276 FRESNEL ON DOUBLE REFRACTION. 



we see that the curve represented by this equation always cuts 

 the axis of {z) in the same points for every instant {t), that these 

 are the points for which 



z = 0, z = —\,z = K,z——K, &c. 



The greatest displacements of the molecules, or the greatest values 

 of?/, correspond, on the contrary, to the values of r, which contain 



— A an uneven number of times. Considering now the changes 



undergone by the curve from one moment to another, consequent 

 on the different values of t, we see that the ordinates always pre- 

 serve the same proportion to each other as in the oscillations of 

 a vibrating cord ; and the preceding formula shows that the ve- 

 locities which animate the molecules at each instant follow also 

 the same law as those of the elements of a vibrating cord. We 

 may therefore assimilate each portion of the medium comprised 

 between two consecutive nodal planes to an assemblage of vibra- 

 ting cords perpendicular to these planes, and attached to them 

 by their extremities ; the tension of these cords would produce 

 the same effect as the elasticity of the medium, since like this 

 latter it would incessantly tend to restore the straight lines which 

 had become curved by the small displacements of the molecules 

 perpendicular to these lines, and that with a force proportional 

 to the angle of contingence. Hence, since the direction of the 

 oscillatory movements, their law and that of the accelerating 

 forces, are the same in the two cases, the rules which apply to 

 the one necessarily apply to the other. Now, we know that in 

 order for a cord to execute always its vibrations in the same 

 time, when its tension varies it is necessary that its length in- 

 crease proportionally to the square root of its tension ; therefore 

 the length of the same luminous waves (which must remain iso- 

 chronous in all mediums which they traverse) is proportional to 

 the square root of the elasticity which urges the molecules of the 

 vibrating medium parallel to their surface ; hence the velocity 

 of propagation of these waves, measured perpendicularly to 

 their surface, is proportional to the square root of this same 

 elasticity. 



Without recurring to the known laws of the oscillations of 

 vibrating cords, it is easy to demonstrate directly, by geometrical 

 considerations, the principle just announced. 



Let ABC (fig. 6) be the curve formed by a row of molecules 



