the Boundary of two Isotropic Transparent Media. 485 
amplitudes of the rays reflected 1, 3,5... times, may be simi- 
larly found, and may be expressed as follows: 
» ? ? u 
u u u u u u u Ve B 
7, ( P du ( du, ( Plus+ | P du du, "dua ing | duUy—... 
c e Uy e Un e Uy e uy et e uy e Vy Ug 
for which we will put 
A { *8 ul fu) 
Ua 
and 
asi - ("i aes) 
o uy 
From the last equation we get 
flu)= aes 
et Uae 
whence (11) or the amplitude of the reflected ray is 
Ug—Ug__ Ug—Uq 
oT ie ea 
Pia va 2s euig—Ua 
And substituting w= —4}logtanz, we get for the amplitude of 
the portion polarized in the plane of incidence, which we will 
call R, 
_, sin (a—£) 
Beran (2+) 
If KR! is the amplitude of the ray polarized perpendicularly to 
the plane of incidence, and if in (12) we substitute u= 4 log sin2z, 
we get 
(18) 
1, tan (a—8) 
Se oe (2+) 
In this case also, therefore, we return to Fresnel’s formule. 
The result is, that even if there be a gradual change of the 
index of refraction between the two media, and consequently an 
infinite number of reflexions at the boundary, Fresnel’s formule 
nevertheless remain true so long as the thickness of the inter- 
mediate layers is infinitely small as compared with the length of 
a wave. If this be not the case, then the retardations of the dif- 
ferent rays must be taken into consideration. 
For the refracted light this correction is very small, and could 
hardly be confirmed by experiment. We shall therefere proceed 
to calculate it in the case of the reflected hght. 
|) 
heal 
