194 REPORT—1885. 
flexions. They have shown tkat, in general, plane polarised light becomes 
elliptically polarised by such reflexion, and have measured the difference 
in phase between the components polarised in and perpendicular to the 
plane of incidence and the ratio of the intensities of these two vibrations. 
MacCullagh! was the first to attempt to express the laws of this 
elliptic polarisation mathematically. He supposes that in the case in 
question the angle of refraction becomes imaginary, so that we have 
sin ¢/= St (cos xX +7 sin x), 
7 
cos p= 2 (cos x’ +7sin x): 
Vt 
He then substitutes these expressions in the values given by Fresnel’s 
theory for the amplitude of the reflected ray, which he shews may be 
written in the form a+b,/—1. 
Thus the intensity of this ray will be represented by a?+b?, and the 
difference of phase between the incident and reflected rays will depend on 
tan~'b/a; a and b are functions of m, m’, x, and x’, and these quantities 
are connected by the equation sin?¢’+cos*¢’=1, which leads to two con- 
ditions, giving m’ and x’ in terms of m and x. 
The final formule are :— 
(1) Light polarised in the plane of incidence. 
__ D? + cos? ¢ — 2D cos ¢ cos (x— x’) 
_— 
D? + cos? @ + 2D cos 9 cos (x— x’) ae 
6 2D cos¢sin (x-—x’) 
(2) Light polarised perpendicular to the plane of incidence. 
= m4 cos? ¢ + D? — 2Dm? cos ¢ cos (x + x’) (48) 
~~ mi! cos? @ + D? + 2Dm? cos 9 cos (x + x’) 
& __ 2Dm? cos ¢ sin (x +x’) 
tan Qn = ee a (49) 
Where Dt = m4 + sin*t ¢ —2m? sin? ¢ cos ey 54 
and D? sin 2 (x— ’) = m? sin 2x ets 
These formule are simplified in the case of metals from the considera- 
tion of the fact that the proportion of light reflected at normal incidence 
is nearly unity. It follows from this that m is very large and y’ very 
small, so that we may put siny’ =0, cosy/=1 in the equations, and 
hence m’ = cos ¢/cos ¢’, 
And for Case I.— 
oe m? + m'? — 2mm! cos x 
m? + m'* + 2mm’ cos x 
2 : : 51 
t 275__ 2mm’ sin x GY 
QM, esate oe 
a m2 —m? 
1 MacCullagh, Proc. Irish Acad. vol. i. pp. 2, 159; vol. ii. 375 ; Trans. Irish Acad. 
1 xxviii. Pt. I. 
