482 M. L. Lorenz on the Reflexion of Light at 
the angle of incidence being called x, and that of refraction z,, 
the ratio of the amplitudes of the incident, refracted, and reflected 
light, according to Fresnel, is 
2cosasinz, sin (#—z2z,) 
"sin (w+ 2) ‘sin (w@+a,) (2) 
1 1 
For the light polarized perpendicularly to the plane of incidence, 
the ratio of the same three amplitudes is 
2cosx sin 2, _ _ tan (e@—2,) (2) 
"sin (@+2,)cos(w—az,) © tan (w@+ 2) 
Assume now that these formule are correct when the difference 
between x and 2, is infinitely small, so that 2, =x+dz. - Sub- 
stituting this value of x, the above expressions become - 
dz dx 
sin 22 ) sin 2a" 
de dz 
sin 2a ° tan 2a” 
L:1+ 
1+ (4) 
We suppose that the incident ray approaches the bounding sur- 
face of the media at an angle a, and that its direction is there 
gradually changed by having to traverse successive parallel refrac- 
tive layers, until it emerges completely into the other medium 
at the constant angle £. 
In order to simplify the calculation, we will in the first in- 
stance neglect the retardation of the ray. 
Let A be the amplitude of the incident ray, and let this 
become y and y+dy for the refracted ray, when the angle of 
incidence, a, becomes xz and x+dz. Then, whatever may be 
the polarization, we have, according to (3) and (4 (4), 
dy 22 dx 
y sin Qa? 
from which, by integrating and determining the constants, 
we get 
tan @ 
a tan a 
The ray reflected from this layer, if it be polarized in the plane 
of incidence, has, according to (3), the amplitude a 
and if polarized perpendicularly, it has, according to (4), the 
amplitude OY These two values we indicate by xdu, 
‘x 
tan 22 
where, in the first case, 
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