FRESNEL ON DOUBLE REFRACTION. 279 



For one and the same degree of elasticity of the vibrating me- 

 dium, d^y measures the energy of the foi'ce which tends to bring 

 back the point M to P, and [y) is the space which this point 

 must describe. Hence for equal displacements of the point M, 



the accelerating force is proportional to —^ ; therefore the dura- 



A 



tion of its oscillation will be proportional to X. Consequently 

 the duration of the vibrations of the assemblage of particles re- 

 presented by the curve ABC {concamerations (is proportional to 



— —, denoting by (e) the elasticity of the medium. Now as this 



duration must remain constant for the same luminous waves, 

 whatever medium they traverse, it is necessary that the length 

 of an undulation (A) or the velocity of pi'opagation be propor- 

 tional to the square root of the elasticity put in play. It is suf- 

 ficient therefore to determine the law according to which this 

 elasticity varies in one and the same medium, to know all the 

 velocities of propagation with which light may be affected in it. 

 The law which I have found for the case where the axes of 

 elasticity have parallel directions throughout the whole extent of 

 the medium, is founded on the theorems of general statics which 

 have been demonstrated, and on the following principle : — The 

 elasticity put into play by the relative displacements of molecules 

 remains always the same in the same medium, so long as the direc- 

 tion of these displacements does not change ,and whatever moreover 

 may be that of the plane of the wave. I shall now endeavour to 

 give the theoretical reason of this principle, the accuracy of which 

 I have moreover verified by very precise experiments. 



The elasticity put into play by Luminous Vibrations depends 

 solely on their direction, and not on that of the Waves. 

 Let us consider the molecules comprised in one and the same 

 plane parallel to the surface of the wave : they preserve always 

 the same relative positions, and the resultant of all their actions 

 upon one of their number does not tend to impress on it any 

 movement. The same is not the case for the action of the next 

 stratum of the medium on this molecule, which being no longer 

 in its primitive position of equilibrium with regard to it, exerts 

 upon it a small action parallel to the plane of the wave. Con- 



VOL. V. PART XVI II. U 



