52 Proceedings of the Royal Society of Edinburgh. [Bess. 
If (p be the angle of incidence we have 
a m n , — 
6 "7 — — j= VaK. 
sm cos r 
It is evident from the form of equations 3, 4, and 5 that if X and a 
are given for any medium the electric and magnetic vectors in that 
medium are completely specified. In the three media therefore we have 
respectively 
IXj = Ae lK ( m 2/+ M i 2 + ci ) + A 'e iic ( m y- n i z + ct ) 
7 I X 2 = J± ie iK(my+n z z+ct) + ^ e iK(my-7i 2 z+ct) 
I X 3 = A s e ilc ( m y+ n xZ+a+ ct ) 
8 a i ^^ K ( m y +n i z k ct ) + T)'e iK ( m y~ n i z + ct \ 
with similar forms for « 2 and a 3 . 
Using equation 3, we have the following form for /3j and similar forms 
for /3 2 and f3 3 : 
9 . . fii = — ^l-J^Qi^my+n^+ct) + Qin{my - riyZ+ct) ^ 
Pl c P i c 
The boundary conditions are 
10 
11 
— X 2 , Yj — Y 2 , otj — a 2 , — ^ 2 , at % — 0. 
X 2 — X 3J Y 2 — Y 2 , ot 2 = otg; f3 9 = /? 3 , sit 2 == a. 
From these equations we obtain 
12 . 
. A + A — Aj + A 2 ; Pi(A A)— p 2 (A 1 + A 2 ) 
and 
13 . 
A 1 e~ ie + A 2 e*> = A 3 ; p 2 {k Y e~ ie - A 2 e ie ) = p 3 A 3 
where 
and 
p, = -^l., etc. 
Pi c 
A 27T 
0 = K.an 2 . arc 2 . 
A 
Similar equations hold for the “ B ” coefficients if p v p 2 , p s are replaced 
by q v q 2 > q z where 
C71, , 
( h — ®tc. 
Solving equations 12 and 13, we obtain 
Pi A 
Pi A' 
P 2 (Pi + Ps) cos e + *(jPiPs + P 2 2 ) sin 0 P 2 (Pi ~ Ps) cos 0 + APiPz ~ P 2 ) sin 0 
1 _ 
(P2+Ps) el ° (Pl-PsY 
Ag 
2 Pi 
14 
