FRESNEL, ON DOUBLE REFRACTION. 255 



cos^ i . sin'^ 2w .t . + sin^ i . sin^ 2 7r It ) (A) . 



From this formula may also be obtained the displacements of the 

 vibrating molecule relative to its position of rest, by changing 

 the time {t) by a quarter of a circumference, or the common 

 point of departure by a quarter of an undulation ; for these dis- 

 placements follow the same law as the velocities, with this dif- 

 ference only, that the velocity is nothing at the moment of the 

 molecule's being at its greatest distance from its position of rest, 

 and that the instant of its passing through this position is that 

 of maximum velocity. 



For the same reason the displacements of the vibrating mole- 

 cule, measured parallel to the rectangular directions O O' and 

 EE', are proportional to the expressions 



cos i . cos 2 71 .t, and sin i . cos 2 tt (i ). 



If we wish to find the curve described by the molecule referred 

 by parallel coordinates to OO' and EE', it is sufficient to write 



cos i . cos 2 7: t = X, and sin i . cos 2 7r li )=yj 



and to eliminate {t) between these two equations, which gives 



x^ sin^ i + y^ . cos^ i . — 2 x y . sin i . cos i . cos - — '— 



. 2 • 2 • ■ o'^Tt .a 

 = sm"' e . COS'' ^ . sm'' , 



an equation of a curve of the second degree referred to its centre. 

 Without discussing this equation, we are certain beforehand 

 that the curve can only be an ellipse, since the excursions of the 

 molecule in the direction of x and y have for limits the constant 

 quantities sin i and cos i. This curve becomes a circle when 

 i = 45°, and (a) contains the fourth part of an undulation an 

 uneven number of times; or, in other words, when the two 

 systems of waves polarized at right angles have the same in- 

 tensity, and differ in their route by an uneven number of quarter- 

 undulations. We have then 



• • /l ^ ^ 1 • (i 



sm I = cos ? = A / _, cos 2 TT . — = and sin 2 tt . — = 1, 



V 2 ^ ^ 



which reduces the above equation to 



x^ + f = i. 



