the Boundary of two Isotropic Transparent Media. 483 
and in the second, 
uo led sin Dye eee eee 2 6} 
The amplitude of the reflected ray is therefore 
tan @ 
xXdu= A A/a a 
and when this ray encounters a layer whose angle of refraction 
is @,, its amplitude becomes 
qf 2 ay / 22 ay 
bane 
At the boundary between this layer and the following, where 
the angle of refraction is z,—dvz,, a portion of the light is again 
reflected ; and uw, being the same function of z, that w is of iz 
the amplitude of the twice reflected ray 1S 
—A oa du du, 3 
and when this ray ae traversed all the layers until its angle of 
refraction has become constant and equal to @, its amplitude is © 
—A 2s B du dup. 
tan e 
_The angle x, may now have all values between « and x, and 
a all values between « and 8. The sum, therefore, of the am-_ 
plitudes of all the twice reflected rays will be represented by: 
the definite double integral 
VE: 
DEVE eng di a Fr : 
where wu, and ug indicate the me a of u mt x equal to a, and a. 
equal to £. 
In this manner the sum of the amplitudes of the rays reflected 
4, 6,.... times can easily be calculated; and as the sum of the 
different rays that have been 0, 2, 4, 6... times reflected make 
up the whole of the refracted ray, the amplitude of the latter is 
Ee aw i au" du +f ae" anf" "a4 ditg— 
tan @ 
which we shall indicate by 
A tana) 
tan a 
jp @ 
