220 UNDULATORY THEORY OF LIGHT. 



radius; and if, to facilitate conception and comparison, we conceive it to be 

 the diagonal of another parallelopipedon, CMM'FK, &c., tbo three dimensions 

 of this solid will be Ix^JrcosA, Il^frcoal^, and R^yz-cosC. For the sake of 

 symmetry, we will employ for a moment these components, inst(!ad of H'^J'r 

 itself. We will denote, also, as before, the two resultants by p and p' and the 

 angle, GCF, between them, by w. Then, 



p' frVW^'^A+W^^'B+W^^G R2 



Cosw=;— =± — ^ 



_ , a^cos^A+i^cos^B+c^cos^C , ,^ , 



Or, cosai=4- , -- . 47. 



Va'cos^A+i^cos^B+c-^cos^C 



Now the wave front-in which r, having the direction CF, is one of the move- 

 ments, cuts the surface of elasticity in an ellipse, which may be represented by 

 ADBE. The line CG will not usually lie in the plane of this ellipse. If the 

 resultant, p, be decomposed into two forces, one of them equal and opposite to 

 />', or CF, and the other GF, perpendicular to OF, this last tends to turn the 

 movement in CF out of that line, as before. But, as it is not in the plane 

 of the ellipse, ADBE, which is the wave front, in order to undcr.stand more 

 clearly its effect upon the direction of movement in the plane of the Avave 

 (which is all that concerns the question of polar' zation) decompose this force 

 again, by drojiping, from G, the perpendicular Gil" upon the wave front, and 

 joining 11"F. The component, Gil", being normal to to the wave, can produce 

 no effect in the way of polarization. The other component. H"F, tends to turn 

 the movement, as in the former case, alternately, in the direction of the shorter 

 axis, DE, of the elliptic section of the surface of elasticity, and of the longer 

 axis, AB. 



Observe that if the displacement had been originally in the direction of one 

 of these axes, there would have been no deflecting force, 11"F. For this lateral 

 force owes its existence to the inequality of elasticity, or resisting force, on the 

 two sides of the movement of displacement. But, by the law of construction 

 of the surface of elasticity, the squares of its radii are the measures of the 

 elastic forces in their directions ; and at the extremities of the major and minor 

 axis of an ellipse the radii on either side of the axis are equal and symmetri- 

 cally disposed. 



It follows, that whenever a ray of light falls upon a medium of sucb a nature 

 as we have been considering, all its movements will be thrown into parallelism 

 with the two axes of the elliptic section made by its front Avith the surface of 

 elastictity. And thus we have a physical account of the polarization of light 

 by double refraction. 



We have, at the same time, the cause of the unequal velocity of the two 

 waves. For, by the construction of the surface of elasticity, all its radii are 

 measures of the velocities of undulations whose molecular movements coincide 

 with them in direction. The two velocities will, accordingly, be to each other 

 as the major and minor axes of the elliptic section of the surface of elasticity 

 made by the Avavc front. 



We have also the cause of the polarization of the two rays in planes at right 

 angles to each other. This is so, because the two axes of the ellipse are in 

 that relation. 



Since the two velocities are both iniiform, though unequal, a plane wave is 

 transformed into two plane waves, by double refraction. Supposing the I'efract- 

 ing surface to be also plane and of indefinite extent, and that a plane wave 

 enters it obliquely, the intersection of the wave front with the surface will be a 

 straight line, and will advance along the surface parallel with itself, as the wave 

 advances. The refracted waves necessarily both intersect the refracting surface 



