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 § 45. Equation ($$IV_b$$) admits of a similar application. Before, I give the remark that no magnetic forces exists, as long the ions are at rest, thus that $$\mathfrak{H}$$ is of same order as the velocities $$\mathfrak{v}$$. In ($$VII_b$$) we can consequently neglect the last term; it becomes $$\mathfrak{F}=\mathfrak{E}$$, consequently by ($$IV_b$$) for the interior of a body

$$Rot\ \mathfrak{\overline{E}}=-\dot{\overline{\mathfrak{H}}}$$,

and for the borderline

$$\overline{\mathfrak{E}}_{h(1)}=\overline{\mathfrak{E}}_{h(2)}$$.

At last it still follows from ($$V_b$$) and ($$VI_b$$)

and

The equations of motion for ions.
§ 46. So far everything was quite simple. Yet great difficulties arise, when we also want to form the equations of motion for the oscillating ions themselves. To express in these equations the relations, which are the basis for dispersion, birefringence, and circular polarization, it would be required an understanding of molecular processes that wasn't achieved by us by far. We want to restrict ourselves, to derive from a very simple presupposition the most probable shape of the sought relations, and then help on ourselves as good as possible. It is of course an advantage, that for this task we have to consider the interior of the homogeneous body, since (regarding the borderlines) the already derived equations enclose all required conditions.

The mentioned assumption is now, that any of the mutual completely equal molecules, only contains a single movable ion, while all others are fixed.

Let m be the mass of a movable ion, $$\mathfrak{K}$$ the total


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 force acting on it, N the number of molecules in unit volume. For the equations

$$m\frac{d^{2}\mathfrak{q}_{x}}{dt^{2}}=\mathfrak{K}_{x}$$, etc.

it follows, when we take the averages of second kind and multiply them by eN

$$m\frac{\partial^{2}\mathfrak{M}_{x}}{\partial t^{2}}=eN\overline{\overline{\mathfrak{K}_{x}}}$$, etc.

As regards $$\mathfrak{K}$$, it is at first to note, that by our assumption the fixed parts of the molecule are acting upon the ion by a certain force, that is exactly caused by the displacement $$\mathfrak{q}$$. Let the components of this force be linear, homogeneous functions of $$\mathfrak{q}_{x}, \mathfrak{q}_{y}, \mathfrak{q}_{z}$$, or rather, since only this is relevant for the following, let the averages of those components be given by

{{MathForm2|(55)|$$\left.\begin{array}{c} -\left(s_{1.1}\overline{\overline{\mathfrak{q}_{x}}}+s_{1.2}\overline{\overline{\mathfrak{q}_{y}}}+s_{1.3}\overline{\overline{\mathfrak{q}_{z}}}\right),\\ -\left(s_{2.1}\overline{\overline{\mathfrak{q}_{x}}}+s_{2.2}\overline{\overline{\mathfrak{q}_{y}}}+s_{2.3}\overline{\overline{\mathfrak{q}_{z}}}\right),\\ -\left(s_{3.1}\overline{\overline{\mathfrak{q}_{x}}}+s_{3.2}\overline{\overline{\mathfrak{q}_{y}}}+s_{3.3}\overline{\overline{\mathfrak{q}_{z}}}\right),\end{array}\right\}$$}}

in which certain constants are denoted by s.

We also assume for these forces, that they won't be changed by the translation $$\mathfrak{p}$$, at least not as regards magnitudes of first order.

§ 47. In consequence of the electric motions, also the aether exerts an action upon the ions. This can be derived from formula ($$V_b$$), since $$\mathfrak{E}=\mathfrak{F}$$ as we have seen. If it would be allowed, to put for the electric force $$\mathfrak{E}$$ everywhere the average $$\overline{\mathfrak{E}}$$, that has the same magnitude and direction at all points of an ion, then we would have to add into the expressions (55) only the terms

But this matter isn't all that simple. First, the oscillating ion itself causes a value of $$\mathfrak{E}$$, that is not the same in all points of the particle, so that we could find the corresponding part of $$\mathfrak{K}$$ only by an

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 integration over the space in which the ion is located. Second, even if we could neglect this, for the calculation of $$\overline{\overline{\mathfrak{K}}}$$ the average $$\overline{\overline{\mathfrak{E}}}$$ is of relevance, not the average $$\overline{\mathfrak{E}}$$, and it is not allowed, to mutually interchange both. Of course nothing would be in the way, in so far the motions of ions that cause the electric force $$\mathfrak{E}$$, take place in the distance P from the considered point that is much greater as the distance of the molecules, but $$\mathfrak{E}$$ is partly caused by molecules that are located more nearly — we want to say, by the oscillations within the sphere I drawn around P — and an inequality $$\overline{\mathfrak{E}}$$ and $$\overline{\overline{\mathfrak{E}}}$$ is very well possible for a irregular distribution of the thus produced states in the aether.

When we now, in agreement with these remarks and to obtain $$\overline{\overline{\mathfrak{K}}}$$, add to the expressions (55) not only the values (56) but also certain supplementary terms

$$\mathfrak{k}_{x},\ \mathfrak{k}_{y},\ \mathfrak{k}_{z}$$

and thus put

then we can maintain for the magnitudes $$\mathfrak{k}$$, that they only depend on processes within sphere I. Additionally it is given, that also the supplementary terms only exist during the displacement of the ions from their equilibrium positions and — since $$\mathfrak{q}$$ can be considered as infinitely small — they must be linear, homogeneous functions of the magnitudes $$\mathfrak{q,\dot{q}}$$, etc., or rather of their averages. In consequence of equations (48), also $$\mathfrak{k}$$ are homogeneous, linear functions of the values of $$\mathfrak{M}_{x},\mathfrak{M}_{y},\mathfrak{M}_{z},\dot{\mathfrak{M}}_{x}$$, etc. in the various points of the spherical space I. Eventually we still have to consider, that all these values can be expressed by application of 's theorem by the values, which will be assumed by $$\mathfrak{M}_{x},\mathfrak{M}_{y},\mathfrak{M}_{z},\dot{\mathfrak{M}}_{x}$$, etc., and the derivatives with respect to x, y, z in the considered point P, the center of the sphere. All these values thus are linearly included into the expressions for $$\mathfrak{k}_{x},\mathfrak{k}_{y},\mathfrak{k}_{z}$$.

To which extend these latter ones must contain the translation velocity $$\mathfrak{p}$$,


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 remains undecided for now. In any case, since we neglect magnitudes of second order, only the first powers of $$\mathfrak{p}_{x},\mathfrak{p}_{y}, \mathfrak{p}_{z}$$ will occur. If we also consider now, that in formulas (57) the magnitudes $$eN\overline{\overline{\mathfrak{q}_{x}}}$$, etc., could be replaced by $$\mathfrak{M}_{x}$$, etc., and if we think of these equations as solved with respect to $$\overline{\mathfrak{E}_{x}}$$, etc., then we can see, that these components of the electric force can be represented as linear, homogeneous functions of $$\mathfrak{M}_{x},\mathfrak{M}_{y},\mathfrak{M}_{z}$$ and their derivatives with respect x, y, z, t, and that the coefficients in these functions can linearly contain the velocities $$\mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$.

For brevity, the equations that would result in a completely developed theory for $$\overline{\mathfrak{E}_{x}},\overline{\mathfrak{E}_{y}},\overline{\mathfrak{E}_{z}}$$, may be summarized in the formula

As regards any of the vectors $$\mathfrak{M,\dot{M},\ddot{M}}$$, ..., we also have to consider the derivatives of its components with respect to the coordinates.

If we now eventually let fall our simplifying presupposition, and consider any molecule as a formation of, maybe, very complicated structure that contains several movable ions, then it is near at hand to assume, that still a relation like the one represented in (58) does exist. Our next task shall be, to simplify as much as possible the relation by means of certain, general considerations.

Simplification for transparent bodies.
§ 48. If a certain motion exists in a system, then, as it was shown in § 18, also the inverse motion is possible, as soon as forces of non-electric origin are the same for a certain location of the ions as well, as in the original case. From this it directly follows, that all motions in a body, that besides ions also contain uncharged mass particles,


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 can be reversed, in case all molecular forces are determined by the configurations and not, for example, depending on the velocities.

During the inversion of motions all velocities obtain an opposite direction, thus also the translation $$\mathfrak{p}$$. Furthermore we can easily see, — look at formulas of §§ 43 and 44 —, that in the new state at time t, the vectors

have the same direction and magnitude, as the vectors

in the original state at time -t.

Obviously, the transparent bodies, namely only those, in which the light motions are reversible in the alluded sense, and it may be clearly emphasized, that the circular polarizing substances form no exception from this rule.

We now want to see, which simplification of equation (58) is obtained from this reversibility; there, terms without and with $$\mathfrak{p}$$ shall be considered separately.

§ 49. If $$\mathfrak{p}=0$$, then it must be possible to express $$\overline{\mathfrak{E}}_{x},\overline{\mathfrak{E}}_{y},\overline{\mathfrak{E}}_{z}$$ as homogeneous, linear functions of the magnitudes $$\mathfrak{M}_{x},\dot{\mathfrak{M}}_{x},\ddot{\mathfrak{M}_{x}}$$, etc., and their derivatives with respect to the coordinates; the relations that serve for this, must stay unchanged, when we pass to the inverse motion. As to this motion we now have (at time t) $$\overline{\mathfrak{E}}_{x},\overline{\mathfrak{E}}_{y},\overline{\mathfrak{E}}_{z}$$, and also the components $$\mathfrak{M}_{x},\mathfrak{M}_{y},\mathfrak{M}_{z}$$, as well as their derivatives with respect to the coordinates, have the same value and the same sign as with the original motion (at time -t). The same is true for all even derivatives with respect to time. The uneven derivatives with respect to t have, however, the same magnitude as regards the two motions, but opposite signs, and thus these derivatives


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 cannot occur in the relations between $$\overline{\mathfrak{E}}$$ and $$\mathfrak{M}$$. To indicate this, we replace (58) for resting bodies by

If we again allow the translation, then we have to add to $$F_1$$ still another vector, whose components are linear and homogeneous functions of $$\mathfrak{M,\dot{M},\ddot{M}}$$, ..., and which contain in any term one of the factors $$\mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$; also this new vector must stay unchanged when passing to the inverse motion. As in this case the components $$\mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$ contain opposite signs, thus they can only be multiplied by such magnitudes which also change the sign, i.e. by uneven derivatives with respect to time. The equations (58) therefore generally assume the form

An additional simplifications we can achieved, by considering a certain kind of homogeneous light, i.e. by considering goniometric functions of time of a certain period T. Then

If we in (60) express in this way all even derivatives by $$\mathfrak{M}$$ and all uneven by $$\dot{\mathfrak{M}}$$, it will be given

The components of $$F_1$$ are now homogeneous functions of $$\mathfrak{M}_{x},\mathfrak{M}_{y},\mathfrak{M}_{z}$$ and its derivatives with respect to x, y, z, while $$F_2$$ depends in a similar way on $$\dot{\mathfrak{M}}$$. The coefficients of this function may well depend on the oscillation period T, since we have introduced the values (61) into (60).


