Any Arbitrary Spectral Distribution of Energy. Entropy and Temperature of Monochromatic Radiation

91. We have so far applied Wien’s displacement law only to the case of black radiation; it has, however, a much more general importance. For equation (95), as has already been stated, gives, for any original spectral distribution of the energy radiation contained in the evacuated cavity and radiated uniformly in all directions, the change of this energy distribution accompanying a reversible adiabatic change of the total volume. Every state of radiation brought about by such a process is perfectly stationary and can continue inﬁnitely long, subject, however, to the condition that no trace of an emitting or absorbing substance exists in the radiation space. For otherwise, according to Sec. 51, the distribution of energy would, in the course of time, change through the releasing action of the substance irreversibly, i.e., with an increase of the total entropy, into the stable distribution corresponding to black radiation.

The difference of this general case from the special one dealt with in the preceding chapter is that we can no longer, as in the case of black radiation, speak of a deﬁnite temperature of the radiation. Nevertheless, since the second principle of thermodynamics is supposed to hold quite generally, the radiation, like every physical system which is in a deﬁnite state, has a deﬁnite entropy, $S = V s$ . This entropy consists of the entropies of the monochromatic radiations, and, since the separate kinds of rays are independent of one another, may be obtained by addition. Hence

(113)

$s = \int_{0}^{\infty} s d ν, S = V \int_{0}^{\infty} s d ν,$

where $s d ν$ denotes the entropy of the radiation of frequencies between $ν$ and $ν + d ν$ contained in unit volume. $s$ is a deﬁnite function of the two independent variables $ν$ and $u$ and in the following will always be treated as such.

92. If the analytical expression of the function $s$ were known, the law of energy distribution in the normal spectrum could immediately be deduced from it; for the normal spectral distribution of energy or that of black radiation is distinguished from all others by the fact that it has the maximum of the entropy of radiation $S$ .

Suppose then we take $s$ to be a known function of $ν$ and $u$ . Then as a condition for black radiation we have

(114)

$δ S = 0,$

for any variations of energy distribution, which are possible with a constant total volume $V$ and constant total energy of radiation $U$ . Let the variation of energy distribution be characterized by making an inﬁnitely small change $δ u$ in the energy $u$ of every separate deﬁnite frequency $ν$ . Then we have as ﬁxed conditions

(115)

$δ V = 0 and \int_{0}^{\infty} δ u d ν = 0 .$

The changes $d$ and $δ$ are of course quite independent of each other.

Now since $δ V = 0$ , we have from (114) and (113)

$\int_{0}^{\infty} δ s d ν = 0,$

or, since $ν$ remains unvaried

$\int_{0}^{\infty} \frac{\partial s}{\partial u} δ u d ν = 0,$

and, by allowing for (115), the validity of this equation for all values of $δ u$ whatever requires that

(116)

$\frac{\partial s}{\partial u} = const .$

for all different frequencies. This equation states the law of energy distribution in the case of black radiation.

93. The constant of equation (116) bears a simple relation to the temperature of black radiation. For if the black radiation, by conduction into it of a certain amount of heat at constant volume $V$ , undergoes an inﬁnitely small change in energy $δ U$ , then, according to (73), its change in entropy is

$δ S = \frac{δ U}{T} .$

However, from (113) and (116),

$δ S = V \int_{0}^{\infty} \frac{\partial s}{\partial u} δ u d ν = \frac{\partial s}{\partial u} V \int_{0}^{\infty} δ u d ν = \frac{\partial s}{\partial u} δ U$

hence

(117)

$\frac{\partial s}{\partial u} = \frac{1}{T}$

and the above quantity, which was found to be the same for all frequencies in the case of black radiation, is shown to be the reciprocal of the temperature of black radiation.

Through this law the concept of temperature gains signiﬁcance also for radiation of a quite arbitrary distribution of energy. For since $s$ depends only on $u$ and $ν$ , monochromatic radiation, which is uniform in all directions and has a deﬁnite energy density $u$ , has also a deﬁnite temperature given by (117), and, among all conceivable distributions of energy, the normal one is characterized by the fact that the radiations of all frequencies have the same temperature.

