from Bradley to Fresnel. 127 



it is well known that #1 and v z are the roots of the equation in v 



; --0; 



1 . 1 ,1 



tf tf v- 



ti 2 3 



and accordingly Fresnel conjectured that the roots of this 

 equation represent the velocities, in a biaxal crystal, of the two 

 plane-polarized waves whose normals are in the direction 

 (I, m, n). 



Having thus arrived at his result by reasoning of a purely 

 geometrical character, he now devised a dynamical scheme to 

 suit it. 



The vibrating medium within a crystal he supposed to be 

 ultimately constituted of particles subjected to mutual forces ; 

 and on this assumption he showed that the elastic force of 

 restitution when the system is disturbed must depend linearly 

 on the displacement. In this first proposition a difference is 

 apparent between Fresnel's and a true elastic-solid theory ; for 

 in actual elastic solids the forces of restitution depend not on 

 the absolute displacement, but on the strains, i.e., the relative 

 displacements. 



In any crystal there will exist three directions at right 

 angles to each other, for which the force of restitution acts in 

 the same line as the displacement : the directions which possess 

 this property are named axes of elasticity. Let these be taken 

 as axes, and suppose that the elastic forces of restitution for 

 unit displacements in these three directions are 1/5], l/c 2 , l/s 

 respectively. That the elasticity should vary with the direction 

 of the molecular displacement seemed to Fresnel to suggest that 

 the molecules of the material body either take part in the 

 luminous vibration, or at any rate influence in some way the 

 elasticity of the aether. 



A unit displacement in any arbitrary* direction (a, )3, 7) can 

 be resolved into component displacements (cos a, cos /3, cos 7) 

 parallel to the axes, and each of these produces its own effect 



