must be allowed to have any arbitrary value. This can also be expressed as: If we decompose the light vector into two components, of which one has the direction of the beam, while the other is perpendicular to it, then the latter can be arbitrarily rotated around the beam, and in every direction, the ratio is determined between the two.
The state of motion is thus completely known, once the nature of the body, the translation, the relative period, the ray direction and finally the direction and magnitude of the "transverse" component of the light vector, are given. At the places where we will later speak of the light vector, we will only think of that transverse component.
Now, if this vector in the incident light is perpendicular to the plane of incidence, it must also have the same direction in the reflected and transmitted beams; in the same way, also the light vector in these beams must be parallel to the plane of incidence, as soon as the light vector of the incident light lies in that plane. To justify these theorems, we only have to consider the mirror image of the entire state of motion in relation to the first plane of symmetry. For example, the light vector of the incident light might have the first of those directions. In the transition to the mirror image, this vector gets the opposite direction, or, as it can also be said, the opposite phase; the light vector of the other two light beams now must be changed in the same way, hence the accuracy of the above claim follows immediately.
The problem is now reduced to the two main cases, i.e. that the light vectors are everywhere perpendicular to the plane of incidence, or are everywhere located in its interior. In the course of the further investigation, we always have to think of one of these cases; however, it applies to one case as well as to the others.
As regards each light path, we call a certain direction of the light vector positive, and namely, this direction shall be the same for all the light paths in the first main-case, while in the second main-case the positive directions chosen for 2 and 4 are mirror images of those adopted for 1 and 3 with respect to the second plane of symmetry.
Eventually, in order to represent the vibrations conveniently,
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