Any change in the energy distribution consists of a passage of energy from one monochromatic radiation into another, and, if the temperature of the ﬁrst radiation is higher, the energy transformation causes an increase of the total entropy and is hence possible in nature without compensation; on the other hand, if the temperature of the second radiation is higher, the total entropy decreases and therefore the change is impossible in nature, unless compensation occurs simultaneously, just as is the case with the transfer of heat between two bodies of different temperatures.

94. Let us now investigate Wien’s displacement law with regard to the dependence of the quantity $s$ on the variables $u$ and $ν$ . From equation (101) it follows, on solving for $T$ and substituting the value given in (117), that

(118)

$\frac{1}{T} = \frac{1}{ν} F (\frac{c^{3} u}{ν^{3}}) = \frac{\partial s}{\partial u}$

where again $F$ represents a function of a single argument and the constants do not contain the velocity of propagation $c$ . On integration with respect to the argument we obtain

(119)

$s = \frac{ν^{2}}{c^{3}} F_{1} (\frac{c^{3} u}{ν^{3}})$

the notation remaining the same. In this form Wien’s displacement law has a signiﬁcance for every separate monochromatic radiation and hence also for radiations of any arbitrary energy distribution.

95. According to the second principle of thermodynamics, the total entropy of radiation of quite arbitrary distribution of energy must remain constant on adiabatic reversible compression. We are now able to give a direct proof of this proposition on the basis of equation (119). For such a process, according to equation (113), the relation holds:

(120)

$\begin{array}{l} δ S & = \int_{0}^{\infty} d ν (V δ s + s δ V) \\ = \int_{0}^{\infty} d ν (V \frac{\partial s}{\partial u} δ u + s δ V) . \end{array}$

Here, as everywhere,

s

should be regarded as a function of

u

and

ν

, and

δ ν = 0

Now for a reversible adiabatic change of state the relation (95) holds. Let us take from the latter the value of $δ u$ and substitute. Then we have

$δ S = δ V \int_{0}^{\infty} d ν \{\frac{\partial s}{\partial u} (\frac{ν d u}{3 d ν} - u) + s\} .$

In this equation the differential coeﬃcient of $u$ with respect to $ν$ refers to the spectral distribution of energy originally assigned arbitrarily and is therefore, in contrast to the partial differential coeﬃcients, denoted by the letter $d$ .

Now the complete differential is:

$\frac{d s}{d ν} = \frac{\partial s}{\partial u} \frac{d u}{d ν} + \frac{\partial s}{\partial ν} .$

Hence by substitution:

(121)

$δ S = δ V \int_{0}^{\infty} d ν \{\frac{ν}{3} (\frac{d s}{d ν} - \frac{\partial s}{\partial ν}) - u \frac{\partial s}{\partial u} + s\} .$

But from equation (119) we obtain by differentiation

(122)

$\frac{\partial s}{\partial u} = \frac{1}{ν} Ḟ (\frac{c^{3} u}{ν^{3}}) and \frac{\partial s}{\partial ν} = \frac{2 ν}{c^{3}} F (\frac{c^{3} u}{ν^{3}}) - \frac{3 u}{ν^{2}} Ḟ (\frac{c^{3} u}{ν^{3}}) .$

Hence

(123)

$ν \frac{\partial s}{\partial ν} = 2 s - 3 u \frac{\partial s}{\partial u} .$

On substituting this in (121), we obtain

(124)

$δ S = δ V \int_{0}^{\infty} d ν (\frac{ν}{3} \frac{d s}{d ν} + \frac{1}{3} s)$

or,

$δ S = \frac{δ V}{3} {[ν s]}_{0}^{\infty} = 0,$

as it should be. That the product $ν s$ vanishes also for $ν = \infty$ may be shown just as was done in Sec. 83 for the product $ν u$ .

96. By means of equations (118) and (119) it is possible to give to the laws of reversible adiabatic compression a form in which their meaning is more clearly seen and which is the generalization of the laws stated in Sec. 87 for black radiation and a supplement to them. It is, namely, possible to derive (105) again from (118) and (99b). Hence the laws deduced in Sec. 87 for the change of frequency and temperature of the monochromatic radiation energy remain valid for a radiation of an originally quite arbitrary distribution of energy. The only difference as compared with the black radiation consists in the fact that now every frequency has its own distinct temperature.

