2O4 Notes on some Points in the Theory of Light. 



wherein ft - - expresses the ratio of the semiaxes of the elliptic 

 vibration, and 



At ^-1 / * 9/1 T\f *^ -i 9/1 



.4 = ~ 5- S/ sin 0, .D = 2<7 sin a, 

 O r = -j-; SA sin 20. 



Equating the two values of s 2 , we get, for the determination of 

 k, the following quadratic : 



# + ^ ~ ff k + l = 0. (5) 



Now making the substitutions (3) in equations (1), page 197, 

 we have 



2 A rif 2 7? (K\ 



\ \ yfc' 



and thence 



a result which is perfectly inconsistent with the former, since 

 the two roots of (5) have the same sign, if they are not imagi- 

 nary, while those of (7) have opposite signs, and cannot be 

 imaginary. If, therefore, one equation agrees with the phe- 

 nomena, the other must contradict them. The last equation 

 indicates that, in the double refraction of quartz, the two 

 elliptic vibrations are always possible, and performed in oppo- 

 site directions, which ig in accordance with the facts ; whereas 

 the equation (5), deduced from M. Cauchy's theory, would 

 inform us that the vibrations of the two rays are either im- 

 possible or in the same direction.* 



To apply the results to a particular instance, let us con- 

 ceive a circularly polarized ray passing along the axis of 

 quartz, or through one of the rotatory liquids, such as oil of 



* This conclusion, which shows that M. Cauchy's Theory is in direct opposition 

 to the phenomena, might have been obtained without any reference to the equa- 

 tions (1). But these equations are necessary in what follows. 