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The dispersion of light.
§ 50. There are two ways of attempting to explain the dispersion of colors, either by (like ) considering from location to location the variation of the equilibrium disturbance, or by considering as relevant the variation with respect to time. In one case it is the wave length, in the other one the oscillation period, that directly determines the propagation velocity, although at the end both have the same result.

If we would take the first path and also reproduce the explanation given by — in its mathematical form — in our theory, then we would have to assume, that the equations summarized in (59) likely contain derivatives with respect to x, y, z, but not such with respect to t, and that namely, due to the smallness of m, the first term in (57) would vanishes. It is clear, that the propagation velocity must change with wave length, as soon as, for example, $$\mathfrak{M}_{x}$$, and $$\frac{\partial^{2}\mathfrak{M}_{x}}{\partial y^{2}}$$ are standing next to one another. Namely, the latter magnitude gains with respect to the first a greater influence, the smaller the wavelength.

The straight opposite assumption would be, that only derivatives with respect to t, but none with respect to x, y, z occur in formula (59). Now, in so far, that the only magnitude of the first kind (whose introduction has proven to be necessary) is the term

in equation (57), we can say that the second mentioned view reduces the phenomenon to the mass of the co-oscillating ion.

That this explanation can really be achieved now, was already proven by and earlier also by me. The new form that I now give to the theory, makes no difference in this respect.


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 As we know, mainly the phenomena of anomalous dispersion speak in favor of the assumption of co-oscillating masses. On the other hand, as regards the derivative with respect to x, y, z, it is the question, whether the terms in which they occur, are really great enough to exert a considerable influence. As we saw, the mentioned terms can only stem from the fact, that the electric moment $$\mathfrak{M}$$ doesn't have in all points of sphere I the same magnitude and direction. Since the radius is much smaller than the wave length, thus the differences are surely very insignificant, and we won't hesitate to neglect them, if it is about an action upon a distant point. Anyway, is would be premature to claim that also this small variation of $$\mathfrak{M}$$ couldn't have an influence on the phenomena in the interior of the sphere. The rotation of the polarization plane, to which we will return too, which presumable can't be understood without the aid of derivatives with respect to x, y, z, must prevent us from denying from the outset an influence of such terms on dispersion.

With more justification we can derive from the phenomena the insignificance of that influence. Namely, if we retain in equations (59) the derivatives with respect to x, y, z, and then simplify the equations, so far it is possible due to the known symmetry relations of crystals, then we are lead to laws for the motion of light, which are more complicated than the ones actually applied, and only go over into them by further simplification of the formulas, for which we cannot give any reason. For example, according to these laws the regular crystals wouldn't be isotropic, but must show a peculiar kind of birefringence.


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 The things said may justify, that we, while preliminarily the circular-polarizing media remain excluded, assume for the other transparent bodies that the relation (62) contains no derivative with respect to x, y, z. We thus put

and here we understand by $$(\mathfrak{\dot{M},p})_{x},\ (\mathfrak{\dot{M},p})_{y},\ (\mathfrak{\dot{M},p})_{z}$$ expressions, which are linear and homogeneous with respect to $$\dot{\mathfrak{M}}_{x},\ \dot{\mathfrak{M}}_{y},\ \dot{\mathfrak{M}}_{z}$$ as well as to $$\mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$. The coefficients in these expressions, as well as the factors $$\sigma$$ are to be viewed as functions of T.

Now I will prove, that for a very general class of bodies, the terms $$(\mathfrak{\dot{M},p})_{x}$$, etc. will vanish; at the same time we reach on that occasion also a simplification of the terms independent from $$\mathfrak{p}$$.

Bodies with three mutually perpendicular planes of symmetry.
§ 51. Let A be an arbitrary body, and $$A'$$ a second body that is the mirror image of the first one with respect to a certain plane E, and namely down to the smallest parts, thus also for the distribution of the smallest particles. If the molecular forces depend in such way from the configurations, that the vectors, by which they were represented in A and $$A'$$, behave like objects and their mirror images, then ion motions can occur in the two bodies in connection with state changes of aether (§ 18), so that also regarding these phenomena, one system is forever the mirror image of the other one. When passing from the first system to the second, the vectors $$\mathfrak{\overline{E},M}$$ and $$\mathfrak{p}$$ are transformed into their mirror images.

The inner construction of body A can only be such, by appropriate choice of the plane E, so that A and $$A'$$ with respect


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 to the same coordinate system have the same properties, i.e. that the properties in A and $$A'$$ can be expressed by the same equations, without change of a constant or sign. In this case we call E a plane of symmetry. The bodies which we now consider, and to which we will restrict ourselves preliminarily, are those, for which three mutually perpendicular symmetry planes of this kind exist.

We give the coordinate planes the direction with of the symmetry-planes, and consider at first the mirror image with respect to the yz-plane. When passing to this image, $$\overline{\mathfrak{E}}_{x},\mathfrak{M}_{x}$$ and $$\mathfrak{p}_{x}$$ change their sign, while the other components of $$\mathfrak{\overline{E},M}$$ and $$\mathfrak{p}$$ remain completely unchanged. This is only possible, when (after $$(\mathfrak{\dot{M},p})_{x}$$, etc. are represented as functions of $$\dot{\mathfrak{M}}_{x},\ \dot{\mathfrak{M}}_{y},\ \dot{\mathfrak{M}}_{z},\ \mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$) the index x appears in every term of the first formula once, or in every term of the second and third either not at all, or two times. To a similar conclusion we come also with respect to indices y and z. If we additionally consider the mirror images with respect to the zx- and the xy-plane, then we find, that not a single term as $$(\mathfrak{\dot{M},p})_{x}$$ is applicable, and that from the nine coefficients $$\sigma$$, only $$\sigma_{1.1}$$, $$\sigma_{2.2}$$ and $$\sigma_{3.3}$$ can be different from zero.

Thus we obtain

or

If we add these formulas to the three summarized in (53), and put

then

where, for a certain type of light, $$\varkappa_{1}, \varkappa_{1}$$ and $$\varkappa_{3}$$ are constants.


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Summary of the equations.
§ 52. Neglecting the primes over the letters — since was continue to only speak about averages — we summarize the equations of motion now in the following way.

In the interior of any body it is given

{{MathForm2|$$(V_c)$$|$$\left.\begin{array}{c} \varkappa_{1}\mathfrak{E}_{x}=4\pi V^{2}\mathfrak{D}_{x}+[\mathfrak{p.H}]_{x},\ \varkappa_{2}\mathfrak{E}_{y}=4\pi V^{2}\mathfrak{D}_{y}+[\mathfrak{p.H}]_{y}\\ \varkappa_{3}\mathfrak{E}_{z}=4\pi V^{2}\mathfrak{D}_{z}+[\mathfrak{p.H}]_{z}\end{array}\right\}$$}}

and

since, neglecting magnitudes of second order, we may replace, by the relation (53), $$4\pi\overline{\mathfrak{d}}$$ by $$\mathfrak{E}/V^{2}$$ in equation (54).

At the borderline the conditions apply

If there is no translation, then $$\mathfrak{H}'$$ falls into $$\mathfrak{H}$$; then the equations ($$III_c$$) and ($$V_c$$) go over into

and the last of the limiting conditions ($$VIII_c$$) into

Thus for this case, the known equations of motion and limiting conditions of the electromagnetic theory of light are given. From formulas $$(I_c), (II_c), (III'_c), (IV_c)$$ and $$(V'_c)$$ we derive (when $$\varkappa_1, \varkappa_2, \varkappa_3$$ are different from each other) the laws of light motion in crystals of two axes, while the assumption $$\varkappa_1=\varkappa_2=\varkappa_3$$ leads back to isotropic bodies. Besides, since $$\varkappa_1, \varkappa_2$$ and $$\varkappa_3$$ depend on the


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 oscillation period, also the explanation of the dispersion of light is contained in the formulas.

Also the case of the pure aether is not excluded. Since no electric moments $$\mathfrak{M}$$ exist in it, then we have to set by (64) $$\sigma_{1.1}=\sigma_{2.2}=\sigma_{3.3}=\infty$$ and thus $$\varkappa_{1}=\varkappa_{2}=\varkappa_{3}=1$$. The equations $$V_c$$) and ($$V'_c$$) thereby are transformed into

We can easily see, that the equations which we obtain in this way for the aether, are in agreement with formulas (I)-(V) or ($$I_b$$)-($$VII_b$$)

It is self-evident, as regards the interior of the pure aether, that the connection between the various magnitudes is always the same, the ponderable matter may be in motion or not.

Circular polarizing media.
§ 53. Bodies, which turn the polarization plane, were excluded above. It's not feasible to form a thorough theory for them until now; nevertheless some general consideration, as required by our purpose, may find their place here.

Since the rotation of the polarization plane is connected with the fact, that the medium is not in accordance in all its properties with its mirror image, then the things said in § 51 are not applicable anymore. Nevertheless, everything becomes quite easy when we restrict ourselves to isotropic media.

If we assume, in the relation between $$\overline{\mathfrak{E}}$$ and $$\mathfrak{M}$$, that no derivatives with respect to x, y, z are present, then we have to understand under $$F_{1}(\mathfrak{M})$$ of equation (62), a vector that is completely determined even by $$\mathfrak{M}$$, namely the isotropy requires that the figure consisting of $$\mathfrak{M}$$ and $$F_{1}(\mathfrak{M})$$ can be rotated in an arbitrary manner, without that $$F_{1}(\mathfrak{M})$$ ceases to fit to $$\mathfrak{M}$$. If we now choose the direction of $$\mathfrak{M}$$ itself as the rotation axis, then $$\mathfrak{M}$$


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 always remains the same vector; thus $$F_{1}(\mathfrak{M})$$ must remain unchanged, which is only possible when this vector has the direction of $$\mathfrak{M}$$. With respect to the linear character of the sought relation, we consequently have to set

where $$\sigma$$ is a scalar constant.

The second vector $$F_{2}(\mathfrak{\dot{M},p})$$ occurring in (62), has the following properties. First, its components are homogeneous, linear functions of $$\dot{\mathfrak{M}}_{x},\ \dot{\mathfrak{M}}_{y},\ \dot{\mathfrak{M}}_{z}$$ as well as from $$\mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$. Second, after an arbitrary rotation of the figure consisting of the three vectors $$\mathfrak{\dot{M},p}$$ and $$F_{2}(\mathfrak{\dot{M},p})$$, $$F_{2}(\mathfrak{\dot{M},p})$$ must still fit to $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}$$. By that we derive If we decompose $$\mathfrak{p}$$ into two components $$\mathfrak{p}_{1}$$ and $$\mathfrak{p}_{2}$$, then it follows from the first mentioned property of $$F_{2}(\mathfrak{\dot{M},p})$$

$$F_{2}(\mathfrak{\dot{M},p})=F_{2}(\mathfrak{\dot{M},p_{1}})+F_{2}(\mathfrak{\dot{M},p_{2}})$$

It is assumed, that $$\mathfrak{p}_{1}$$ falls into the direction of $$\dot{\mathfrak{M}}$$, and $$\mathfrak{p}_{2}$$ is perpendicular to it. If we now rotate the figure (consisting of $$\dot{\mathfrak{M}},\mathfrak{p}_{1}$$ and $$F_{2}(\mathfrak{\dot{M},\mathfrak{p}_{1}})$$) around an axis that falls into $$\dot{\mathfrak{M}}$$, $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{1}$$ stay were they are, and thus $$F_{2}(\mathfrak{\dot{M},\mathfrak{p}_{1}})$$ may not change as well. Consequently, this vector must have the direction of $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{1}$$. That

can be shown in addition, by means of a rotation of 180° around an axis perpendicular to $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{1}$$. In the course of this rotation, the vector $$F_{2}(\dot{\mathfrak{M}},\mathfrak{p}_{1})$$ would obtain the opposite direction; yet it shouldn't be changing, because both vectors $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{1}$$ change their sign.