Moreover it follows from (119) and (99b) that

(125)

$\frac{s'}{ν'^{2}} = \frac{s}{ν^{2}} .$

Now $s d ν V = S d ν$ denotes the radiation entropy between the frequencies $ν$ and $ν + d ν$ contained in the volume $V$ . Hence on account of (125), (99a), and (99c)

(126)

$S' d ν' = S d ν,$

i.e., the radiation entropy of an inﬁnitely small spectral interval remains constant. This is another statement of the fact that the total entropy of radiation, taken as the sum of the entropies of all monochromatic radiations contained therein, remains constant.

97. We may go one step further, and, from the entropy $s$ and the temperature $T$ of an unpolarized monochromatic radiation which is uniform in all directions, draw a certain conclusion regarding the entropy and temperature of a single, plane polarized, monochromatic pencil. That every separate pencil also has a certain entropy follows by the second principle of thermodynamics from the phenomenon of emission. For since, by the act of emission, heat is changed into radiant heat, the entropy of the emitting body decreases during emission, and, along with this decrease, there must be, according to the principle of increase of the total entropy, an increase in a different form of entropy as a compensation. This can only be due to the energy of the emitted radiation. Hence every separate, plane polarized, monochromatic pencil has its deﬁnite entropy, which can depend only on its energy and frequency and which is propagated and spreads into space with it. We thus gain the idea of entropy radiation, which is measured, as in the analogous case of energy radiation, by the amount of entropy which passes in unit time through unit area in a deﬁnite direction. Hence statements, exactly similar to those made in Sec. 14 regarding energy radiation, will hold for the radiation of entropy, inasmuch as every pencil possesses and conveys, not only its energy, but also its entropy. Referring the reader to the discussions of Sec. 14, we shall, for the present, merely enumerate the most important laws for future use.

98. In a space ﬁlled with any radiation whatever the entropy radiated in the time $d t$ through an element of area $d σ$ in the direction of the conical element $d Ω$ is given by an expression of the form

(127)

$d t d σ cos θ d Ω L = L sin θ cos θ d θ d φ d σ d t .$

The positive quantity $L$ we shall call the “speciﬁc intensity of entropy radiation” at the position of the element of area $d σ$ in the direction of the solid angle $d Ω$ . $L$ is, in general, a function of position, time, and direction.

The total radiation of entropy through the element of area $d σ$ toward one side, say the one where $θ$ is an acute angle, is obtained by integration with respect to $φ$ from $0$ to $2 π$ and with respect to $θ$ from $0$ to $\frac{π}{2}$ . It is

$d σ d t \int_{0}^{2 π} d φ \int_{0}^{\frac{π}{2}} d θ L sin θ cos θ .$

When the radiation is uniform in all directions, and hence $L$ constant, the entropy radiation through $d σ$ toward one side is

(128)

$π L d σ d t .$

The speciﬁc intensity $L$ of the entropy radiation in every direction consists further of the intensities of the separate rays belonging to the different regions of the spectrum, which are propagated independently of one another. Finally for a ray of deﬁnite color and intensity the nature of its polarization is characteristic. When a monochromatic ray of frequency $ν$ consists of two mutually independent29 components, polarized at right angles to each other, with the principal intensities of energy radiation (Sec. 17) $K_{ν}$ and $K'$ , the speciﬁc intensity of entropy radiation is of the form

(129)

$L = \int_{0}^{\infty} d ν (L_{ν} + L') .$

The positive quantities $L_{ν}$ and $L'$ in this expression, the principal intensities of entropy radiation of frequency $ν$ , are determined by the values of $K_{ν}$ and $K'$ . By substitution in (127), this gives for the entropy which is radiated in the time $d t$ through the element of area $d σ$ in the direction of the conical element $d Ω$ the expression

$d t d σ cos θ d Ω \int_{0}^{\infty} d ν (L_{ν} + L'),$

and, for monochromatic plane polarized radiation,

(130)

