1821.] Crystallized Bodies on Homogeneous Light, 129 



Let n be the number of periods (each consisting of a double 

 alternation) and parts of a period performed by the elementary 

 pencil C, in its passage through the medium : then, according to 

 the theory of M. Biot, when n is 0, 1, 2, 3, &c. ad inf. the penciF 

 will wholly pass into the ordinary image ; but when the values 

 of n are -i, I, f, &c. it will wholly ^ be thrown into the extraor-^ 

 dinary 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.^ {n %) 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 intermediate values of w, will a^ree sufficiently 

 with the judgment of the eye to warrant its adoption. f The part 

 of the elementary pencil C then, which enters into the extraor- 

 dinary image, will be C. sin.'^ {n tt). Let us denote by S {C. sin.* 

 {n tt) } the aggregate of all such elements from one extremity of 

 the spectrum to the other, or take 



S{C. sin.^ {n tt)} = C. sin.2 {n tt) + C\ sin.« {n' tt) + &c. 



Then will this expression represent the tint developed in the 

 extraordinary image, and, consequently, S {C. cos.® (w tt)} that 

 in the ordinary one. 



Now, Uj the number of periods performed depends, first, on 

 the nature of the ray, or on c ; secondly, on the intrinsic energy 

 of the action of the medium on that ray ; and thirdly, on the 

 direction 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.~ (M /c. tt)} 



In the theory of the Newtonian colours of thin plates and the 

 polarised rings in crystals with one axis, the multipHer 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 



* 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 maxi- 

 mum. 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.*^ n tt, will have an addi- 

 tional 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 hrightness in the isochromatic lines, and the black cross or hyperbolic, 

 branches which intersect them, depend. But it is not my intention at present to enter 

 on this part of the subject, for reasons to be explained further on. 



-'- No part of our subsequent reasoning depends on iheforryi of this function. ^ It i* 

 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 precise as- 

 sumption. 



^ew Series, vol. i. i 