To find out the direction of $$F_{2}(\dot{\mathfrak{M}},\mathfrak{p}_{2})$$, we turn the figure (which is formed by this vector with $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{2}$$) around an axis perpendicular to the plane $$(\dot{\mathfrak{M}},\mathfrak{p}_{2})$$ or $$(\mathfrak{\dot{M},p})$$, namely around 180°. Here, $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{2}$$ go over into $$-\dot{\mathfrak{M}}$$ and $$-\mathfrak{p}_{2}$$; the vector $$F_{2}(\dot{\mathfrak{M}},\mathfrak{p}_{2})$$ thus may not be changed, which is only possible when it has the direction of the axis.

Thus the vector $$F_{2}(\dot{\mathfrak{M}},\mathfrak{p}_{2})$$ — and thus by (67) also the vector $$F_{2}(\mathfrak{\dot{M},p})$$ — is perpendicular to the plane $$(\mathfrak{\dot{M},p})$$; its magnitude is proportional to the values of $$\dot{\mathfrak{M}}$$ and $$\mathfrak{p}_{2}$$. Both we have expressed in (66).

where k is a positive or negative constant, which by the way, as $$\sigma$$ above, can also depend on the oscillation period T.


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 § 54. The presupposition, that no derivatives with respect to x, y, z occur, has led us to equation (65), from which the rotation of the polarization plane does not arise. Thus it is necessary, as it was already indicated earlier to assume (at least in the expression $$F_{1}(\mathfrak{M})$$) derivatives with respect to the coordinates. The most simple is, to add to the second term of (65) another vector $$\mathfrak{R}$$, whose components do linearly and homogeneously depend on the first derivatives of $$\mathfrak{M}_{x},\ \mathfrak{M}_{y},\ \mathfrak{M}_{z}$$. Magnitude and direction will now again be closely determined by isotropy. Namely, if we imagine at any point of space a line, that represents the vector $$\mathfrak{M}$$, and in addition in the considered point the vector $$\mathfrak{R}$$, then after an arbitrary rotation of that entire figure, $$\mathfrak{R}$$ must still fit to $$\mathfrak{M}$$. Only the assumption

$$\mathfrak{R}=j\ Rot\ \mathfrak{M}$$,

is in agreement with this,


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 where j is a certain constant and which we want to add for resting bodies (65) to

$$F_{1}(\mathfrak{M})=\sigma\mathfrak{M}+j\ Rot\ \mathfrak{M}$$

Now, we could introduce (into the term $$F_{2}(\dot{\mathfrak{M}},\mathfrak{p})$$) derivatives with respect to x, y, z; however, we will omit this, since the things already said are sufficient for our purpose. By that we have (when we omit the prime over $$\mathfrak{E}$$ from now on) to put for isotropic, circular-polarizing media

§ 55. It is not without interest, to consider for a moment the mirror image of a motion to which the found equation applies. The vectors that apply to this new motion, which may be called $$\mathfrak{E'},\mathfrak{M'},\dot{\mathfrak{M}}'$$ and $$\mathfrak{p}'$$, are mirror images of the vectors $$\mathfrak{E,M,\dot{M}}$$ and $$\mathfrak{p}$$. From that if follows, that the mirror images of $$Rot\ \mathfrak{M}$$ and $$[\mathfrak{\dot{M}.p}]$$ don't fall into $$Rot\mathfrak{\ M'}$$ and $$[\mathfrak{\dot{M'}.p'}]$$, but into $$-Rot\mathfrak{\ M'}$$ and $$-[\mathfrak{\dot{M'}.p'}]$$. Now, since the linear relation between four vectors expressed in (68), also then remains when we replace any of them by its mirror image, hence

$$\mathfrak{E'}=\sigma\mathfrak{M'}-j\ Rot\ \mathfrak{M'}-k[\mathfrak{\dot{M'}.p'}]$$

By that we see, that the processes that can occur in the mirror image of the considered body, don't satisfy the relation (68) anymore, but a relation in which the terms with j and k have difference signs. Thus it is confirmed, that these terms are likely be connected with the fact, that body and its mirror image have different properties; we may expect, that a rotation of the polarization plane will actually be in agreement with them.

I postpone the details about this. Here, it only shall be remarked that the magnitude $$j\ Rot\ \mathfrak{M}$$ (we will make that the natural rotation of the polarization plane will depend on it) has much similarity with the terms, that were assumed by various physicists in the equations of motion of light,


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 to explain circular-polarization. Indeed I regard, in the absence of a theory that explains the phenomenon more deeply, the introduction of the term $$j\ Rot\ \mathfrak{M}$$ as neither better nor worse than the hypotheses of those physicists.

The last term in (68) has a peculiar meaning. Namely a rotation of the polarization plane would correspond to it, that would be caused in a body (that is different from its mirror image) by the motion of earth.


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Reduction to a resting system.
§ 56. The specification of the influence, that the motion of ponderable bodies exerts on the phenomena of light, can be achieved in a very simple manner, if we neglect circular polarization, as it will always take place in this section.

Namely we want, as we did it earlier (§ 31) already, by continuing omission of magnitudes of second order, to introduce (instead of t) the "local time"

as an independent variable; besides we want (instead of $$\mathfrak{D}$$) consider a new vector $$\mathfrak{D}'$$, which we define by the formula

If we consider an arbitrary magnitude as a function of x, y, z and t' , then (as before (§81)) we denote the partial derivative by

Furthermore, according to this notation, we shall understand by

the expression


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 and by

$$Rot'\mathfrak{A}$$

a vector with the components

$$\left(\frac{\partial\mathfrak{A}_{z}}{\partial y}\right)^{'}-\left(\frac{\partial\mathfrak{A}_{y}}{\partial z}\right)^{'}$$ etc.

The introduction of t'  and $$\mathfrak{D}'$$ gives the advantage, that (as I will show now) the equations ($$I_c$$) — ($$V_c$$) assume the same form as the formulas that apply to $$\mathfrak{p}=0$$.

§ 57. At first we obtain, by consideration of formulas (35),

$$Div\ \mathfrak{D}=Div'\ \mathfrak{D}-\frac{1}{V^{2}}\left(\mathfrak{p}_{x}\dot{\mathfrak{D}}_{x}+\mathfrak{p}_{y}\dot{\mathfrak{D}}_{y}+\mathfrak{p}_{z}\dot{\mathfrak{D}}_{z}\right)$$,

or by ($$III_c$$), if we replace (in the terms multiplied by $$\mathfrak{p}_{x},\mathfrak{p}_{y},\mathfrak{p}_{z}$$) $$\mathfrak{H}'$$ by $$\mathfrak{H}$$ and Div by $$Div'$$

Hence the equation ($$I_c$$) becomes

In a similar way

$$Div\ \mathfrak{H}=Div'\ \mathfrak{H}-\frac{1}{V^{2}}\left(\mathfrak{p}_{x}\dot{\mathfrak{H}}_{x}+\mathfrak{p}_{y}\dot{\mathfrak{H}}_{y}+\mathfrak{p}_{z}\dot{\mathfrak{H}}_{z}\right)$$

i.e., by ($$IV_c$$),

so that it can be written for ($$II_c$$)

Now let us turn to formula ($$III_c$$). In this one,


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 three equations are summarized, namely in the first of them on the left side, the expression

is stated. For that, we can write with respect to (35)

and thus for the equation itself

The two other equations admit of a similar transformation, and therefore we have

Furthermore, as regards the first of equations $$IV_c$$), this one goes over, since

into

so that ($$IV_c$$) is equivalent with

Eventually it follows from

§ 58. To introduce the new variables also into the limiting conditions, we consider the perpendicular n for the considered point, and also two directions h and k that are perpendicular to one another and to n. There, the direction n shall correspond to a rotation by a right angle from h to k. Consequently it follows from (IX) (§ 56)

Now, since $$\mathfrak{D}_{n,}\ \mathfrak{H}'_{k}$$ and $$\mathfrak{H}'_{h}$$ are steady, then this must also be so for $$\mathfrak{D}'_{n}$$.


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 In a similar manner we derive from the continuity of $$\mathfrak{H}{}_{n},\ \mathfrak{E}_{h}$$ and $$\mathfrak{E}_{k}$$, by means of the relation to be derived from ($$VI_c$$)

the continuity of $$\mathfrak{H}'_{n}$$.

If we also notice the other equations ($$VIII_c$$), then it is clear, that all limiting conditions are contained in the formulas

in which h can be now any arbitrary direction in the border surface.

§ 59. The equations $$(I_d) - (V_d)$$ and ($$VIII_d$$) differ from the equations which apply to stationary bodies by § 52, only by the fact that

has taken the place of

This coincidence opens for as a way, to treat problems regarding the influence of Earth's motion on optical phenomena, in a very simple way.

Namely, if a state of motion for a system of stationary bodies is known, where

are certain functions of x, y, z and t, then in the same system, if it is displaced by the velocity $$\mathfrak{p}$$, there can exist a state of motion, where

are exactly the same functions of x, y, z and t' [that is, $$t-\frac{1}{V^{2}}\left(\mathfrak{p}_{x}x+\mathfrak{p}_{y}y+\mathfrak{p}_{z}z\right)$$].

Although we have given (in the previous consideration) to the coordinate axes the directions of the symmetry axis, the derived theorem applies to any right-angled coordinate system. We can easily recognize this, when we consider, that for local time $$t'$$ it can also be written

where r is the line drawn from the coordinate origin to the point (x, y, z), and $$t'$$ is independent of the direction of the coordinate axes.


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 We may remember the fact, that in a moving system we always have to understand by x, y, z the coordinates with respect to the axes that share the translation.

If the magnitudes (70) are known as functions of x, y, z and t' , thus also as functions of x, y, z and t, then $$\mathfrak{D}_{x},\mathfrak{D}_{y},\mathfrak{D}_{z},\ \mathfrak{H}_{x},\mathfrak{H}_{y},\mathfrak{H}_{z}$$ can be calculated fron the equations (IX) and ($$VI_c$$).

Different applications.
§ 60. We want to call the two states of motion — in the stationary and in the moving system of bodies —, of which we have spoken so far, corresponding states. Now, they shall be mutually compared more precisely.

a. If in a stationary system the magnitudes (69) are periodic functions of t with the period T, then in the other system the magnitudes (70) have the same period with respect to $$t'$$, thus also with respect to t, when we let x, y, z remain constant. When interpreting this result, we have to consider, that two periods must be distinguished in the case of translation (see §§ 37 and 38), which we accordingly can call absolute and relative period. We are dealing with the absolute one, when we consider the temporal variations in a point that has a fixed position against the aether; but we are dealing with the relative one, when we consider a point that moves together with ponderable matter. The things found above can now be expressed as follows:

If a state of oscillation in the moving system shall correspond to a state in the stationary system, then the relative oscillation period in the first mentioned case must be equal to the oscillation period in the second mentioned case.

b. In the stationary system, no motion of light may take place at an arbitrary location, i.e., $$\mathfrak{D}, \mathfrak{E}$$ and $$\mathfrak{H}$$ may vanish at this place. At the corresponding location of the moving bodies it is consequently $$\mathfrak{D}'=0,\mathfrak{E}=0,\mathfrak{\mathfrak{H}}'=0$$, thus also $$\mathfrak{D}=0,\mathfrak{H}=0$$, so that at this place the motion of light is missing as well.