$d t d σ cos θ d Ω L_{ν} d ν = L_{ν} d ν sin θ cos θ d θ d φ d σ d t .$

For unpolarized rays $L_{ν} = L'$ and (129) becomes

$L = 2 \int_{0}^{\infty} L_{ν} d ν .$

For radiation which is uniform in all directions the total entropy radiation toward one side is, according to (128),

$2 π d σ d t \int_{0}^{\infty} L_{ν} d ν .$

99. From the intensity of the propagated entropy radiation the expression for the space density of the radiant entropy may also be obtained, just as the space density of the radiant energy follows from the intensity of the propagated radiant energy. (Compare Sec. 22.) In fact, in analogy with equation (20), the space density, $s$ , of the entropy of radiation at any point in a vacuum is

(131)

$s = \frac{1}{c} \int L d Ω,$

where the integration is to be extended over the conical elements which spread out from the point in question in all directions. $L$ is constant for uniform radiation and we obtain

(132)

$s = \frac{4 π L}{c} .$

By spectral resolution of the quantity $L$ , according to equation (129), we obtain from (131) also the space density of the monochromatic radiation entropy:

$s = \frac{1}{c} \int (L + L') d Ω,$

and for unpolarized radiation, which is uniform in all directions

(133)

$s = \frac{8 π L}{c} .$

100. As to how the entropy radiation $L$ depends on the energy radiation $K$ Wien’s displacement law in the form of (119) affords immediate information. It follows, namely, from it, considering (133) and (24), that

(134)

$L = \frac{ν^{2}}{c^{2}} F (\frac{c^{2} K}{ν^{3}})$

and, moreover, on taking into account (118),

(135)

$\frac{\partial L}{\partial K} = \frac{\partial s}{\partial u} = \frac{1}{T} .$

Hence also

(136)

$T = ν F_{1} (\frac{c^{2} K}{ν^{3}})$

(137)

$K = \frac{ν^{3}}{c^{2}} F_{2} (\frac{T}{ν}) .$

It is true that these relations, like the equations (118) and (119), were originally derived for radiation which is unpolarized and uniform in all directions. They hold, however, generally in the case of any radiation whatever for each separate monochromatic plane polarized ray. For, since the separate rays behave and are propagated quite independently of one another, the intensity, $L$ , of the entropy radiation of a ray can depend only on the intensity of the energy radiation, $K$ , of the same ray. Hence every separate monochromatic ray has not only its energy but also its entropy deﬁned by (134) and its temperature deﬁned by (136).

101. The extension of the conception of temperature to a single monochromatic ray, just discussed, implies that at the same point in a medium, through which any rays whatever pass, there exist in general an inﬁnite number of temperatures, since every ray passing through the point has its separate temperature, and, moreover, even the rays of different color traveling in the same direction show temperatures that differ according to the spectral distribution of energy. In addition to all these temperatures there is ﬁnally the temperature of the medium itself, which at the outset is entirely independent of the temperature of the radiation. This complicated method of consideration lies in the nature of the case and corresponds to the complexity of the physical processes in a medium through which radiation travels in such a way. It is only in the case of stable thermodynamic equilibrium that there is but one temperature, which then is common to the medium itself and to all rays of whatever color crossing it in different directions.

In practical physics also the necessity of separating the conception of radiation temperature from that of body temperature has made itself felt to a continually increasing degree. Thus it has for some time past been found advantageous to speak, not only of the real temperature of the sun, but also of an “apparent” or “effective” temperature of the sun, i.e., that temperature which the sun would need to have in order to send to the earth the heat radiation actually observed, if it radiated like a black body. Now the apparent temperature of the sun is obviously nothing but the actual temperature of the solar rays,30 depending entirely on the nature of the rays, and hence a property of the rays and not a property of the sun itself. Therefore it would be, not only more convenient, but also more correct, to apply this notation directly, instead of speaking of a ﬁctitious temperature of the sun, which can be made to have a meaning only by the introduction of an assumption that does not hold in reality.

