268 



FRESNEL ON DOUBLE REFRACTION. 



It might at first sight be thought that the nine constants 

 a, b, c, a' V c', a" b" c" are independent, but it is easy to perceive 

 that there exists amongst them a relation which reduces their 

 number to six. 



In fact, let Ax, Ay, A z (fig. 5.) be the three rectangular axes 

 along which the molecule A is successively displaced by a very 

 small quantity equal to unity ; let A P be the direction on the pro- 

 longation of which is situated another material point M, which 

 acts on A, and which I always suppose separated from this point 

 by a quantity very great relative to the extent of the displace- 

 ments. 



Let us first suppose that it is displaced along the axis of x by 

 a quantity A B equal to unity ; this small displacement will cause 

 to vary at the same time the direction and the intensity of the 

 force exerted by the point M by bringing the other molecule 

 nearer;- if from the point B the perpendicular B Q be dropped 



on the direction A P M, A Q will be 

 the variation of the distance, and 

 B Q may be considered as propor- 

 tional to the variation of the direc- 

 tion. The former variation will 

 produce a differential force A x A Q 

 along the direction A P M, and the 

 second a differential force B x B Q 

 in the direction B Q, the coefficients 

 A and B remaining constant so long 

 as we consider the action exerted 

 by the same molecule M. 

 To fix the direction in which these differential forces push the 

 point A, suppose the molecule M to exert a repulsive action on 

 this point. The distance A M being diminished by A Q, this 

 action is increased, and the differential A x A Gl acts in the 

 direction MA; in the same manner the differential B x B Q, 

 resulting from the small change of direction of the force, acts in 

 the direction Q B. If then we regard as positive the directions 

 of action Ax, Ay, Az for forces parallel to the axes of coordi- 

 nates, the component parallel to x of this second differential 

 will be negative, whilst the components parallel to y and z will 

 be positive, as well as the three rectangular components of the 

 fii'st differential. 



Let us now seek for the components of the two differential 



