176 On a New Case of Interference 
ject, lead to an opposite conclusion, and tend to ascribe the effect which is found to 
take place to reflexion at the surface of the denser medium. In fact, it would ap- 
pear from Fresnex’s conclusions, that the sign of the vibratory movement is in all 
cases changed by reflexion at the surface of the denser medium, the angle of incidence 
exceeding the polarizing angle ; and it can readily be shown that this change of sign 
. 1s equivalent to the addition of + 7 to the phase. 
The present case of interference seems to support this view. It follows, as we have 
seen, from theory, that if the light undergoes no change of phase by reflexion, the 
distances of the successive dark fringes from the edge of the shadow will be as the odd 
numbers 1, 3, 5, &c.; so that the distance of the first dark band from the edge will be 
half the interval between each succeeding pair of dark bands. But it appears, on the 
contrary, from the phenomena, that the distance is, as far as the eye can judge, exactly 
equal to the succeeding intervals ; or that the bands are all shifted from the edge by 
the amount of half an interval. The phenomena, therefore, require us to suppose 
that the phase of the reflected wave is accelerated, and that the amount of this accele- 
ration is exactly half a phase, or z. For the general expression for the shifting 
of the bands is {A cotan a; and as this is found to be equal to 1A cotan a, it follows 
that »=1, or the acceleration equal to 7. It appears then that when light is reflected 
at the surface of a denser medium, the wave—at the limiting incidence at least—gains 
half an undulation at the instant of reflexion. 
In order to satisfy myself more fully of the effects of reflexion upon the phase, I 
repeated the experiment with polarized light. The light was polarized, before it 
reached the aperture in the screen, by transmission through a good tourmaline; and 
the fringes were observed in various positions of the plane of polarization with respect 
to the plane of reflexion. I could detect, however, no sensible difference in the posi- 
tion of the fringes under all these changes of circumstance ; and, in particular, the 
distance of the first dark band from the edge of the shadow seemed, as before, to be 
precisely equal to the intervals of the succeeding bands, whether the light was polarized 
in the plane of incidence, or the plane perpendicular to it. 
This result seems to be just what might be expected from Fresnex’s theory of re- 
flexion. From this theory it appears that if + a@ be the coefficient of the displacement 
of the incident ray, or the amplitude of the vibration, and 7 and? the angles of inci- 
dence and refraction, the coefficients of displacement of the reflected ray will be 
sin (7—7’) tan(7—7’) 
sin(Z+7) ’ oF 9 tani 7)? 
according as the plane of polarization coincides with the plane of reflexion, or is per- 
pendicular to it. Now the former quantity is always negative, so long as @ is greater 
than 7, or the ray incident on the surface of the denser medium. Under the same 
circumstances, the latter quantity is positive or negative, according as i+? is less or 
