the Boundary of two Isotropic Transparent Media. 489 
reflexions, while those whose index of refraction is greater than 
that amount give positive reflexions. Between these are bodies 
which at the angle of polarization produce a sudden change of 
phase from 0 to 7. These remarkable relations between the 
difference of phase and index of refraction could not have been 
anticipated from Cauchy’s theory, while on the other hand they 
can be immediately deduced from the theory above enunciated. 
Let p be the distance of the first intermediate layer from that 
in which the angle of refraction is z, and dp the distance of the 
latter from the next whose angle of refractionisa+dz. It will 
easily be seen that if a wave be reflected at these two consecutive 
layers, the difference of path of the two reflected waves will 
be equal to 2dp cos z. 
If ae / be the wave-length in the first medium, and there- 
length in the layer under consideration, we 
have for the difference of phase corresponding to this difference 
of path, 
Qc sin « ds 
Teng 4 cos aes dx. 
Instead of « we might introduce, as a new variable, the square 
of the index of refraction. If this variable be called v, we have 
_ sinta 
~ sin? av 
The limits of the variables p and v are, for e=a, 0 and 1; and 
for c= they may be denoted by p, and v,, where p, is the 
thickness of all the intermediate layers, and v, the square of the 
observed index of refraction. 
If now the new values of és and x be substituted in (16) and 
dx 
.(17), and these expressions be integrated by parts, we have 
tan A = <7. 5 ("pdty Sh een eee nae Sue TEE) <2) 
one _ sin? 87) 
2 j@: (19 
4 
T Ee 
ae 
— Tv, cos? —aeccere | ? 
From which it is obvious that tan A is always positive, 
whereas tan A! may either be positive or negative. 
In one particular case A’ passes suddenly. from 0 to +7 at the 
angle of polarization; this is when for this angle | 
*% [eost 8 sin? 8 
Wal FT cine |a=o; 
vl vy 
