64 Mr. J. F. W. Herschel on the action of 
transmission and reflection, or the extent of one pulse, on the 
undulatory hypothesis, in vacuo, and at a perpendicular inci- 
dence for any homogeneous ray, and let C denote its colour 
and proportional intensity or illuminating power in the pris- 
matic spectrum. Then will the formula representing a beam 
of white light intromitted into the crystal, be 
C + C' + C"-j-&c. 
from one end of the spectrum to the other. 
Let n be the number of periods (each consisting of a double 
alternation) and parts of a period performed by the elemen- 
tary pencil C, in its passage through the medium : then, ac- 
cording to the theory of M. Biot, when n is o, 1,2,3, &c. 
ad inf. the pencil will wholly pass into the ordinary image ; 
but when the values of n are -f, y, y, &c. it will wholly* be 
thrown into the extraordinary one, and in the intermediate 
states of n, partly into one and partly into the other. These 
conditions are satisfied if we represent by sin. 3 (n tt) the 
intensity of the ray in the ordinary image, taking unity for 
its original intensity ; and it will I believe be found, that the 
gradation of intensity given by this formula for the inter- 
* The amplitude or total extent of each oscillation of the plane of polarisation is 
here supposed 90°, in which case the contrast of colour in the two pencils is at its 
maximum. This is the case in the situation we are considering, but in general the 
intensity of the extraordinary ray, instead of being represented by sin. 1 n it, will have 
an additional factor, a function of the azimuth A of the principal section of 
the crystallized plate and the position of the refracted ray, and which becomes unity 
when A=45°, and the plane of incidence is that of the principal section. It is on 
this factor that the gradation of brightness in the isochromatic lines, and the black 
cross or hyperbolic branches which intersect them, depend. But it is not my inten- 
tion at present to enter on this part of the subject for reasons to be explained farther 
on. See note in page 84. 
