THROUGH TRANSPARENT MEDIA, AND ON DOUBLE REFRACTION. 529 



cl<p 11 I \ d'<b 1 1 



Hence, ^^ • „ + „, = A^ "77 • and by substituting the value of - + - , obtained above, 

 (is: \si Jx ! ass Jt H 



, d-d) / d'Fi d-F, 



dz' ^ KF^dw- F.,dy- 

 For a point on the axis of x: this equation becomes — ^ + tvcp = 0, the constant n' being 



such that if \ be the breadth of the waves, w = — . Hence, substituting - n'd) for ? in 



the foregoing equation, the result is 



if-F, <fF,. 



F,dx- F.,df 



X ' ^ ^ ■ d^ 



+ kn^ = 0. 



By taking account of the equality f = F,'' . F.^' , we obtain by substitution in the above 

 equation, 



. d\f d\f h . (A - 1) df /.(/-I) df 



If now for tiie same reasons as those given in p. 369 of the Paper on Luminous Rays, the 



(// J df ^ , ^ , 



terms involving -r^^, and — Y~; "^ neglected, we have, finally, 

 fdx- Jdy- 



^ • ^4 + ^ • ^ + /"»7 = 0. 



dor dy~ • 



The general result fi-ora this course of reasoning is, that a ray of which the condensation in the 

 transverse direction is defined by a function of x and y, which satisfies this equation, may be 

 propagated in a medium whose elasticity varies with the direction of propagation. The reasoning, 

 iiowever, only applies to a function of ac and y, which is the product of a function of w and 

 a function of y. It is evident that / cannot be a function of x^ + y^, and, consequently, that the 

 ray cannot be one of common light. 



8. It is found by experience that a po/nrized ray may be transmitted in certain transparent 

 crystallized media. I shall assume that in these media the retardation of the propagation produced 

 by the presence and inertia of the atoms, is such as corresponds to an apparent diminution of the 

 elasticity of the a;ther, different in degree in different directions. I shall assume also that there 

 are three rectangular axes of elasticity, and tliat, in accordance with the result contained in 

 Art. 8. of this Paper, the surface of elasticity is an ellipsoid. On these suppositions the ray 

 cannot be one of common light, because the eft'ective elasticity is different in different directions 

 transverse to the axis of the ray. But the suppositions are consistent with the transmission 

 of a polarized ray. For according to the Theory of Polarization contained in my Paper in the 

 Cfimh. Phil. Tranaactions, (Vol. vui. Part in. p. 372), the condensations for a polarized ray must 

 be disposed symmetrically witli reference to two planes at riglit angles to each other ])assing 

 through the a.\is of the ray. Consequently the force of retardation and tiic effective elasticity 

 must act symmetrically with reference to two such planes. And this will evidently be the case : 

 for any section through the centre of the surface of elasticity is an ellipse, the radii of which 

 drawn from its centre, are symmetrically disposed with reference to its axes. It is possible that 

 the function / for a polarized ray may ho sue!) as that sup])osed in the preceding Article, 

 namely, the product of a function of .v and a function of y. All, however, that can be alllinu-d 

 respecting this function from the reasoning in the Paper above referred to is, that for small 

 distances from the axis of the ray, it is a function of one co-ordinate only, the axis of .r and // 



.'i y 2 