From that it directly follows, that a surface that forms the border of a space filled with light within a stationary body,


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 can have the same meaning when the body is moving.

In a stationary, homogeneous medium, for example, light bundles are possible which are limited by cylindric surfaces, if it is only assumed that the dimensions of the bisections are much greater than the wave length. By our theorem, such bundles also can exist in a moving system.

The described lines of the mentioned cylindric surfaces we call light rays, and in the case of translation: relative light rays. The cylinders we have to imagine as rigidly connected with ponderable matter; thus they form the paths for the propagation of light relative to that matter.

c. A cylindric light bundle falls upon a plane limiting-surface in a stationary system, and it will be mirrored and refracted by it, — for generality we want to say: bi-refracted. The new light bundles have a cylindric border as well. If we now apply the things said under a and b to the corresponding case of the moving system, then we come to the theorem:

In the moving system, relative light rays of relative oscillation period T were mirrored and refracted by the same laws, as rays of the oscillations period T in the stationary system.

d. Let in the stationary system be a transparent body of arbitrary form, that was hit by a cylindric light bundle, and by that an arbitrary interference- or diffraction-phenomenon occurs. If dark strips do occur on that occasion, then they must appear in the corresponding state of the moving system at exactly the same locations.

An extreme case of a diffraction-phenomenon is the unification of all light in a focus. By the preceding, the laws by which a light ray of certain cylindric limitation is concentrated by a telescope objective, won't be changed at all by a translation.

e. While in corresponding states the lateral limitation of a light ray is the same, the wave normals have different directions. If it is set, for example, that plane waves are propagating with the velocity W in the stationary system whose perpendicular has the direction ($$b_{x}, b_{y}, b_z$$), so that


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 the deviation from equilibrium is a function of

$$t-\frac{b_{x}x+b_{y}y+b_{z}z}{W}$$

then for the moving system, similar functions of

$$t'-\frac{b_{x}x+b_{y}y+b_{z}z}{W}=t-\left\{ \left(\frac{b_{x}}{W}+\frac{\mathfrak{p}_{x}}{V^{2}}\right)x+\left(\frac{b_{y}}{W}+\frac{\mathfrak{p}_{y}}{V^{2}}\right)y+\left(\frac{b_{z}}{W}+\frac{\mathfrak{p}_{z}}{V^{2}}\right)z\right\}$$

occur. The direction constants $$b'_{x}, b'_{y}, b'_z$$ of the wave normal will thus be determined for this system by the condition

$$b'_{x}:b'_{y}:b'_{z}=\left(b_{x}+\frac{W\ \mathfrak{p}_{x}}{V^{2}}\right):\left(b_{y}+\frac{W\ \mathfrak{p}_{y}}{V^{2}}\right):\left(b_{z}+\frac{W\ \mathfrak{p}_{z}}{V^{2}}\right)$$,

or, in the case of a propagation in pure aether, by

$$b'_{x}:b'_{y}:b'_{z}=\left(b_{x}+\frac{\mathfrak{p}_{x}}{V}\right):\left(b_{y}+\frac{\mathfrak{p}_{y}}{V}\right):\left(b_{z}+\frac{\mathfrak{p}_{z}}{V}\right)$$.

From this equation is is given in reverse

The aberration of light.
§ 61. Let $$b'_{x}, b'_{y}, b'_z$$ be the direction constants of the line drawn from a stationary celestial body to earth, thus also the direction constants of the perpendicular with respect to the plane waves that arrive in the vicinity of earth. So when we, to investigate the following path of propagation, relate the motion of light to a coordinate system, that shares the motion of earth, then of course the direction constants of the wave normal remain $$b'_{x}, b'_{y}, b'_z$$, while that one comes into play as the relative oscillation period $$T'$$ (§ 37), which was modified by 's law. As we have seen, the motion (as regards the lateral limitation of a light bundle cut out by a diaphragm, the concentration through lenses, and the passage through other transparent bodies) will correspond to a motion in a stationary system, for which the oscillation period


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 is $$T'$$, and the perpendicular to the incident waves has the direction constants $$b_{x}, b_{y}, b_z$$ that are to be determined by (71).

''Thus all phenomena happen exactly in such a manner, as if the earth were at rest, the oscillation period ware T', and the celestial body, as seen from earth, would be located not in the direction ($$-b'_{x}, -b'_{y}, -b'_z$$), but in the direction ($$-b_{x}, -b_{y}, -b_z$$). ''

Now, aberration exactly consists of the latter. That the magnitude and direction, which we find for it, actually corresponds to the known rule which is in accordance with observations, follows immediately from equation (71). Namely, we obtain a vector of direction ($$b_{x}, b_{y}, b_z$$), when we compose a vector of direction ($$b'_{x}, b'_{y}, b'_z$$), whose length represents the velocity of light, with a second one which is equal and opposite to Earth's velocity $$\mathfrak{p}$$.

By the way, in our theorem also lies the explanation for the fact, that during the observation by a lens system, always that aberration arises which is determined by the previously mentioned rule, as well as the explanation for the known experiments of by a prism, and for the experiment proposed by  and executed by , in which the tube of a telescope was filled with water.

Observations by sun light.
§ 62. The trajectory of Earth deviates as little from a circle,


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 that, when we are dealing with sun rays, we can neglect the velocity component $$\mathfrak{p}_{r}$$, on which the variation of the oscillation period depends (§ 37). Experiments with these rays must have the result, as if the earth were at rest, and the sun were in a direction changed by aberration, and would emanate types of light of the same oscillation period, as in reality

From that it immediately follows, that (as regards a certain line of during a refraction in a prism, or the diffraction through a lattice) we don't register any influence of Earth's motion, thus it cannot make any difference, whether the direction of light (that falls upon the prism of the lattices) would form this or that angle with the translation of earth. As regards the lattice-spectra, this result was confirmed by the careful experiments of. This physicist has additionally demonstrated by certain experiments, that as regards the mentioned spectra, the deflection for a certain line fully agrees with the deflection for the corresponding rays of a terrestrial light source.

Moving light sources.
§ 63. Above, in § 61, the celestial body was assumed to be at rest. Yet also for a moving body we arrive at a simple result. We already know (§ 36), that the perpendicular to the waves arriving at Earth, is directed to location P, where the light source


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 was present in the instant when the light was emitted. Now the motion of Earth causes, that we observe the star not at this place P, but at another place $$P'$$, namely the displacement from P to P'  can be derived by the ordinary rule for aberration. By the consideration of § 61 its prove is at hand.

Eventually a simple figure shows, that P' falls into the true place at the time of observation, as soon as the velocity of the light source agrees in magnitude and direction with that of earth.

Experiments with terrestrial light sources.
§ 64. From the results previously obtained it directly follows, that we will see a distant terrestrial object always in the direction, where it is actually located. We also have already seen, that for a light sources rigidly connected with earth, no difference exists between the true and the observed oscillation period.

In general, the motion of Earth will never have an influence of first order on the experiments with terrestrial light sources.

To justify this theorem, we want at first (by application of the superposition principle (§ 7)) derive from the formulas of § 33 other ones, which are valid for an arbitrary system of luminous molecules. On that occasion we assume, that they have the common translation $$\mathfrak{p}$$, and we choose the local time $$t'$$ specified by (34), and the relative coordinates (§ 19), as independent variables.

Let

be the locations of the molecules, and

{{MathForm2|(72)|$$\left.\begin{array}{c} \mathfrak{m}_{x(1)}=f_{1}(t'),\ \mathfrak{m}_{y(1)}=g_{1}(t'),\ \mathfrak{m}_{z(1)}=h_{1}(t'),\\ \mathfrak{m}_{x(2)}=f_{2}(t'),\ \mathfrak{m}_{y(2)}=g_{2}(t'),\ \mathfrak{m}_{z(2)}=h_{2}(t'),\\ \mathrm{etc.}\end{array}\right\} $$}}


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 or

{{MathForm2|(73)|$$\left.\begin{array}{c} \mathfrak{m}_{x(1)}=f_{1}\left(t-\frac{\mathfrak{p}_{x}}{V^{2}}\xi_{1}-\frac{\mathfrak{p}_{y}}{V^{2}}\eta_{1}-\frac{\mathfrak{p}_{z}}{V^{2}}\zeta_{1}\right),\ \mathrm{etc.},\\ \mathfrak{m}_{x(2)}=f_{2}\left(t-\frac{\mathfrak{p}_{x}}{V^{2}}\xi_{2}-\frac{\mathfrak{p}_{y}}{V^{2}}\eta_{2}-\frac{\mathfrak{p}_{z}}{V^{2}}\zeta_{2}\right),\ \mathrm{etc.},\\ \mathrm{etc.}\end{array}\right\} $$}}

be the electric moments that occur within.

The condition that was caused by a single molecule in the point (x,y,z) of the aether, will be determined by equations (39) and (40). The latter one we additionally want to transform (to subsequently apply the theorem of § 59 more conveniently) by introducing the expressions $$\mathfrak{D}$$ and $$\mathfrak{D}'$$ for the aether. For this medium, as we know, $$\mathfrak{D}$$ is equal to $$\mathfrak{d}$$, and thus by (IX) (§ 56), $$4\pi V^{2}\mathfrak{D}'$$ is equal to

By means of equation $$V_b$$) we may replace $$\mathfrak{F}$$ by $$4\pi V^{2}\mathfrak{D}'$$ in (40).

Furthermore, if we denote by $$\Sigma$$ the sum of terms, any of them stemming from a luminous molecule, then we obtain from (39) and (40) the following formulas for the condition in the aether caused by ion oscillations (72):

{{MathForm2|(74)|$$\left.\begin{array}{c} \mathfrak{H}'_{x}=\frac{\partial}{\partial t'}\left(\frac{\partial}{\partial y}\right)^{'}\left\{ \sum\left(\frac{m_{z}}{r}\right)\right\} -\frac{\partial}{\partial t'}\left(\frac{\partial}{\partial z}\right)^{'}\left\{ \sum\left(\frac{\mathfrak{m}_{y}}{r}\right)\right\} ,\ \mathrm{etc.},\\ \\4\pi\mathfrak{D}'_{x}=\left(\frac{\partial S}{\partial x}\right)^{'}-\Delta'\left\{ \sum\left(\frac{\mathfrak{m}_{x}}{r}\right)\right\} ,\ \mathrm{etc.},\\ \\S=\left(\frac{\partial}{\partial x}\right)^{'}\left\{ \sum\left(\frac{\mathfrak{m}_{x}}{r}\right)\right\} +\left(\frac{\partial}{\partial y}\right)^{'}\left\{ \sum\left(\frac{\mathfrak{m}_{y}}{r}\right)\right\} +\left(\frac{\partial}{\partial z}\right)^{'}\left\{ \sum\left(\frac{\mathfrak{m}_{z}}{r}\right)\right\} .\end{array}\right\} $$}}

Here, r denotes the distance of point (x, y, z) from the location ($$\xi, \eta, \zeta$$) of one of the luminous molecules, while $$\mathfrak{m}_{x},\mathfrak{m}_{y},\mathfrak{m}_{z}$$ represent the moments of this molecule at local time $$t'-\frac{r}{V}$$. The two first members of the sum

are for example


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when $$r_1$$ and $$r_2$$ are the distances between (x, y, z) and the two first molecules.

