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



wherein k = - expresses the ratio of the semiaxes of the elliptic 



P 

 vibration, and 



A' = ^ Vf sin 2 0, V = ^ ^g sin'0, 



C f = - SA sin 20. 

 4ir z 



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

 k, the following quadratic : 



F + A ~ B + 1 = 0. (5) 



G 



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

 we have 



o _ A ^ Tlf Q 2 - 7? ^ (K\ 



s -A-^Ck, s --B-yp 



and thence 



A-B)k-l = 0, (7) 



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 is 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. 



