150 



Equating the two values of *^, we get, for the deteriniiialion 

 of k, the following quadratic : 



Ir-i^ ^' ~/ k +1=0. (5) 



Now making the substitutions (3) in equations (1), page 142, 

 we have 



s^.--ck, s-=B---, (6) 



and thence 



^■'-^^(a-b)>?:-1=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 jjossihle, and performed in op- 

 posite directions, which is in accordance with the facts ; 

 whereas the equation (o), deduced from M. Cauchy's theory, 

 would inform us that the vibrations of the two rays are 

 either impossible 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 

 turpentine ; the position of the coordinates x and y, in the 

 plane of the wave, being now, of course, arbitrary. In each 

 of these cases we have l- = ± \, and a = b z= a^ so that the 

 value of s^ in equation (6) is expressed by the constant a^, 

 plus or ?ninus a term which is inversely proportional to the 



* 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 tliese equations are necessary in what follows. 



