150 



Equating the two values of «^, we get, for the determiiialion 

 of/', the following quadratic : 



F 4- '''~^' /t + 1 = 0. (5) 



c 



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



we have 



27r , 2irc .^. 



s- zz A - — cA; s- = B — y ^, (b) 



and thence 



F---(a-b)/--1=0, (7) 



~7rc 



a result which is perfectly inconsistent with the former, since 

 the two roots of (5) have the satrie si<rn, if they are not imagi- 

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

 imaginary. If, therefore, one equation agiees 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 ^jo.y.«7>/e, and performed in op- 

 posite directions, which is in accordance with the facts ; 

 whereas the equation (.")), 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 / = ± 1, and a = b z= ci^, so that the 

 value ofs^ in equation (6) is expressed by the constant rt^, 

 phis or minus 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 e(|iia- 

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