§ 65. From the preceding formulas, others immediately arise, which apply to a stationary light source when we simply erase all accents. If in this case in the luminous molecules the moments exist

{{MathForm2|(75)|$$\left.\begin{array}{c} \mathfrak{m}_{x(1)}=f_{1}(t),\ \mathfrak{m}_{y(1)}=g_{1}(t),\ \mathfrak{m}_{z(1)}=h_{1}(t),\ \\ \mathfrak{m}_{x(2)}=f_{2}(t),\ \mathfrak{m}_{y(2)}=g_{2}(t),\ \mathfrak{m}_{z(2)}=h_{2}(t),\ \\ \mathrm{etc.},\end{array}\right\} $$}}

then we have in the aether

{{MathForm2|(76)|$$\left.\begin{array}{c} \mathfrak{H}{}_{x}=\frac{\partial}{\partial t}\frac{\partial}{\partial y}\left\{ \sum\left(\frac{\mathfrak{m}_{z}}{r}\right)\right\} -\frac{\partial}{\partial t}\frac{\partial}{\partial z}\left\{ \sum\left(\frac{\mathfrak{m}_{y}}{r}\right)\right\} ,\ \mathrm{etc.},\\ 4\pi\mathfrak{D}{}_{x}=\frac{\partial S}{\partial x}-\Delta\left\{ \sum\left(\frac{\mathfrak{m}_{x}}{r}\right)\right\} ,\ \mathrm{etc.},\\ S=\frac{\partial}{\partial x}\left\{ \sum\left(\frac{\mathfrak{m}_{x}}{r}\right)\right\} +\frac{\partial}{\partial y}\left\{ \sum\left(\frac{\mathfrak{m}_{y}}{r}\right)\right\} +\frac{\partial}{\partial z}\left\{ \sum\left(\frac{\mathfrak{m}_{z}}{r}\right)\right\} .\end{array}\right\}$$}}

where $$\mathfrak{m}_{x},\mathfrak{m}_{y},\mathfrak{m}_{z}$$ are now the moments of a molecule at time $$t-\frac{r}{V}$$, so that e.g. the two first members of the sum

have the values

Of course, $$\xi, \eta, \zeta$$, x, y, z are now the coordinates related to stationary axes.

§ 66. The two cases considered in §§ 64 and 65 (with or without translation) shall be compared to one another. Here, we imagine that the spatial arrangement of the luminous molecules is the same in the two cases, i.e. that all $$\xi, \eta, \zeta$$ have the same value; the latter we also assume for x, y, z, with the result that we consider the state of the aether in a point


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 that has a particular location with respect to the light source. Eventually we understand by $$f_{1}, g_{1}, h_{1}, f_2$$ etc. the same function-sign for both cases.

A look upon the formulas (74) and (76) let us recognize, that we are dealing with corresponding states, on which the theorem of § 59 is applicable. If the light is incident on a non-transparent screen with one opening, then the limitation of light and shadow, or the location of dark diffraction fringes behind of it, will be the same in both cases. Also no difference in the spatial distribution of light and dark will be seen, when the rays were mirrored or refracted at an arbitrary transparent body, or when a lens concentrates them, or when some interference phenomena occur.

Of course, motions that are present in the light source itself, which generate these corresponding states, are not quite the same. In one case they will be determined by (73), and in the other case by (75). If we put

then we may thus also say:

A moving light source, in which ion motions as represented by

{{MathForm2|(77)|$$\left.\begin{array}{c} \mathfrak{m}_{x(1)}=f_{1}'(t),\ \mathfrak{m}_{y(1)}=g_{1}'(t),\ \mathfrak{m}_{z(1)}=h_{1}'(t)\\ \mathrm{etc.}\end{array}\right\} $$}}

take place, generates the same phenomena as a stationary light source, to which the formulas

{{MathForm2|(78)|$$\left.\begin{array}{c} \mathfrak{m}_{x(1)}=f_{1}'\left(t+\frac{\mathfrak{p}_{x}}{V^{2}}\xi_{1}+\frac{\mathfrak{p}_{y}}{V^{2}}\eta_{1}+\frac{\mathfrak{p}_{z}}{V^{2}}\zeta_{1}\right),\\ \\\mathfrak{m}_{y(1)}=g_{1}'\left(t+\frac{\mathfrak{p}_{x}}{V^{2}}\xi_{1}+\frac{\mathfrak{p}_{y}}{V^{2}}\eta_{1}+\frac{\mathfrak{p}_{z}}{V^{2}}\zeta_{1}\right),\\ \\\mathrm{etc.}\end{array}\right\} $$}}

apply.

If we are dealing with oscillations, then


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 the difference between (77) and (78) is reduced to a variation of the phases, namely this will be determined for an arbitrary molecule by

$$\frac{\mathfrak{p}_{x}}{V^{2}}\xi+\frac{\mathfrak{p}_{y}}{V^{2}}\eta+\frac{\mathfrak{p}_{z}}{V^{2}}\zeta$$

consequently it is not equal for the various molecules.

It is now to be noticed, that the molecules of a light source, e.g. a flame, must be considered as totally independent from one another, so that, as it is ordinarily expressed, the rays emanated by two of these particles cannot mutually interfere. From that if follows, that arbitrary variations in the phases of the single molecules cannot have any influence on the observable phenomena. The stationary light source with motions (78) will give nothing other than a stationary source (also at rest) with motions (77), and thus we may claim:

''If we set a light source into translation, without changing anything of the oscillations of their ions, then the observable phenomena in bodies rigidly connected with them, remain as they were. ''

§ 67. Numerous experiments have proven, that when using terrestrial light sources, the phenomena are indeed independent of the orientation of the devices with respect to the direction of Earth's motion. Here, the observations of, , and  on refraction do belong, as well as the experiments of the three last mentioned physicists on interference phenomena. We are indebted to for an investigation on the inner reflection and the refraction at calcite prisms. and


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 to for an investigation on the interference fringes that appear at calcite plates in polarized light.

The entrainment of light waves by ponderable matter.
§ 68. In a stationary, isotropic or anisotropic body a bundle of plane light waves propagate, as to which the components of $$\mathfrak{D}$$ and $$\mathfrak{H}$$ can be expressed by expressions of the form

thus W is the velocity of propagation. This magnitude can depend on $$b_{x}, b_{y}, b_{z}$$, and T. After we have given to the body the velocity $$\mathfrak{p}$$, a state of motion can occur in it, for which expressions like

or

apply. The direction constants $$b'_{x}, b'_{y}, b'_{z}$$ of the wave normal are now proportional to the magnitudes

If we consequently put

then (80) becomes

for which we can see, that $$W'$$ is the velocity by which the waves of relative oscillation period T are propagating


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 in the direction ($$b'_{x}, b'_{y}, b'_{z}$$) within the moving body.

From (81) we find

and for that we can write, neglecting magnitudes of second order,

Here, $$\mathfrak{p}_{n}$$ is the component of velocity into the direction of the wave normal, with which $$W'$$ is related. Eventually

§ 69. Up to now, the investigation was general. Now it shall be assumed, that the body be isotropic. The velocity W is thus independent of the direction of the waves, and also the ratio

the absolute refraction index of the stationary body, only depends on T.

When interpreting formula (82), which now passes to

we have to remember, that we have used a coordinate system for the description of the phenomena, that moves together with ponderable matter. Thus (83) is the velocity of the light waves relative to that matter. If we wish it know the relative velocity $$W$$ with respect to the aether'', we have to compose the velocity (83), which has the direction of the wave normal, with the component $$\mathfrak{p}_{n}$$ of the translation velocity (that exactly falls in that direction). By that we obtain

which is in agreement with the known assumption of.


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 As regards this result, two things shall be remarked. First, the given derivation applies to every value of T, thus for every kind of light, and second, this has to be understood, that the substitution of the values of N and W, which belong in the stationary body to a particular T, gives the value of $$W$$ for the relative oscillation period T''.

§ 70. If the considered body is birefringent, than it may not be forgotten, that W and W'  in equation (82) are related to different directions of the wave normal, namely W to direction ($$b_{x}, b_{y}, b_{z}$$), and W to direction ($$b'_{x}, b'_{y}, b'_{z}$$). Concerning the question, as to how the velocities in stationary and in moving bodies are mutually different for a given direction of the waves, the equation doesn't directly give an answer. To a simple theorem, however, leads the introduction of light rays.

In a stationary birefringent body, to any direction of the wave normal (as soon as one of the two possible oscillation directions is chosen) belongs a particular direction for the light-rays, i.e., for the describing lines of a cylindric limiting-surface of a light bundle. For the points of such a line, it is now, when $$c_{x}, c_{y}, c_{z}$$ are the direction constants, and s means the distance of a fixed point ($$x_{0}, y_{0}, z_{0}$$) of the line,

By that, when we put

and understand by $$B'$$ a new constant, the expression (79) is transformed into


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 The magnitude $$U$$ is, what we usually call the velocity of the light ray.

If we now pass to the corresponding motion in the progressing body, then the considered line remains (§ 60, b) a light ray, and we obtain for the determination of the deviations of equilibrium, in the different points of it, expressions as

or, by (34) and (85),

where $$\mathfrak{p}_{s}$$ is the component of $$\mathfrak{p}$$ in the direction of the light ray, while the new constant $$B''$$ has the value

The expression (86) goes over into

and here, $$U'$$ is the velocity of the light ray in the moving body, when we put

From that we conclude

a formula, whose shape agrees with (82), in which $$U$$ and $$U'$$ are now related to light rays of the same direction.

§ 71. Formula (84) has found a nice confirmation by the experiment, that were first executed by and later


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 repeated by and , on the propagation of light in streaming water. The arrangement of them should be sufficiently known, so that we can restrict ourselves to compare (still more deeply than it is usually happening) the results with the theory.

To apply the formula, we first have to derive the relative period from the experimental conditions, and then (from the dispersion formula for stationary water) the refraction exponent N corresponding with this period. The value of V/N calculated in this way, we eventually have to substitute into (82) for W. However, as regards the relative period, a more closer consideration is required.

It's known that, as regards these experiment, two tubes were used which are closed by glass plates and which are lying next to one another, through which the water was flowing with the same velocity, but in different direction; since the base tubes were present entirely at the edges, we may assume, that at all places (at least in the middle parts of the bisection) the same velocity $$\mathfrak{p}$$ occurred. The two light bundles, which should mutually interfere, passed through the device, so that one was propagated in both tubes in the direction of the water stream, and the other one steadily in the opposite direction.

