50 
SIR DAVID BREWSTER ON THE PHENOMENA OF THIN PLATES 
If we now calculate by these formulae the values of x for the different angles of inci- 
dence in the preceding Table, and subtract them from 90°, we shall have the numbers 
in the third column of the Table, which agree with those observed within the limits 
of the errors of observation. In the case of water and glass, too, where the azimuth 
of disappearance was observed to be about 79° or 11°, the formula gives 79° 28', or 
10° 32', at an incidence of 56° 45'. 
In order to ascertain the relation between the mutual inclination of the planes of 
polarization of the interfering pencils when they produced black-centred or white- 
centred rings, I have computed the following Table for an incidence of 56° 45'. 
Azimuth of Polarized Light. 
o / 
+ <p 
o / 
-r 
o / 
Film of water and glass. 
Inclination of Planes (j> and 
O / 
90 
87 
30 
90 0 
74 43 
90 0 
82 45 
180 0 
157 28 
White-centred 
> 
85 
0 
49 30 
75 4 
124 34 
J 
rings. 
79 
28 
28 26 
61 34 
90 0 
No rings. 
70 
0 
15 28 
43 19 
58 47' 
45 
0 
5 45 
18 57 
24 42 
35 
0 
4 3 
13 
3 
17 6 
Black- centred 
> 
20 
0 
2 6 
7 
7 
9 13 
rings. 
0 
0 
0 0 
0 0 
0 0 V 
By taking <p positive , or on the right- hand side of the plane of reflexion, then <p"' must 
be negative, or on the left-hand side of that plane* ; hence + <p, — <p"' will be the mu- 
tual inclinations of the planes of polarization of the interfering pencils, and we obtain 
the important law, 
That when two polarized pencils rejlected from the surfaces of a thin plate lying on 
a reflecting surface of a different refractive power interfere, half an undulation is not 
lost, and white -centred rings are produced, provided the mutual inclination of their 
planes of polarization is greater than 90 ° ; and that when this inclination is less than 
90°, half an undulation is lost, and black -centred rings are produced; when the inclina- 
tion is exactly 90°, the pencils do not interfere, and no rings are produced. 
At an incidence of 45° upon water and glass, where the signs of <p and <p"' are the 
same, the maximum difference in the planes of polarization is 23° 12', which takes 
place in azimuth 79° 30'; and at an incidence of 10° the greatest difference is 2° 16', 
which takes place at an azimuth of about 45°. 
In the case of soap and plate glass, where the black-centred rings appear beyond 
the incidence of 71° 45', the difference of inclination in the planes of the two pencils 
is also less than 90°. 
I was now desirous of examining the phenomena of a perfect system of rings when 
the film had a greater refractive power than the substance upon which it was laid ; 
* See Philosophical Transactions, 1830, p. 70, fig. 1. 
