282 FRESNEI. ON DOUBLE REFRACTION. 



topaz, it becomes easy to compare the elasticities put into play- 

 by two vibratory movements which have different directions, 

 and belong to two systems.of luminous waves making any angle 

 with each other. For this it is sufficient to compare in the first 

 place the elasticity put into play by the former system with the 

 elasticity put into play by vibrations whose directions are always 

 in its plane, but parallel to the intersection of the planes of the 

 two systems of waves ; then, changing the plane of the waves 

 without changing the direction of these new displacements, we 

 shall compare in the plane of the second system of waves the 

 elasticity which they develope with that excited by the vibrations 

 of this second system. In one word, the variations of inclination 

 of the surface of the waves relatively to the axes of the vibrating 

 medium, causing no change in the elastic force so long as the 

 direction of the molecular displacements remains the same, the 

 problem always reduces itself to the comparison of the elastici- 

 ties put in play by two systems of waves whose surfaces are 

 parallel, and whose vibrations make with each other any angle 

 whatever. Now, the elasticities excited by two systems of similar 

 waves which coincide as to their surfaces, but whose vibrations 

 are performed in different directions, are evidently to each other 

 as the forces produced by the successive displacements of a 

 single molecule along the former and the latter direction. In 

 fact, consider the stratum situated in the primitive position of 

 equilibrium, and with regard to which the parallel strata have been 

 displaced; in both cases it is the same strata of the medium which 

 have become displaced and by equal quantities, but according 

 to two different directions. Now, on considering these two modes 

 of displacement, we may apply to the influence exerted on each 

 molecule of the immovable stratum by one of the other strata, 

 the theorems we have demonstrated for the action of any mole- 

 cular system whatever on a material point which has been 

 slightly disturbed from its original position, since this is equiva- 

 lent to leaving this point fixed and displacing all the other mole- 

 cules of the system by the same quantity. Thus we may calcu- 

 late and compare, according to these theorems, the actions 

 exerted by any stratum on the fixed stratum ; and the actions 

 of the other strata will be in the same ratio, since their displace- 

 ments are supposed equal in the two cases. Consequently the 

 elasticities put into play by the two undulatory movements are 

 to each other as the elasticities which would be excited by the 