We now consider a fixed point P in the interior of one of the tubes. The conditions, under which the light is propagating from the source to this point, obviously remain — when the water stream is stationary — constantly the same, and namely this applies to both ways, on which the rays can reach point P. Impulses, which emanate by certain periods from the source, will arrive with the same periods in P, and when T is


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 the oscillation period of the light source, then this is also the absolute oscillation period in P.

From that if follows for the relative oscillation period related to the water

where $$W'$$ is exactly the sought velocity of the waves, while (as also in the following formulas) the above or below sign is to be applied, depending on whether the considered light bundle is propagating in the direction of the water motion, or in the opposite direction.

We always neglect magnitudes of second order and thus we may put instead of (88)

Under $$W$$ in equation (82) — and also in this expression (89) itself — we have now to understand the value, which belong to period (89) in the stationary body. The corresponding refraction exponent is

in case we denote the refraction exponent for period T by n; consequently we have to substitute

or, when we replace W by $$\frac{V}{n}$$ in the last member,

Furthermore it is in (82)

so that we find

and for the relative velocity with respect to the aether, thus also with respect to the closing plates of the tubes,


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 § 72. The mentioned physicists have compared their observations, not with that formula, but with another in which the last term is missing; a satisfying agreement occurred at this place. Namely, if we put

thus the coefficient ε can be derived from the experiments. Now, while and  found in this manner

"with a possible error of $$\pm0,02$$", $$1-\frac{1}{n^{2}}$$ for D-light has the value 0,438.

By our theory it should be

or, if we consider n as a function of wave length λ in air,

For the line D, this becomes

Thus formula (90) somewhat further deviates from the observations than the simpler equation

however, the observation were possibly not as exact for allowing us to put weight to this condition.

If it should be achieved, which namely appears to be difficult but not impossible, to experimentally distinguish between the equations (90) and (91), and if the first one should be justified, then we would have observed the variation of the oscillation period for an artificially generated velocity. It is only by consideration of this variation, that we have derived equation (90).

§ 73. It is hardly necessary to recall at this place the importance of the role, which is played by formula (84) in the theory of aberration and the related


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 phenomena. based his explanation of 's prism experiment upon the value $$1-\frac{1}{N^{2}}$$ of the dragging coefficient. Subsequent scientists have applied this equations to many other cases, and have derived from it, that the motion of earth, as regards most of experiments with terrestrial light sources, is without influence, and that experiments with the light of a celestial body must give a result, as if the direction altered by aberration would be the real one. How easy the theoretical considerations are formed, when we look, not upon the direction of the waves, but on the path of light rays, I have demonstrated (following the example of ) in my treatise of the year 1887. At that time, I restricted myself to isotropic bodies, since it wasn't known to me yet, how to extend 's law for crystals. Now, since it was demonstrated, that the propagation velocity of light rays obeys in these bodies the simple law expressed in formula (87), it is easy to show, that also the birefringence of rays is independent of Earth's motion. For this purpose we can start with a simple theorem that follows from the principle of, and I allow myself to shortly state it at this place.

Let A and B two arbitrary, which may lie within different mutually adjacent media. In general, only a restricted amount of light rays can travel from one to another. If we now form (for one such ray, as well as for other ways between A and B with only small deviations) the integral

in which U means the velocity for a light rays that follows the line element ds,


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 then (by the referred theorem) the integral for the light ray is a minimum.

However, I neither want to dwell more closely on these considerations, nor on further applications of formulas (82) and (87), since we have solved the question concerning the influence of Earth's motion in different cases, already above in a much simpler way.

Closer consideration of light bundles of plane waves.
§ 74. In the application of the general theorem found in § 59, I was as brief as possible and I didn't dwell more into the details, as it just was required. For further explanation is seems justified, however, to show by some examples, as to how all details of the light motions are given from that theorem as well.

At first, we consider a light bundle of plane waves, that propagates in the aether, after it went through an additional opening in a non-transparent screen which is connected to Earth. For a moment we neglect the motion of Earth. Let:

 * l, m, n the direction constants of the wave normal,
 * q a constant,
 * f, g, h the direction constants of the dielectric displacement,
 * a the "amplitude" of the latter.

Consequently, the light motion can be represented by the equations


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 with the condition

We can easily see, that these values satisfy all equations of motion. The vectors $$\mathfrak{d}$$ and $$\mathfrak{H}$$ are perpendicular to one another and to the wave normal; the direction of the light rays (§ 60, b) falls into the latter.

§ 75. If the Earth is moving, then by the theorem of § 59 a condition is possible, which (related to a moving coordinate system), will be represented by

By $$\mathfrak{d}'$$ we have to understand a vector $$\mathfrak{D}'$$ for the pure aether, which is defined by (IX) (§ 56).

While the light rays, which determine the lateral limitation of the bundle, have still the direction (l, m, n), the wave normal deviates from it. Its direction constants l', m', n'  satisfy, as it can be seen from (98), the conditions

We will neglect all magnitudes of second order again. Then, by denoting the components of $$\mathfrak{p}$$ in the direction of the rays by $$\mathfrak{p}_{s}$$, we have

by which (98) is transformed into

While T is now the relative oscillation period, we find for the absolute one (§§ 60 and 37)


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 For the determination of $$\mathfrak{d}$$ and $$\mathfrak{H}$$, the formulas (IX) (§ 56) and ($$VI_b$$) (§ 20) can serve, which we may replace by

and

If follows

or, if we put by (99)

and if we consider (95).

By that we see, that $$\mathfrak{d}$$ and $$\mathfrak{H}$$ are both perpendicular to the wave normal, as it was expected. Additionally, both vectors are mutually perpendicular, which can be seen most easily, when by replace (100) by

Furthermore, we can conclude, that the vector $$[\mathfrak{d.H}]$$ which is present in 's theorem, falls into the wave normal. We can easily convince ourselves, that it has the direction, in which the waves are propagating, and we find for its magnitude

The energy flux through a plane which is parallel to the waves, thus amounts for the unit of area and time

§ 76. From a light bundle as the one considered above, others of the same kind can arise by refraction and mirroring at plane limiting surfaces.


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 Here, we only consider such ones, that are again propagating in the aether, and we represent (for the case that the earth is at rest) one of the bundles, which emerge from the incident motion considered in § 74, by the following formula

§ 77. With this motion only that corresponds, which (in case Earth is moving together with the reflecting or refracting body) emerges from the light represented by (96)-(98). For this new state of motion we can thus write

from which it again follows — see (100) and (101) —

In these equations $$l_{1}, m_{1}, n_{1}$$ determine the direction of the rays, which we also want to denote by $$s_{1}$$.

§ 78. In the course of mirroring or refraction, the absolute period will be changed in general, while, as it nearly goes without saying and as it is also expressed by our formulas, the relative period is the same for all relevant light bundles. The absolute period of the incident motion is (§ 75)

Also, as regards the bundle considered in the previous paragraph it becomes


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 Thus it has changed in the ratio of 1 to $$1+\frac{\mathfrak{p}_{s}-\mathfrak{p}_{s1}}{V}$$.

If e.g. the rays are falling perpendicular upon a plate, which retreats by the velocity $$\mathfrak{p}$$ in the direction of the perpendicular, then for the incident light $$\mathfrak{p}_{s}=\mathfrak{p}$$, and for the reflected light $$\mathfrak{p}_{s1}=-\mathfrak{p}$$. The variation of the absolute oscillation period during reflection will consequently be determined by the ratio $$1+\frac{2\mathfrak{p}}{V}$$.

Also in the ratio between the amplitudes of the incident and the mirrored or refracted light, an influence of Earth's motion can be seen. The amplitude of the dielectric displacement $$\mathfrak{d}$$ is namely with respect to the states of motion considered in §§ 74, 75, 76 and 77

The ratio just mentioned is

in case the earth is at rest, and

if it is moving.

In the case previously considered, where the rays are falling perpendicular to the retreating plate, the latter expression becomes

the reflected light will thus be weakened by the motion of the plate. Of course, the opposite motion would strengthen it.

Now the important question emerges, whether these variations of intensity are in accordance with the conservation of energy. To decide this matter, we have to consider, that the aether (due to the motion of light), is acting by certain forces on the mirroring or refracting body


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 (§ 17), and that these forces do some work, as soon as the body is displaced by the velocity $$\mathfrak{p}$$.

Now, we imagine (limited by plane surfaces and surrounded by aether) a transparent body K, upon which a system of plane waves is falling, and from which reflected and refracted light-bundles are emanating again. Let us put a fixed, closed surface $$\sigma$$ around it, and calculate for a time interval which is equal to the relative period T,

1°. the amount of energy A, which is flows rather in- than outwards through $$\sigma$$,

2°. the growth B of the electric energy within the surface, and

3°. the work C of the forces mentioned above.

For simplification we assume, that the amplitudes be constant, and that the body will be continuously hit by rays in the same way, which is the case, when the light source, or the diaphragm that serves to limit a bundle of sunlight, shares the translation of K. After expiration of time T, the energy itself has again the original value within the body, and even the energy located in $$\sigma$$ wouldn't be changed, when also the surface would be displaced by the velocity $$\mathfrak{p}$$. As regards the calculation of B, consequently only the energy in certain parts of space that lie in the direct vicinity of $$\sigma$$, come into consideration.

Eventually, we will find

by which its is proven, that we were always (as regards our developments) in agreement with the energy theorem.

However, I don't want to hinder myself with the verification of equation (104), since it might be preferable to treat the question more generally.


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The conservation of energy in a more general case.
§ 79. An arbitrary transparent body K shall be hit by a homogeneous light motion, whose intensity remains constant; consequently, a certain motion arises in the body and in the aether in its vicinity.

Here, when Earth is at first imagined as stationary, the components of $$\mathfrak{d}$$ and $$\mathfrak{H}$$ in the aether are certain functions of x,y,z,t, and namely as regards the last variable, goniometric functions with the period T. During a complete period, e.g. in the time interval from $$t_{0}-T$$ to $$t_0$$, equal quantities of energy must flow in- and outwards through an arbitrary surface σ that surrounds the surface, which can be expressed by 's theorem by

By assuming, that this condition is fulfilled, we want to show, that also state of motion that corresponds with that above, which can exist in the case of a translation $$\mathfrak{p}$$, satisfies the energy theorem.

If we replace in the functions, which apply to $$\mathfrak{d}_{x},\ \mathfrak{H}_{x}$$, etc. when the Earth is at rest, the time t by the "local time" $$t'$$ (§ 31), and if we understand in those functions by x, y, z the coordinates with respect to a movable system, then we obtain values of $$\mathfrak{d}'_{x},\ \mathfrak{H}'_{x}$$, etc. for the new state. From (105) it thus directly follows, that

if it is presupposed, that we choose for σ a surface, which shares the motion of the body.

§ 80. However, now the flux of energy through a fixed surface $$\sigma$$ shall now be considered. The energy flux related to its unit shall be

or, as we find from the formulas (IX) and ($$VI_b$$) (§§ 56 and 20),


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 under continuing omission of magnitudes of second order,

If we want, on that basis, to calculate the energy which flows more out- than inwards between the times $$t_{0}-T$$ and $$t_0$$, and consequently, by remarking the latter, integrate with respect to time. As regards the two latter terms, we can also think of a surface, that progresses with velocity $$\mathfrak{p}$$.

