crystallized bodies on homogeneous light. 65 
mediate values of n, will agree sufficiently with the judgment 
of the eye to warrant its adoption.* The part of the elemen- 
tary pencil C then, which enters into the extraordinary image, 
will be C. sin 2 (w7r). Let us denote byS[c. siri 2 (« 7 r)| 
the aggregate of all such elements from one extremity of the 
spectrum to the other, or take 
S jc. sin* ( n tt) j = C. sin* ( n 7 r) + C'. sin 2 (»V) + &c. 
Then will this expression represent the tint developed in 
the extraordinary image, and consequently, S j C. cos 2 [n tt) j 
that in the ordinary one. 
Now, n, the number of periods performed depends, 1st. on 
the nature of the ray, or on c ; 2dly, on the intrinsic energy of 
the action of the medium on that ray ; and 3dly, on the di- 
rection of its course, the thickness of the plate, and whatever 
other cause or limit of periodicity may happen to prevail. 
Hence we may take n = M x k, k being a function of c, de- 
pendent only on the nature of the body through which the 
ray C passes, and M being a certain multiplier whose form 
we shall consider presently. This substitution made, the 
expression for the tint becomes S { C. sin 2 ( M k. vr) j 
In the theory of the Newtonian colours of thin plates and 
the polarised rings in crystals with one axis, the multiplier 
M is independent on c, varying only with the direction of the 
ray and the thickness of the plate. It is therefore the same 
for all the coloured rays, and the tint, for any value of M, 
will be 
* No part of our subsequent reasoning depends on the form of this function. It 
is sufficient to know that it must be a periodical, and even function of n. It is only 
in the computation of numerical values that it is necessary to make any more pre- 
cise assumption. 
MDCCOXX. 
K 
