76 
DIt. BREWSTER ON THE LAW OF 
In these experiments the average error does not exceed half a degree. The 
third column is computed by the formula tan <p = (tan 26° 27') tan x. 
From these experiments it appears that the formula expresses with great 
accuracy all the changes in the planes of polarization which are produced by a 
single reflexion, and we may therefore apply it in our future investigations. 
Let us now suppose that a beam of common light composed of two portions 
A, B, (Fig. 2.) polarized + 45° and — 45° to the 
plane of reflexion, is incident on a plate of glass at 
such an angle that the reflected pencil composed of 
C and D has its planes of polarization inclined at 
an angle <p to the plane M N. When a rhomb of 
calcareous spar has its principal section in the plane 
M N, it will divide the image C into an extraordinary 
pencil E and an ordinary one F ; and the same will 
take place with D, G being its extraordinary and H its ordinary image. If we 
represent the whole of the reflected pencil or C + D by 1, then C = 
D = E + F = 1 , and G + H = 1 . But since the planes of polarization of 
C and D are each inclined <p degrees to the principal section of the rhomb, the 
intensity of the light of the doubly refracted pencils will be as sin 2 cp : cos 2 <p ; 
that is, the intensity of E will be £ sin 2 <p, and that of F, £ cos 2 <p. Hence it 
follows that the difference of these pencils, or ^ sin 2 <p — -i cos 2 <p, will express 
the quantity of light which has passed from the extraordinary image E into 
the ordinary one F, that is, the quantity of light apparently polarized in the 
plane of reflexion M N. But as the same is true of the pencil D, we have 
2 (^ sin 2 <p — £ cos 2 <p ) or sin 2 <p — cos 2 <p for the whole of the polarized light 
in a pencil of common light C + D. Hence, since sin 2 <p -f- cos 2 (p — 1 
and cos 2 <p = 1 — sin 2 <p, we have for the whole quantity of polarized light 
Fig. 2." 
Hut 
Q = 1 — 2 sin 2 <p. 
Tan (p = tan x 
cos (z + t) 
cos (z — z 7 ) 
Tan 2 (p 
sin 2 p 
cos 'Z' 
and sin 2 (p + cos 2 <p = 1, 
And as 
