274 FRESNEL ON DOUBLE REFRACTION. 



but they have been displaced relative to the rest of the vibrating 

 medium, or if you like, it is this medium which has been dis- 

 placed relative to them, but not by the same quantity for the 

 different strata or layers of molecules ; the neighbouring stratum 

 is the least displaced, and the molecules of the succeeding strata 

 are found so much the more displaced from their positions cor- 

 responding to those of the molecules comprised in the first plane, 

 as they are further off from it. If we consider all the molecules 

 which were originally situated on the same straight line perpen- 

 dicular to this plane or to the surface of the wave, they will be 

 found transported, in consequence of the vibratory movement, 

 along a " sinusoidal" curve on one side and the other of this 

 perpendicular, which will be the axis of the curve; its ordinates 

 parallel to the wave, that is to say the small displacements of 

 the molecules, will be proportional to the sines of the correspond- 

 ing abscissae ; such at least will be the nature of this curve in all 

 cases where the illuminating particle which has produced the 

 waves, having been slightly displaced from its position of equili- 

 brium, is brought back to it by a force proportional to the dis- 

 placement. Confining ourselves then to the hypothesis of small 

 movements, we may represent the absolute velocity which ani- 

 mates an aetherial molecule after a time [t) by the formula 



u-= a . sin 1i:[t jjin which (?<) represents this velocity, (a) 



a constant coefficient which depends on the energy of the vibra- 

 tions, (2 tt) the circumference to radius unity, {x) the distance of 

 the molecule from the luminous point, (A) the length of an un- 

 dulation, and {t) the time elapsed since the origin of the motion. 

 If we suppose that these plane and indefinite waves are totally 

 reflected at a plane parallel to their surface, that is to say that 

 on this plane the aetherial molecules are restrained to remain 

 completely immovable, then the reflected waves will have the 

 same intensity as the incident waves, to which they will more- 

 over be parallel ; so that the same coefficient [a) must be em- 

 ployed in expressing the absolute velocities caused by these 

 waves in the aetherial molecules. Calling [z] the distance of the 

 direct wave from the i*eflecting plane, and (c) the constant 

 distance of this plane from the source of movement, the space 

 described by the direct wave is (c — z), and the space described 

 by the reflected wave which comes to meet it is (c + ;r). Hence 

 the velocities, brought in the same time and to the same point 



