Page:Elementary Text-book of Physics (Anthony, 1897).djvu/484

470 only necessary to take account of the time that elapses between the impulse in the direction $$cd$$ and the corresponding impulse in the direction $$ab.$$ It is sufficient to consider any particle as actuated by two vibratory motions in the directions $$cd$$ and $$ab$$ at right angles, and differing in phase. In Fig. 149, one side of the rectangle represents the greatest displacement in the direction $$cd,$$ and the other side the displacement occurring at the same instant in the direction $$ba.$$ The point $$r$$ will represent the actual position of the vibrating particle. Constructing now the successive displacements of the particles in the directions $$cd$$ and $$ba$$ and combining these, we have the elliptical path as shown. As the light penetrates farther and farther into the plate the relative phases of the two vibrations change continually, and the ellipse passes through all its forms from the straight line $$yy$$ to the straight line $$xx$$ at right angles to it and back to the straight line $$yy.$$ The direction of the path of the particle in the surface of the plate as the light emerges will be the direction of the path of all the particles in the polarized beam beyond the plate. If the component vibrations be in the same phase, that is, if they reach their elongations in the directions $$ba$$ and $$cd$$ (Fig. 149) at the same instant, the resultant vibration is in the line $$yy$$ and the light is plane polarized exactly as it left the polarizer. This will occur when the retardation of light in the plane of ha with respect to that in the plane of $$cd$$ is one, two, or more whole wave lengths. When the retardation is one half, three halves, or any odd number of half wave lengths, the phases of the two vibrations are as shown in Fig. 150, and the resultant is a plane polarized beam the vibrations of which are at right angles to those of the beam from the polarizer. A case of special interest is shown in Fig. 151, in which the difference of phase is one fourth a period, and the resultant vibration is a circle. A difierence of three fourths will give a circle also, but with the rotation in the