FRESNEL ON DOUBLE REFRACTION. 273 



thev are moreover perpendicular to each other, for the product 

 of the two values of («) when multiplied by each other is equal 

 to the last term ( — 1) of the second equation. Therefore there 

 exist always three rectangular axes of elasticity for every material 

 point in any molecular system whatever, and whatever may he the 

 laws and the nature of the actions which these material points 

 exert on each other. 



If we suppose that in a homogeneous medium the corre- 

 sponding faces of the particles or the homologous lines of the 

 molecular groups are all parallel to each other, the three axes of 

 elasticity for each material point will have the same direction 

 throughout the whole extent of the medium. This is the most 

 simple case of a regular arrangement of molecules, and that 

 which seemingly should be always exhibited by crystallized sub- 

 stances according to the idea one forms of regular crystallization ; 

 nevertheless the needles of rock-crystal present optical pheno- 

 mena which show that this condition of parallelism of homolo- 

 gous lines is not always rigorously fulfilled by it. It is in fact 

 conceivable that there may be without this condition many dif- 

 ferent sorts of regular arrangements ; but as yet I have sought 

 only the mathematical laws of double refraction, on the sujjpo- 

 sition that the axes of elasticity have the same direction through- 

 out the whole extent of the vibrating medium, and consequently 

 shall confine myself to the consideration of this particular case, 

 the most simple of all, and which appears to be that of the 

 greater number of crystallized substances ; for as yet rock-cry- 

 stal is, I believe, the only known exception to this rule. 



Application of the preceding Theorems to the complex displace- 

 ment of the Vibrating Molecules which constitutes Luminous 

 Waves. 



Hitherto we have only considered the displacement of a mate- 

 rial point, supposing all the other molecules immovable ; we 

 have been allowed to suppose, without altering the problem in 

 any way, that it is the medium which displaces itself and the 

 material point alone w"hich remains fixed. But the relative dis- 

 placements of the molecules in which consist the vibrations of 

 luminous waves are more complex. Let us first consider the 

 most simple case, that of an indefinite plane wave. All the mole- 

 cules comprised in the same plane parallel to the sui*face of the 

 Mave, have remained in the same positions relative to each other, 