§ 81. To also arrange the integration of the first term in such a way, that we have to deal with such a movable surface, we at first set for the increase of the integral $$V^{2}\int[\mathfrak{d}'.\mathfrak{H}']_{n}d\sigma$$ at certain t, when we displace the surface $$\sigma$$ in the direction of $$\mathfrak{p}$$ about an infinitely small distance $$\epsilon$$, the sign

$$\varkappa\ \epsilon$$,

where $$\varkappa$$ is of course a very special function of t. Furthermore, we think of a surface $$\sigma_0$$, which falls into $$\sigma$$ at time $$t_0$$, yet which is rigidly connected with earth. Then, at time t the "distance" of $$\sigma$$ and $$\sigma_0$$ has the value $$\mathfrak{p}(t_{0}-t)$$, which is to be considered as infinitely small, and our integral for the fixed surface $$\sigma$$ amounts

$$\mathfrak{p}\varkappa(t_{0}-t)$$,

more than for $$\sigma_0$$. The time integral, about which we speak eventually, is thus about

greater than the time integral taken for $$\sigma_0$$, and, since the latter vanishes by (106), we have only to deal with the value (108).

By the way, in $$\varkappa$$ we don't have to consider the magnitudes containing $$\mathfrak{p}$$, and thus we may understand, since with this omission


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 $$V^{2}\int[\mathfrak{d}'.\mathfrak{H}']_{n}d\sigma$$,

is the energy flux, under

$$\varkappa\ \epsilon$$

the difference calculated for the unit time, and under

$$\varkappa\ \epsilon\ dt$$

the difference calculated for the element dt, of the energy fluxes through two fixed surfaces that are mutually distant by the length ε.

Now, let $$Q\epsilon$$ be the energy, which at time t is more surrounded by our surface $$\sigma$$ in its fixed location, as when this surface would be displaced by $$\epsilon$$ in the direction of $$\mathfrak{p}$$; then we immediately see, that

By that, and furthermore by partial integration, (108) is transformed into

$$\mathfrak{p}\int_{t_{0}-T}^{t_{0}}\frac{dQ}{dt}(t_{0}-t)dt=-\mathfrak{p}TQ_{t=t_{0}-T}+\mathfrak{p}\int_{t_{0}-T}^{t_{0}}Q\ dt$$,

or

$$-\mathfrak{p}TQ_{t=t_{0}}+\mathfrak{p}\int_{t_{0}-T}^{t_{0}}Q\ dt$$,

since, except magnitudes of order $$\mathfrak{p}$$, Q has again the original value after the expiration of time T.

§ 82. Until now, we only spoke about the first member in (107). If we denote the two other members by A, then we have

$$-\mathfrak{p}TQ_{t=t_{0}}+\mathfrak{p}\int_{t_{0}-T}^{t_{0}}Q\ dt+\int_{t_{0}-T}^{t_{0}}dt\int A\ d\sigma$$

the complete value of the energy, that travelled outwards through $$\sigma$$. If we add the increase of the energy in the interior of $$\sigma$$, and the work of the forces, by which the


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 aether is acting on the ponderable body, then we must, if the energy theorem shall be satisfied, obviously obtain zero.

The increase energy in a full period T would be zero, if the surface $$\sigma$$ together with the body K would be displaced over the distance $$\mathfrak{p}T$$, and at this place would have taken the location $$\sigma$$; it factually consists of the energy amount, which, at time $$t_0$$, is more contained in $$\sigma$$ than in $$\sigma$$. This is now, as it follows form the definition given for Qε, exactly

$$\mathfrak{p}TQ_{t=t_{0}}$$.

The work mentioned above can be expressed, as we will see soon, by an expression of the form

$$\int_{t_{0}-T}^{t_{0}}dt\int S\ d\sigma$$;

thus the energy law requires that

$$\mathfrak{p}\int_{t_{0}-T}^{t_{0}}Q\ dt+\int_{t_{0}-T}^{t_{0}}dt\int A\ d\sigma+\int_{t_{0}-T}^{t_{0}}dt\int Sd\sigma=0$$

If it is additionally achieved, to represent Q as an integral over $$\sigma$$, e.g. in the form

$$Q=\int\ q\ d\sigma$$

and to show, that

then we have achieved our goal.

§ 83. From the definition given for $$Q\epsilon$$ we derive, that by $$q\epsilon d\sigma$$ we have to understand the energy content of the space, which is traversed by the element during the displacement ε, and namely we have, depending on whether the displacement takes place with respect to the inner- or the outer-side of $$\sigma$$, to apply the positive or the negative sign. Thus we have

$$q\epsilon d\sigma=-\epsilon\cos(\mathfrak{p},n)\left(2\pi V^{2}\mathfrak{d}^{2}+\frac{1}{8\pi}\mathfrak{H}^{2}\right)d\sigma$$,

and

$$\mathfrak{p}q=-\mathfrak{p}_{n}\left(2\pi V^{2}\mathfrak{d}^{2}+\frac{1}{8\pi}\mathfrak{H}^{2}\right)$$.


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 Second, as regards the work, we don't have to care about the last member in equation (15) and the analogous formulas. Only the "tensions" come into account, and

is the work of the tension with respect to $$d\sigma$$. The components of this tension are

from which it follows

Eventually, A means the sum of the two last members in (107).

The given values now actually satisfy the condition (109).


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The rotation of the polarization plane.
§ 84. As the equations of motion of light for an isotropic body that has not the same properties as its mirror image, we have to assume by the considerations of the 4th section:

where by $$\mathfrak{E,d,H}$$ and $$\mathfrak{H}'$$ we have to understand averages.

We now want to presuppose, that the velocity $$\mathfrak{p}$$ would have the direction of the x-axis, and to study the propagation of plane waves, whose normal coincides with that axis as well.

§ 85. To find a particular solution of the equations corresponding to such waves, we put

where a, ν, n and m are constants. Already by that, the condition ($$II_e$$) is satisfied.


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 Now, the equation ($$IV_e$$) will be satisfied by us, by putting

and then it follows from ($$V_e$$), ($$VI_e$$) and ($$III_e$$), one after the other,

whose latter values are also in agreement with condition ($$I_c$$).

Eventually we derive from (X)

and then we only have to satisfy condition (XI).

The first one of the herein summarized relation gives nothing new, while the second and third ones read:

and

Now, since by the reported formulas

it can be written for (110) and (111)

and

thus at first we find


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 and furthermore

Now, if $$\sigma, j, k$$ and n are given, we can determine m from this equation, namely we obtain two values, depending on whether we apply the above, or the below sign.

§ 86. We put

$$n = i n',\ m = i m'$$,

by that, equation (112) is transformed into

from which two real values are given from $$m'$$, which we want to denote by $$m'_1$$ and $$m'_2$$.

For $$\nu=+i,\ m'=m'_1$$, it becomes now

$$\mathfrak{H}_{y}=ae^{i(n't-m'_{1}x)},\ \mathfrak{H}_{z}=iae^{i(n't-m'_{1}x)}$$,

and for $$\nu=-i,\ m'=m'_2$$,

$$\mathfrak{H}_{y}=ae^{i(n't-m'_{2}x)},\ \mathfrak{H}_{z}=-iae^{i(n't-m'_{2}x)}$$.

If we eventually take the real parts, we arrive at the following two particular solutions

which obviously represent two opposite, circular-polarized light beams of propagation velocities $$n'/m'_1$$ and $$n'/m'_2$$.

The composition of these states of motion leads in a known way to a beam of linear-polarized light, whose oscillation direction gets rotated. Namely, addition of the values (114) and (115) gives the solution

The rotation $$\omega$$ of the polarisation plane related to unit volume, consequently amounts

$$\omega=\frac{1}{2}(m'_{1}-m'_{2})$$.


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 § 87. If we replace in equation (113), $$\mp j$$ by $$\alpha$$, and $$\mp k\mathfrak{p}_{x}$$ by $$\beta$$, it follows

$$4\pi V^{2}\frac{n'}{m'}=(\sigma+\alpha m'+\beta n')\left(V^{2}\frac{m'}{n'}-\frac{n'}{m'}-2\mathfrak{p}_{x}\right)$$.

Since the terms with $$\alpha, \beta$$ and $$\mathfrak{p}_{x}$$ are in any case very small, the value of m following from it, can be represented by a row that progresses with respect to the powers of $$\alpha, \beta$$ and $$\mathfrak{p}_{x}$$. The first term independent of these magnitudes, has the value

$$m'_{0}=n'\sqrt{\frac{4\pi}{\sigma}+\frac{1}{V^{2}}}$$,

and then we also find

where we didn't calculated the three latter terms more closely, and we have neglected all higher powers of $$\alpha$$ and $$ \beta$$, as well as all terms that include $$\mathfrak{p}_{x}^{2}$$. To these latter ones, also the terms with $$\beta^2$$ and $$\beta\mathfrak{p}_{x}$$ do belong, since $$\beta=\mp k\mathfrak{p}_{x}$$.

Now, we obtain $$m'_1$$, or $$m'_2$$, depending on whether we put $$\alpha=-j,\ \beta=-k\mathfrak{p}_{x}$$, or $$\alpha=+j,\ \beta=+k\mathfrak{p}_{x}$$. The sought rotation of the polarization consequently becomes

$$\omega=\frac{2\pi}{\sigma^{2}}n'^{2}\left(1+\frac{n'}{m'_{0}}\frac{\mathfrak{p}_{x}}{V^{2}}\right)j+\frac{2\pi}{\sigma^{2}}\frac{n'^{3}}{m'_{0}}\mathfrak{p}_{x}k$$,

or, when we denote the propagation velocity $$\frac{n'}{m'_{0}}$$ by W,

$$\omega=\frac{2\pi}{\sigma^{2}}n'^{2}\left(1+\frac{W\mathfrak{p}_{x}}{V^{2}}\right)j+\frac{2\pi}{\sigma^{2}}n'^{2}W\mathfrak{p}_{x}k$$.

The natural rotation of the polarization plane in stationary bodies would consequently be

if we were allowed to consider as constant $$\sigma$$ and j, then it would be, as it follows from the meaning of $$n'$$, proportional to the square of the oscillation time. It's known that all bodies deviate


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 more or less from this law; but we already know, that $$\sigma$$ is changing with the duration of oscillation, and j probably might depend on it as well.

The translation has two influences by our equation. First, it changes the already existing rotation in the ratio

and furthermore it additionally causes a rotation.

The theory cannot give a relation between this value and (116); probably such a relation doesn't exist at all, and cases could exist, in which j is very small, while k nevertheless has a noticeable value.

By the way, it is probably not required to be remarked, that the phenomena represented by (118) are similar to the polarization in so far, as it also only arises by an exterior influence, namely by the translation, and most strongly emerges, when this influence has the direction of the light rays.

§ 88. Experiments on the rotation of the polarization plane at different orientation, as far as I know, were only undertaken by He was unable to conclude a change of rotation with respect to quartz, when the light rays have, on one hand, the direction of Earth's motion, and on the other hand, the opposite direction. For the observation it had to be concluded, that the change in any case didn't amount the 20000th part of the rotation, and as regards a certain direction of the light rays, the rotation was altered by Earth's motion by less than 1/40000.

