826 MR. G. A. SCHOTT ON THE REFLECTION AND REFRACTION OF LIGHT. 
equations (1.) for the electromagnetic theory. These two theories will, therefore, be 
considered together. 
We shall suppose the medium in which the disturbances take place to be perfectly 
continuous, though its qualities may vary from place to place. It follows that 
7], I, u, V, IV must be continuous functions of {xyz), as well as their first differential 
coefficients, and this condition must replace boundary conditions at places where the 
nature of the medium changes, however rapidly it may do so. 
§ 2. Waves in a Variable Layer hetiveen tivo Media. 
For our purpose it is only necessary to consider the very special case when the 
heterogeneous medium is arranged in plane layers, perpendicular to Oa: suppose, and 
we shall further suppose the variable portion to be a thin layer separating two media 
of different but constant quality, into each of which the layer passes continuously. 
We shall suppose plane waves incident in the first medium, which will give rise to 
plane reflected and refracted waves. Take Oz perpendicular to the plane of incidence ; 
then Oy will be parallel to the intersections of the plane of incidence with the plane 
layers, and since the traces of all the waves on the plane layers must move along these 
layers at the same rate, the coefficient of y must be the same in the expressions for 
the different waves. 
Let now X be the wave-length in vacuo of the light e “ jx its refractive index from 
vacuum into the variable layer, the values for the two media on either side. 
O')) -r/i 
Then we have (or If i be the angle the wave-normal makes with 
47r“ 
''TT • • 
Oa:, the coefficient of y in the expression of the wave will be — p sin i, which, 
A 
2tt 
l^eing everywhere the same, we shall write — v. 
Jc • 
Also write —^- =; m®, so that the velocity of the pressural wave is m time§ that 
of the transverse wave. 
By what- has been said above, p, in are functions of x only. 
And the displacements, &c., are independent of «, and proportional to ^0. 
First medium. 
This is of constant quality (p^, m^^ and extends from x — — oo to .a: = 0 ; the 
equations (II.) become 
3 A / A I A'\ I A /A _ M I A" 3 — 0 I 
dx \0.r 3;// dy \3;/ dx) Vibrations parallel to plane 
a /3 
hi 
dr 
3 /du 
3?/ \dx di/j dx \ 03 / dx 
3r 
4-77^ 
of incidence. 
