248 FRESNEL ON DOUBLE REFRACTION. 



paths described, when the two interfering beams have their 

 planes of polarization perpendicular to each other. 



Thus, in this case, the sum of the three foregoing expressions 

 remains the same for all values of (a,' — x) . We must therefore 

 have 



c« + Z,2 + c2 + «'2 -f b'^ + c'2 + 2 aa' coii2n (u - u' -\- "^—^^ 



v—v' + - j +2cc'.cofi2'!rlw — iv'-{ -j = C, 



an equation in which the only vai-iable is (x' — x). 



Now, since this equation must be satisfied whatever be the 

 value of {x' — x"), it is clear that all the terms containing (x' — x) 

 must disappear, since otherwise we should obtain from the equa- 

 tion particular values of (a;' — .r). Therefore we have 



ad = 0, bb' = 0, cc' = 0. 

 The two polarized beams which interfere differ only in the 

 azimuths of their planes of polarization ; that is to say, if we turn 

 one of them about its axis so that its plane of polarization may 

 be parallel to that of the other, these two luminous beams will 

 present in every direction exactly the same properties ; they will 

 be reflected and refracted in the same manner and in the same 

 proportions at the same incidences. We must therefore admit 

 that if one has no vibratory movements perpendicular to the 

 waves, no more has the other. Now (a) and (a') are the con- 

 stant coefficients of the absolute velocities normal to the waves 

 in these two beams ; and since aa' = 0, which requires that W'e 

 have at least c = or «' = 0, we must conclude from this that 

 both (a) and [a') are equal to zero. 



There cannot therefore be in polarized light any other than 

 vibratory movements parallel to the surface of the waves. 



Let us now consider the other two equations, bb' = and 

 cc' — 0, which contain the constant coefficients of the velocities 

 perdendicular to the rays, or, more generally, parallel to the 

 waves : [b) is for the first luminous beam the component parallel 

 to its plane of polarization, and (c) that which is perpendicular 

 to it ; whilst for the second, {b') being parallel to {b), is perpen- 

 dicular to the plane of polarization, and (c') is parallel to it : thus 

 {b') and (c') are respectively for the second beam that which 

 (c) and (b) are for the first. Therefore, according to the remark 

 just made on the perfect similitude between the properties of 

 the two interfering beams, if in the former 5 = 0, in the second 