Due to the lack of a theory applicable for anisotropic bodies, we maybe also apply the above reported formulas to quartz. Now, since the refractive index is 1,55, and $$\mathfrak{p}_{x}/V=1/10000$$, then the value of the second member in (117) becomes


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 0,000064. The change of rotation caused by that, could not have been overlooked by, and thus his negative result can only by explained by the assumption, that, in the formula for $$\omega$$, k has a value comparable with $$j/V^2$$, and the opposite sign of j.

Now, whether (for quartz and other bodies) the two terms containing $$\mathfrak{p}_{x}$$ will mutually be cancelled, or whether an observable influence of Earth's motion remains finally, has to be decided by additional investigations.

The interference experiment of Michelson.
§ 89. As it was first noticed by, and which follows from a very simple calculation, the time required by a light ray to travel forth and back between two points A and B must change, as soon as these points are subject to a common displacement, without dragging the aether. Although the variation is a magnitude of second order, it is nevertheless big enough that it can be demonstrated by means of a sensitive interference method.

The experiment was executed by in the year 1881. His apparatus, a kind of interference-refractor, had two equally long, horizontal, mutually perpendicular arms P and Q, and from the two mutually interfering light beams, one went forth and back along arm P and the other one along arm Q. The whole instrument, including the light source and the observation device, could be rotated around a vertical axis, and especially the two locations come into consideration, at which arm P or arm Q had (so far as possible) the direction of Earth's motion. Now, during the rotation from one "main-position" into the other, a displacement of the interference fringes was expected on the basis of 's theory.


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 However, the change in this displacement caused by the variation of the propagation times — we want to call it 's displacement for sake of brevity — was found, and thus thought that he is allowed to conclude that the aether wouldn't remain at rest when the Earth is moving, a conclusion however, whose correctness was soon questioned. By inadvertence, has estimated the change of the phase differences as expected by the theory, to double of the correct value; if we correct this error, we arrive at displacements, which just could be hidden by the observational errors. Together with, has started again the investigation, where (to increase the sensitivity) he let reflect every light beam by some mirrors back and forth. This artifice gave the same advantage, as if the arms of the earlier apparatus would have been considerably extended. The mirror was carried by a heavy stone plate, that floated on mercury and thus was easily rotatable. Altogether, every beam had to traverse a path of 22 meters, and by 's theory, when passing from one main-position to the other, a displacement of 0,4 of the fringe-distance was to be expected. Nevertheless, during the rotation only displacements of at most 0,02 of the fringe-distance were obtained; they probably might stem from observational errors.

Now, is it allowed to assume on the basis of this result, that the aether shares the motion of Earth, and thus ' aberration theory is the correct one? The difficulties, with which this theory is confronted when explaining aberration, seem too great to me as for having that opinion, so I rather should try to remove the contradiction between 's theory and 's result. Indeed


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 this can be achieved by means of a hypothesis, which I already have spoken out some time ago, and to which, as I found out later, also arrived. Of which the hypothesis consists, shall be shown in the next §.

§ 90. For simplification we want to assume, that we would work with an instrument as that during the first experiments, and that with respect to one main-position, the arm P coincides exactly with the direction of Earth's motion. Let $$\mathfrak{p}$$ be the velocity of this motion, and L the length of every arm, thus 2L the path of the light rays. Then by the theory, the translation causes that the time, in which one light-beam travels forth and back along P, is longer by

than the time, in which the other beam completes its path. Exactly this difference would also exist, when (without that the translation would have an influence) arm P would be longer by

than arm Q. Similar things are true for the second main-position.

Thus we see, that the phase difference expected by the theory could also arise, when (during the rotation of the apparatus) sometimes one, sometimes the other arm would have the greater length. From that if follows, that they can be compensated by


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 opposite variations of the dimensions.

If we assume, that the arm lying in the direction of Earth's motion, is shorter by

than the other one, and simultaneously the translation would have an influence which follows from 's theory, then the result of Michelson's experiment is fully explained.

Consequently we have to imagine, that the motion of a rigid body, e.g. a brass rod or of the stone plate used in later experiments, would have an influence on the dimensions throughout the aether, which, depending on the orientation of the body with respect to the direction of motion, is different. E.g, if the dimensions parallel to the direction of motion would be changed in the ratio of 1 to $$1+\delta$$, and the dimensions perpendicular to them by a ratio of 1 to $$1 + \epsilon$$, than it should be

Here, the value of one of the magnitudes $$\delta$$ and $$\epsilon$$ would remain undetermined. It could be $$\epsilon=0,\ \delta=-\frac{\mathfrak{p}^{2}}{2V^{2}}$$, but also $$\epsilon=\frac{\mathfrak{p}^{2}}{2V^{2}},\ \delta=0$$, or $$\epsilon=\frac{\mathfrak{p}^{2}}{4V^{2}}$$, and $$\delta=-\frac{\mathfrak{p}^{2}}{4V^{2}}$$.

§ 91. As strange as this hypothesis would appear at first sight, nevertheless one must admit that it's not so far off, as soon as we assume that also the molecular forces, similarly as we now definitely can say it of the electrical and magnetic forces, are transmitted through the aether. If this is so, then the translation will change the action between two molecules or atoms most likely in a similar way, as the attraction or repulsion between charged particles. Now, since the shape and the dimensions of a fixed body are, in the last instance, determined by the intensity of the molecular effects,


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 then also a change of the dimensions is inevitable.

Thus from a theoretical perspective there is no objection to the hypothesis. As regards the experimental confirmation, it is to be noticed at first, that the relevant elongations and contractions are extremely small. We have $$\mathfrak{p}^{2}/V^{2}=10^{-8}$$, and thus (in case we put $$\epsilon = 0$$) the contraction of one diameter of Earth would amount ca. 6,5 c.M. The length of a meter rod, however, changes by $1/200$ Micron (when we bring it from one main-position into the other). If we would like to observe magnitudes so small, then we probably can hope to succeed only by an interference method. Thus we would have to work with two mutually perpendicular rods, and of two mutually interfering light beams, let one travel back and forth with respect to the first rod, and the other with respect to the second rod. By that we come again, however, to 's experiment, and we wouldn't observe any displacement of the fringes during the rotation. In reverse as we have expressed it earlier, we could say now, that the displacement stemming from the changes of length, is compensated by 's displacement.

§ 92. It is noteworthy, the we are led exactly to the above presupposed changes of dimensions, when we first (without consideration of the molecular motion) assume, that in a rigid body which remains at its own, the forces, attractions or repulsions which act on an arbitrary molecule, are mutually in equilibrium, and second — for which, however, there is no reason — when we apply to these molecular forces the law which we have derived in § 23 for the electrostatic actions. If we understand by $$S_1$$ and $$S_2$$, not two systems of charged particles as in that paragraph, but two systems of molecules, — the second at rest and the first with the velocity $$\mathfrak{p}$$ in the direction of the $$x$$-axis —, between whose dimensions the relation given early exists, and if we assume, that in both systems the x-components of the forces are the same, but the y- and z-components are mutually different by


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 the factors given in § 23, then it is clear, that the forces in $$S_1$$ will be mutually canceled, as soon as this happens in $$S_2$$. Consequently, if $$S_2$$ is the state of equilibrium of a stationary, rigid body, then in $$S_1$$ the molecules have exactly those positions, in which they can remain under the influence of translation. The displacement would of course cause this configuration by itself, and thus by (24) it would cause a contraction in the direction of motion in the ratio of 1 to $$\sqrt{1-\frac{\mathfrak{p}^{2}}{V^{2}}}$$. This leads to the values

$$\delta=-\frac{\mathfrak{p}^{2}}{2V^{2}},\ \epsilon=0$$,

which is in agreement with (119).

In reality the molecules of a body are not at rest, but there exists a stationary motion in every "equilibrium state". As to how this condition is of influence as regards the considered phenomenon, may remain undecided; in any case, due to inevitable observational errors, the experiments of and  let remain a considerable wide margin for the values of $$\delta$$ and $$\epsilon$$.

The polarization experiments of Fizeau.
§ 93. In the oblique passage of a polarized light beam through a glass plate, the azimuth of the polarization changes in general, namely this phenomenon is depending on the nature of the plate, so that the increase or decrease of its refractive index is followed by a rotation of the polarization plane of the emanating light. This fact was the starting point for the experiments with glass columns, executed by, whose results deserve our attention to a high degree. The apparatus employed, consisted of a polarized prism,


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 a number of glass columns located after one another, and an analyzer. At the time of solstice, mostly at noon, the devise was turned at first with the polarizator into the east, and with the analyzer into the west, then they were brought into the opposite direction, while in the whole time, a beam of light rays was sent through by means of appropriately located mirrors. Although some irregularities showed up in the settings of the analyzer, yet altogether, a constant difference between the obtained readings for both locations seemed to exist.

When I developed the present theory, I hoped at first to be able to explain this difference, but soon I found myself disappointed in my expectation. If the equations developed by me are correct, then an influence, as the one expected by, cannot exist. The prove for that should be given by the next paragraph.

§ 94. Since we were working with white light, and the rotation of the polarization plane in the glass columns is not the same for all colors, so it was necessary to compensate the dispersion that arose from it. For that, circular-polarizing fluids were used, e.g. lemon oil or turpentine, and sometimes thin quartz plates that were cut perpendicular to the axis. For simplicity, we want to assume however, that light is homogeneous and therefore that no such substances are available in the apparatus. The theorem derived in § 59, is then readily applicable as it applies to an arbitrary system of refractive or birefringent bodies.

Now, an ideal experiment with respect to a stationary earth shall be compared with a real experiment, in which the apparatus in relation to Earth's motion is oriented in an arbitrary way. In the first case, the polarizer shall receive rays from the direction s and the oscillation period T; here, we imagine the analyzer thus placed that it does not transmit light. In the latter case the "corresponding" state of motion (§ 59) shall exist. For that, the incident light must have the relative oscillation period T(§ 60 a), and still have the ray-direction s (§ 60, b ). Behind the


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 analyzer, it will be dark again (§ 60, b), and we may therefore conclude:

Which direction Earth's motion may have, whether from the polarizer to the analyzer, or vice versa, light will always be erased at the presupposed position of the analyzer, as long as nothing is changed as regards the relative period of oscillation and the direction of the rays in relation to the apparatus.

Obviously, these conditions would have been met by the experiments, when the sun would have emitted white light. The relative oscillation period would thus have been as it is required by 's law, and namely at each position of the apparatus. As for the direction of the beams in relation to the glass columns, it has probably not been exactly the same with respect to the various readings; however, this has not caused an error, since an influence of a small directional change of the incident light would hardly have been overlooked by the observer.

§ 95. The phenomenon that was expected by and what he really believed to have observed, would have to occur even when using homogeneous light. Thus, here we come to a contradiction, that I can not solve. A source of error, of which one could say that it would have caused the differences in the analyzer locations, I could not discover. The activated circular-polarizing substances were probably a little too thick to allow a prominent influence of Earth's motion considered in § 87. Nor is it possible to think of an effect of terrestrial magnetism. The only thing might yet be, that the two mirrors located east and west of the apparatus, have not always received light of the same nature. Namely, to reflect rays of the sun, sometimes by one, and sometimes by the other mirror, the heliostat had to have different positions; between the angles at which he threw back light in both cases, there was a difference depending on the position of the sun, and we know that light reflected by a metal surface, has not the same composition in all directions of incidence.