the Boundary of two Isotropic Transparent Media, 487 
-. We confine ourselves now to the case in which the values of 
dx 
du 
gives by integration, since the last member vanishes, 
and = are very small. The differential equation for > then 
an _ alee ie 2(u— 2(Upg— do 7 
aT oc Seemmmer B ‘G (u—ug) __ 2(ug )) = du. 
If in this expression we substitute the value of wu, it is obvious 
a" dn. 
that for all angles of incidence aa 8 small so long as 7 1S 80, 
which is the only hypothesis. 
If now we substitute for /(w) its values as found in (15), then 
series (15) becomes the real part of 
— Pp la Ug Mga as 
3 eae oe ar bet de 
ela— Up 4 lp Me du . 
and its sum is therefore 
Ug—Ug __ Ug—U, Ug 2(u—Ug) __ 2(ug—u) d, 
: [ cos e+ inf Se SSeS Te eee oF : 
2s > au 
elas 4 Mp Ya Uy e2(u,—Ug)_ _2(U,— Mn) des 
And substituting again in this expression for u its value 
—tlogtanz, we get for R the amplitude of the reflected ray 
polarized in the plane of incidence, 
epee -/) [cos kt+ sin kt tan A] 
sin (2+) ! ‘ey 
_  sindcosa BD ee ee: dé 
tan A= Se (cos? 6 tan 2— sin? cot A sn dz. 
If, on the other hand, for u we substitute $ log sin 2z, we get 
for the amplitude of the reflected ray polarized perpendicularly 
to the plane of incidence, 
1 _ , tan (a—P) : - 
R/=—A tan (2 +8) [cos A¢-+sin kt tan A’) 
sin2asin28 (8 ie 2¢ sin ad db 
SS —— - = — dk. 
sin? 2a—sin?26} Lsm28 sin 2z 
. dz 
From these equations it may be seen that tan A, for all angles 
(17) 
tan A! = 
of incidence, is small provided Ta is so also; while, on the con- 
trary, tan A’ may be infinite, as when sin 2¢=sin 28; that is, 
when the angle of incidence is equal to the angle of polarization, 
in which case # and 6 are complementary. m: 
