s + xr+^/=0, 



480 Prof. Challis on the Theory of Double Refraction 



inasmuch as this condition cannot be generally fulfilled, because 

 the effective elasticity is different in different directions. But 

 such ray-undulations are the exponents of common light. Hence 

 it follows that common light cannot be transmitted through any 

 substance the elasticity of which varies with direction, and there- 

 fore it is incapable of transmission through a doubly refracting 

 medium, of which the change of elasticity with direction is an 

 essential property. This inference is conformable with experience. 

 But & polarized ray is transmitted through such substances. In 

 order to give a theoretical reason for this fact of observation, it 

 will be necessary to recur to the analytical expressions, appli- 

 cable to ray-undulations, contained in the early part of this com- 

 munication. The value of the function / there given being 

 obtained without reference to an arbitrary disturbance, defines the 

 law of transverse velocity and transverse variation of density in 

 rays which have undergone no modification subsequent to their 

 original production, which are always rays of common light. 

 But I have shown under Prop. X. in the Philosophical Magazine 

 for December 1852, that the value of/ is determined generally 

 by integrating the partial differential equation 



dx 2 + dtf 



in which e is a certain constant satisfying the equation e\2 = 4. 

 The integrals, however, of this equation are subject to the limi- 

 tation of applying only to modifications which the original rays 

 may receive by successive disturbances. We may presume, 

 because this equation is linear with constant coefficients, that all 

 the parts into which an original ray-undulation may be resolved 

 will by recomposition form the original ray, and that, too, with- 

 out restriction of the values of x and y. But we are not now 

 concerned with movements and condensations at considerable 

 distances from the axes, which only perform the part of prevent- 

 ing the lateral diffusion of the waves, but with those that are ex- 

 tremely near the axes, which alone are productive of the sensation 

 of light. Respecting these, it is proved in the article just referred 

 to, that an original ray is resolvable into two equal rays polarized 

 in planes at right angles to each other, and that the transverse 

 vibrations in each polarized ray, as also the condensations, are 

 symmetrical with reference to the fixed plane coincident with the 

 axis, to which the motion is perpendicular. 



This being understood, suppose a polarized ray to be propa- 

 gated in any direction from the centre of the surface of elasticity, 

 and conceive this surface to be cut by a plane perpendicular to 

 the direction of propagation. The section will be an ellipse, of 

 which the radius-vectors drawn from the centre are proportional 

 to the effective elasticities of the aether in their respective direc- 



