298 
DR. BREWSTER ON THE PHENOMENA AND LAWS 
When the number of reflexions which begin the series is odd or fractional, 
we must determine, by the preceding formula, the value of <p for the even 
number immediately above it : and calling v the number of odd or fractional 
reflexions, and N the number of even reflexions immediately above v, <p the 
inclination for N reflexions as given by the formula (A), and <p the inclination 
required, we shall have, 
tan = tan x — {v — 2 — N _ 2 — J. (B) 
The truth of this formula will appear from the following Table. 
Silver. 
Inclination of the Plane of 
Number of 
Reflexions. 
Angles of 
Incidence. 
Polarization. 
Observed. Calculated. 
3 . . 
7°9 40 . 
. . 38 28 . 
. 38 33 
5 . . 
77 13 . 
. . 33 10 . 
. 33 36 
5 . . 
84 5 . 
. . 26 0 . 
. 26 24 
Steel. 
3 . . 
77 37 . 
. . 13 15 . 
. 14 11 
5 . . 
84 38 . 
. . 10 30 . 
. 10 23 
The same results will be obtained at the angles of equal phase below the 
maximum polarizing angle. 
This last rule is suited to even as well to odd numbers of reflexions, but it 
does not give precisely the same results for even numbers as the formula (A). 
The difference, however, is far within the limits of the errors of observation. 
The inclination, for example, at 4 reflexions, is by formula (A) 37° 22' for 
silver, whereas by formula (B) it is 37° 34', the difference being only 12 
minutes. 
In circular polarization, therefore, the plane of polarization of the restored 
light continues by successive reflexions to oscillate on each side of the plane 
of reflexion with a never varying amplitude from +45° to —45°; while in 
elliptical polarization, the same plane oscillates with an amplitude continually 
diminishing till it is brought to nothing in the plane of reflexion. 
In steel, as we have seen, the polarization is highly elliptical, and the ampli- 
tude of the oscillations of the plane of restoration is quickly brought to zero; 
