REFLEXION OF POLARIZED LIGHT. 255 
have attempted in any published work to remove the 
difficulty, and it is not to be supposed that they had 
despised it. 
And by the analogy of certain geometrical cases where the multi- 
plication by »/—1 indicates a line differing in angular position by 90°, 
he hazarded the inference that such an interpretation might hold 
good here, and that this expression would be equivalent to one of the 
form, 
; ar ‘ . 2 . 
cos @ sin x (vt—a) + sin 9 sin x (vi—a + 90°) 
which is trigonometrically the same as 
j 2r 
sin ( = (ot—2) +0) 
This applying to the component in the plane of incidence, a similar 
expression would apply to that perpendicular to it, 
orsin (= (tz) + 0 ) 
The difference of these expressions, or the relative retardation of 
the two sets of waves, will be @— 6/ = J. 
In general, 6 having any value, and the plane of polarization being 
inclined at an angle @ to the plane of incidence on the rhomb, the 
components are, 
y = sin asin = (ot—z + 6) (1.) 
2 = 008 asin = (12) (2.) 
This then is precisely the same case as that considered in a former 
note; and exactly in the same way we obtain, 
yf 22 2yz cos d 
sin 2a cos 2a cosa sina 
The general equation to an ellipse. If 6 = 90°, the semi-axes are 
sin a and cos a, parallel and perpendicular to the plane of incidence. 
If a = 45° and 6 variable, it is still an ellipse. If a= 45° and d = 90°, 
it becomes a circle. Thus a ray polarized at an angle a, with the plane 
of incidence, after two internal reflexions in glass, emerges elliptically or 
circularly polarized, according to the above condition. 
From the empirical terms before mentioned, Fresnel derived ex- 
pressions from which he calculated that for crown glass, where “= 
1°51, an internal incidence i = 54° 87/ would give d = 45°. Thus 
experimentally cutting a rhomb of such glass at that angle, so that 
the ray polarized at 45° to the plane of incidence, entering one face 
= sin 20. 
