116 REPORT ON PHYSICAL OPTICS. 



of the displacement, whatever be the nature or laws of the mole- 

 cular action ; and the only assumption which he makes is that these 

 three directions are parallel all throughout the crystal.* These 

 directions Fresnel denominates axes of elasticity. He conceives 

 that they ought also to he axes of symmetry with respect to the 

 crystalline form ; but observes that M. Mitscherlich has noticed 

 some crystals in which this does not hold.f If on each of these 

 axes, and on every line diverging from the same origin, portions 

 be taken which are as the square roots of the elastic forces in their 

 direction, the locus of the extremities of these portions will be a 

 surface which Fresnel calls the surface of elasticity. This surface 

 determines the velocity of propagation of the wave, when the 

 direction of its vibrations is given. For the velocity of undu- 

 latory propagation in an elastic medium, being as the square root 

 of the elastic force, must be represented by the radius- vector of the 

 surface of elasticity in the direction of the vibrations. 



Now let us conceive a plane wave advancing within the crystal. 

 By the principle of transversal vibrations the movements of the 

 ethereal molecules are all parallel to the wave. But the motion 

 of each displaced particle is resisted by the elastic force of the 

 medium, and that force is, in general, oblique to the direction of 

 the displacement. Fresnel shows, however, that the displacement 

 may be resolved in two directions in the plane of the wave, such 

 that the elastic force called into action by each component will be 

 the resultant of two forces, one of which acts in the direction of 

 the displacement itself, while the other is normal to the wave* 

 The latter, by the principle of transversal vibrations, can produce 

 no effect ; and the former will give rise to a wave propagated 

 with a constant velocity. These two directions, he finds, are those 

 of the greatest and least diameters of the section of the surface of 

 elasticity made by the plane of the wave ; and if the original 

 displacement be resolved into two, parallel to them, each component 

 will give rise to a plane wave whose velocity of propagation is 



* This will be the case, if the homologous lines of the groups of particles are all 

 parallel ; an arrangement at once the simplest and most natural, and which appears to 

 be observed in most crystallized bodies. Fresnel admits, however, the possibility of 

 other regular arrangements ; and he conceives that the phenomena of circular polari- 

 zation in rock crystal oblige us to suppose that its molecules are arranged according to 

 some less simple law. 



t See Bulletin de la, Socie'tt Phikmafhi.jne, March, 1824. 