Measurements of the brightness of monochromatic light have recently led L. Holborn and F. Kurlbaum31 to the introduction of the concept of “black” temperature of a radiating surface. The black temperature of a radiating surface is measured by the brightness of the rays which it emits. It is in general a separate one for each ray of deﬁnite color, direction, and polarization, which the surface emits, and, in fact, merely represents the temperature of such a ray. It is, according to equation (136), determined by its brightness (speciﬁc intensity), $K$ , and its frequency, $ν$ , without any reference to its origin and previous states. The deﬁnite numerical form of this equation will be given below in Sec. 166. Since a black body has the maximum emissive power, the temperature of an emitted ray can never be higher than that of the emitting body.

102. Let us make one more simple application of the laws just found to the special case of black radiation. For this, according to (81), the total space density of entropy is

(138)

$s = \frac{4}{3} a^{3} T .$

Hence, according to (132), the speciﬁc intensity of the total entropy radiation in any direction is

(139)

$L = \frac{c}{3 π} a T^{3},$

and the total entropy radiation through an element of area $d σ$ toward one side is, according to (128),

(140)

$\frac{c}{3} a T^{3} d σ d t .$

As a special example we shall now apply the two principles of thermodynamics to the case in which the surface of a black body of temperature $T$ and of inﬁnitely large heat capacity is struck by black radiation of temperature $T'$ coming from all directions. Then, according to (7) and (76), the black body emits per unit area and unit time the energy

$π K = \frac{a c}{4} T^{4},$

and, according to (140), the entropy

$\frac{a c}{3} T^{3} .$

On the other hand, it absorbs the energy

$\frac{a c}{4} T'^{4}$

and the entropy

$\frac{a c}{3} T'^{3} .$

Hence, according to the ﬁrst principle, the total heat added to the body, positive or negative according as $T'$ is larger or smaller than $T$ , is

$Q = \frac{a c}{4} T'^{4} - \frac{a c}{4} T^{4} = \frac{a c}{4} (T'^{4} - T^{4}),$

and, according to the second principle, the change of the entire entropy is positive or zero. Now the entropy of the body changes by $\frac{Q}{T}$ , the entropy of the radiation in the vacuum by

$\frac{a c}{3} (T^{3} - T'^{3}) .$

Hence the change per unit time and unit area of the entire entropy of the system considered is

$\frac{a c}{4} \frac{T'^{4} - T^{4}}{T} + \frac{a c}{3} (T^{3} - T'^{3}) ≧ 0 .$

In fact this relation is satisﬁed for all values of $T$ and $T'$ . The minimum value of the expression on the left side is zero; this value is reached when $T = T'$ . In that case the process is reversible. If, however, $T$ differs from $T'$ , we have an appreciable increase of entropy; hence the process is irreversible. In particular we ﬁnd that if $T = 0$ the increase in entropy is $\infty$ , i.e., the absorption of heat radiation by a black body of vanishingly small temperature is accompanied by an inﬁnite increase in entropy and cannot therefore be reversed by any ﬁnite compensation. On the other hand for $T' = 0$ , the increase in entropy is only equal to $\frac{a c}{12} T^{3}$ , i.e., the emission of a black body of temperature $T$ without simultaneous absorption of heat radiation is irreversible without compensation, but can be reversed by a compensation of at least the stated ﬁnite amount. For example, if we let the rays emitted by the body fall back on it, say by suitable reflection, the body, while again absorbing these rays, will necessarily be at the same time emitting new rays, and this is the compensation required by the second principle.

Generally we may say: Emission without simultaneous absorption is irreversible, while the opposite process, absorption without emission, is impossible in nature.

103. A further example of the application of the two principles of thermodynamics is afforded by the irreversible expansion of originally black radiation of volume $V$ and temperature $T$ to the larger volume $V'$ as considered above in Sec. 70, but in the absence of any absorbing or emitting substance whatever. Then not only the total energy but also the energy of every separate frequency $ν$ remains constant; hence, when on account of diffuse reflection from the walls the radiation has again become uniform in all directions, $u_{ν} V = u' V'$ ; moreover by this relation, according to (118), the temperature $T_{ν}$ ’ of the monochromatic radiation of frequency $ν$ in the ﬁnal state is determined. The actual calculation, however, can be performed only with the help of equation (275) (see below). The total entropy of radiation, i.e., the sum of the entropies of the radiations of all frequencies,

$V' \int_{0}^{\infty} s' d ν,$

must, according to the second principle, be larger in the ﬁnal state than in the original state. Since $T'$ has different values for the different frequencies $ν$ , the ﬁnal radiation is no longer black. Hence, on subsequent introduction of a carbon particle into the cavity, a ﬁnite change of the distribution of energy is obtained, and simultaneously the entropy increases further to the value $S'$ calculated in (82).

