246 FKESNEL ON DOUBLE REFRACTION. 



end of the time (/) will be represented by a . sin 2 tt ( ^ j : («) 



being here a constant coefficient proportional to the amplitude 

 of the oscillations of the aetherial molecules or to the intensity of 

 their absolute velocities*. This being established, let us con- 

 sider one of the two interfering rays. Whatever be the direc- 

 tion of the absolute velocity of the tetherial molecule, we may 

 always decompose this velocity at each instant in three constant 

 directions at right angles to each other; the tirst, for example, 

 being the direction of the normal to the wave, and the other 

 two perpendicular to this being, the one parallel, the other per- 

 pendicular, to the plane of polarization. By the general pi'in- 

 ciple of small motions, we may consider the oscillations per- 

 formed by the a?therial molecule, of whatever nature they may 

 be, as resulting from the combination of three series of recti- 

 linear oscillations whose directions coincide with these three 

 rectangular axes, oscillations which, for the greater generality, 

 we shall suppose to have commenced at different epochs. 



Call {{) the time elapsed since a common epoch, and repre- 

 sent by (w), (f) and {w) that which must be added to {t) to ob- 

 tain the whole time reckoned from the origin of the motion in 

 each of the three modes of rectilinear vibration ; then the abso- 

 lute velocities belonging to the instant we are considering will 

 be 



G . sin 2 7r ( « -H f ), 



6 . sin 2 TT yv + t j, 



c . sin 2 TT ( ?^; -|- t j , 



c, b and c being constant coefficients, which denote the intensity 

 of the absolute velocities in each system of rectilinear oscillation. 

 Now let us consider the second polarized ray, and decompose 

 its absolute velocities in the direction of the same rectangular 

 axes. If we represent by {x') the path which it has passed over 



* A demonstration of these forinulag, and a more detailed explanation of 

 tlieir usage, will be found in the Memoires de VAcademie des Sciences, toni. v. 

 Those readers who are not familiar with the theory of luminous waves, may 

 first study its elementary principles in an article on light in the supplement to 

 the French translation of the fifth edition of Thomson's ' Chemistry.' [This 

 article has been translated by Dr. Young in Brande's ' Quarterly Journal of 

 Science' for Jan. 1827 and following numbers. — Tu. N.] 



