250 FBESNEL ON DOUBLE REFRACTION. 



sent the phaenomena of interference, except so far as their planes 

 of polarization are parallel or approach to parallelism. When 

 these planes are perpendicular, the absolute velocities of the 

 aetherial molecules are also perpendicular to each other ; if, 

 therefore, at each point of the common direction of the two rays 

 we wish to obtain the resultant of the two velocities impressed 

 by them on the molecule of aether, we must take the sum of the 

 squares of the two velocities ; this will be the square of the 

 resultant. The same calculation applies to all the points of the 

 two systems of waves, whatever may be in other respects their 

 difference of route ; thvis the sum of the squares of the absolute 

 velocities impressed on the aetherial molecules by the union of 

 the two systems of waves will always be equal to the sum of the 

 squares of the absolute velocities caused by each of the luminous 

 rays ; or, in other words, the intensity of the whole light will 

 always be equal to the sum of the intensities of the two inter- 

 fering rays, whatever may be their difference of route. Variations 

 therefore in this difference cannot produce those alternations of 

 brightness and obscurity which are observed in ordinary light, 

 or in rays polarized in parallel directions. The ease with which 

 our hypothesis explains the first law of interference of polarized 

 rays is then seen ; and this is what might be expected, since 

 it was from this law itself that we have derived it. 



We may regard it as sufficiently established by the demon- 

 stration just given ; but it will not be without use to show that 

 the same hypothesis agrees quite as well with the other laws of 

 interference of polarized rays which become the immediate con- 

 sequences of it. These theoretical developments on the proper- 

 ties of polarized light will not appear out of place in an essay 

 on double refraction, and will moreover find their application in 

 the memoirs which we intend to publish afterwards on the colours 

 of crystalline plates. 



When the interfering luminous beams have their planes of 

 polarization parallel, their vibratory movements have the same 

 direction, and therefore are added to each other along the whole 

 course of the rays if the difference of route is nothing, or equal 

 to an even number of semi-undulations ; and are subtracted one 

 from the other when the number of semi-undulations is uneven. 

 In general, to obtain in this case the intensity of the light 

 resulting from the concourse of the different systems of waves, 

 we may use the formulae already cited from my Memoir on Dif- 



