MR. G. A. SCHOTT OH THE REFLECTION AND REFRACTION OF LIGHT. 829 
4:7r^ 
^ • {li?' — v~). IV = 0 Vibrations perpendicular to plane of incidence. 
We shall choose u, v, II, iv, so that, when x = 0, u = Uq, v = Vq, n = ITo, 
IV = iVq, and when x = cl, v = Vi, tv = tVi, where 
Wq = - sin ^0 (1 — r) + a^r', = cos Iq (1 + r) + t sin Iq . r', iVq — I r, 
Vy = cos . s + t sin . s', = s. 
The six conditions determining r,r . .. will be 
T-TT"! « d'y _ 27r^Q ^ 2*/i \i * * tito 
When X = 0, -j- = t —— {cos^ • (1 ~ + “o sm in-r }, , = t 
chi \ ^ ^ dx, \ 
7r/.ii, cos ^o 
(1 - r). 
When X —d : u = — (sin + 01.^8'), IIi = ~ {cos^z\. 5 —sin 
clx 
dw 277/1.1 cos L 
-T- = L -. s. 
dx \ 
§ 3. Determination of the Displacements for the Variable Layer. 
It is in general impossible to solve the equations in finite terms; in the physical 
problem the transition layer may be considered thin even in comparison with the 
wave-length, and the equations can be solved in very convergent series, proceeding in 
ascending powers of some small quantity depending on the thickness of the variable 
layer. This quantity we shall take to be 8 = 27rcZ/X. Putting also ^ for xjd, the 
value of ^ will lie between 0 and 1 ; and the equations become 
dil . dv 5/2 
d~v ^ du 
— iOv — 
d^~ af 
du 
0 
t8®. vYi + h^prv = 0 
0 
m-" —r + tSwiV . u + 8n 
Vibrations parallel to plane of 
incidence. .... (III.), 
dhv 
d^ 
fi- 8“. (/A^ — vfw= 0 
Vibrations perpendicular to plane of incidence (IV.), 
and when ^ = 0, u = Uq, v = v^, IT = IIq, tv = iVq ; and when ^ = I, v = v^, w = Wy. 
It will be necessary to treat separately tlie cases of the electromagnetic and 
contractile ether theories on the one hand, for which m, IT are zero, and of the 
elastic solid theory on the other hand, for which m, is very large and IT is finite. In 