104. In Sec. 98 we have found the intensity of entropy radiation of a deﬁnite frequency in a deﬁnite direction by adding the entropy radiations of the two independent components $K$ and $K'$ , polarized at right angles to each other, or

(141)

$L (K) + L (K'),$

where $L$ denotes the function of $K$ given in equation (134). This method of procedure is based on the general law that the entropy of two mutually independent physical systems is equal to the sum of the entropies of the separate systems.

If, however, the two components of a ray, polarized at right angles to each other, are not independent of each other, this method of procedure no longer remains correct. This may be seen, e.g., on resolving the radiation intensity, not with reference to the two principal planes of polarization with the principal intensities $K$ and $K'$ , but with reference to any other two planes at right angles to each other, where, according to equation (8), the intensities of the two components assume the following values

(142)

$\begin{matrix} K {cos}^{2} ψ & + K' {sin}^{2} ψ & = K'' \\ K {sin}^{2} ψ & + K' {cos}^{2} ψ & = K''' . \end{matrix}$

In that case, of course, the entropy radiation is not equal to $L (K'') + L (K''')$ .

Thus, while the energy radiation is always obtained by the summation of any two components which are polarized at right angles to each other, no matter according to which azimuth the resolution is performed, since always

(143)

$K'' + K''' = K + K',$

a corresponding equation does not hold in general for the entropy radiation. The cause of this is that the two components, the intensities of which we have denoted by $K''$ and $K'''$ , are, unlike $K$ and $K'$ , not independent or non-coherent in the optic sense. In such a case

(144)

$L (K'') + L (K''') > L (K) + L (K'),$

as is shown by the following consideration.

Since in the state of thermodynamic equilibrium all rays of the same frequency have the same intensity of radiation, the intensities of radiation of any two plane polarized rays will tend to become equal, i.e., the passage of energy between them will be accompanied by an increase of entropy, when it takes place in the direction from the ray of greater intensity toward that of smaller intensity. Now the left side of the inequality (144) represents the entropy radiation of two non-coherent plane polarized rays with the intensities $K''$ and $K'''$ , and the right side the entropy radiation of two non-coherent plane polarized rays with the intensities $K$ and $K'$ . But, according to (142), the values of $K''$ and $K'''$ lie between $K$ and $K'$ ; therefore the inequality (144) holds.

At the same time it is apparent that the error committed, when the entropy of two coherent rays is calculated as if they were non-coherent, is always in such a sense that the entropy found is too large. The radiations $K''$ and $K'''$ are called “partially coherent,” since they have some terms in common. In the special case when one of the two principal intensities $K$ and $K'$ vanishes entirely, the radiations $K''$ and $K'''$ are said to be “completely coherent,” since in that case the expression for one radiation may be completely reduced to that for the other. The entropy of two completely coherent plane polarized rays is equal to the entropy of a single plane polarized ray, the energy of which is equal to the sum of the two separate energies.

105. Let us for future use solve also the more general problem of calculating the entropy radiation of a ray consisting of an arbitrary number of plane polarized non-coherent components $K_{1}$ , $K_{2}$ , $K_{3}, \dots$ , the planes of vibration (planes of the electric vector) of which are given by the azimuths $ψ_{1}$ , $ψ_{2}$ , $ψ_{3}, \dots$ . This problem amounts to ﬁnding the principal intensities $K_{0}$ and $K'$ of the whole ray; for the ray behaves in every physical respect as if it consisted of the non-coherent components $K_{0}$ and $K'$ . For this purpose we begin by establishing the value $K_{ψ}$ of the component of the ray for an azimuth $ψ$ taken arbitrarily. Denoting by $f$ the electric vector of the ray in the direction $ψ$ , we obtain this value $K_{ψ}$ from the equation

