OF ELLIPTIC POLARIZATION. 
317 
restored to a single plane of polarization at 63° 43', and 79° 40', and conse- 
quently at 1 ^ reflexion it is elliptically polarized at that angle. This is easily 
understood when the number of reflexions is an integer ; but it requires some 
explanation when the number is partly fractional. It has been already stated, 
in page 301, that elliptical polarization may be completed at any fractional 
part of a reflexion ; and since it begins to be restored the instant the polariza- 
tion is complete, and again begins to be elliptically polarized after every resto- 
ration, the points of restoration may take place in the middle of a reflexion ; 
and though we cannot possibly examine what takes place at these points, yet 
the effect must appear when the fractional number of reflexions in the first 
column has been repeated so many times as to become a whole number. Thus 
a ray elliptically polarized by 1^ reflexion will be restored to a single plane 
at 2-f- reflexions at the same angle. It will also be restored at X 2 = 3+, 
and at 2-f- X 3 = 8, in which case its restoration will be seen at the eighth 
reflexion at the same angle ; and also at the sixteenth and twenty-fourth. See. 
90 ° 
In this case the phase P will be -rr — 67^°, R = 33° 45', and <p = 11° 15', 
l 3 
from which we deduce the angles of incidence to be 63° 43', and 79° 40'. In 
order to ascertain the existence of these points of restoration, I made the ex- 
periment at five and seven reflexions as multiples of 2| and 2J, and I found 
the angles to be for five reflexions 68°, and for seven reflexions 7 0 °, in place of 
67° 54', and 69° 29', as computed from the formula. 
The numbers in the third column, with the signs -f and — , are the integer 
multiples of those in the first column, and show the number of reflexions at 
which the elliptically polarized light is restored, the numbers being carried 
the length of twenty reflexions. The sign + shows that the plane of the 
restored ray is to the right, and the sign — that it is to the left of the plane of 
reflexion. In order to determine the sign of the restored ray, we must consider 
that in the same quadrant the signs necessarily alternate. Now at 7 3°, the 
maximum polarizing angle, the signs are— 2, +4,— 6, +8,— 10,+ 12, &c. ; and I 
have also found that all the integer numbers in column 1st, Table I. have their 
signs + or positive, as +3, +5, +7 } +9, & c., and all the even numbers their 
signs — or negative, as —4, —6, —8, —10, &c. ; whereas in Table II. all the 
integer numbers are negative whether odd or even, thus, —3,— 4,— 5,— 6, Sec. 