$f = f_{1} cos (ψ_{1} - ψ) + f_{2} cos (ψ_{2} - ψ) + f_{3} cos (ψ_{3} - ψ) + \dots,$

where the terms on the right side denote the projections of the vectors of the separate components in the direction $ψ$ , by squaring and averaging and taking into account the fact that $f_{1}$ , $f_{2}$ , $f_{3}, \dots$ are non-coherent

(145)

$\begin{matrix} K_{ψ} & = K_{1} {cos}^{2} (ψ_{1} - ψ) + K_{2} {cos}^{2} (ψ_{2} - ψ) + \dots \\ o r K_{ψ} & = A {cos}^{2} ψ + B {sin}^{2} ψ + C sin ψ cos ψ \\ w h e r e A & = K_{1} {cos}^{2} ψ_{1} + K_{2} {cos}^{2} ψ_{2} + \dots \\ B & = K_{1} {sin}^{2} ψ_{1} + K_{2} {sin}^{2} ψ_{2} + \dots \\ C & = 2 (K_{1} sin ψ_{1} cos ψ_{1} + K_{2} sin ψ_{2} cos ψ_{2} + \dots) . \end{matrix}$

The principal intensities $K_{0}$ and $K'$ of the ray follow from this expression as the maximum and the minimum value of $K_{ψ}$ according to the equation

$\frac{d K_{ψ}}{d ψ} = 0 or, tan 2 ψ = \frac{C}{A - B} .$

Hence it follows that the principal intensities are

(146)

$\begin{matrix} K_{0} \\ K' \end{matrix}\} = \frac{1}{2} (A + B \pm \sqrt{{(A - B)}^{2} + C^{2}}),$

or, by taking (145) into account,

(IV.1)

$\begin{matrix} \begin{matrix} K_{0} \\ K' \end{matrix}\} = \frac{1}{2} (K_{1} + K_{2} + \dots \\ \pm \sqrt{\begin{matrix} (K_{1} cos 2 ψ_{1} & + K_{2} cos 2 ψ_{2} + \dots)^{2} \\ + {(K_{1} sin 2 ψ_{1} + K_{2} sin 2 ψ_{2} + \dots)}^{2} \end{matrix}}) . \end{matrix}$

Then the entropy radiation required becomes:

(148)

$L (K_{0}) + L (K') .$

106. When two ray components $K$ and $K'$ , polarized at right angles to each other, are non-coherent, $K$ and $K'$ are also the principal intensities, and the entropy radiation is given by (141). The converse proposition, however, does not hold in general, that is to say, the two components of a ray polarized at right angles to each other, which correspond to the principal intensities $K$ and $K'$ , are not necessarily non-coherent, and hence the entropy radiation is not always given by (141).

This is true, e.g., in the case of elliptically polarized light. There the radiations $K$ and $K'$ are completely coherent and their entropy is equal to $L (K + K')$ . This is caused by the fact that it is possible to give the two ray components an arbitrary displacement of phase in a reversible manner, say by total reflection. Thereby it is possible to change elliptically polarized light to plane polarized light and vice versa.

The entropy of completely or partially coherent rays has been investigated most thoroughly by M. Laue.32 For the signiﬁcance of optical coherence for thermodynamic probability see the next part, Sec. 119.

Chapter IV

29“Independent” in the sense of “non-coherent.” If, e.g., a ray with the principal intensities $K$ and $K'$ is elliptically polarized, its entropy is not equal to $L + L'$ , but equal to the entropy of a plane polarized ray of intensity $K + K ′.$ For an elliptically polarized ray may be transformed at once into a plane polarized one, e.g., by total reflection. For the entropy of a ray with coherent components see below Sec. 104, et seq.

30On the average, since the solar rays of different color do not have exactly the same temperature.

31L. Holborn und F. Kurlbaum, Annal. d. Physik 10, p. 229, 1903.

32M. Laue, Annalen d. Phys. 23, p. 1, 1907.