from Bradley to Fresnel. 129 



displacement), which urges the molecules of the medium parallel 

 to the wave-front. Hence the velocity of propagation of a 

 wave, measured at right angles to its front, is proportional to 

 the square root of the component, along the direction of dis- 

 placement, of the elastic force per unit displacement ; and the 

 velocity of propagation of such a plane-polarized wave as we 

 have considered is proportional to the radius vector of the 

 surface of elasticity in the direction of displacement. 



Moreover, any displacement in the given wave-front can be 

 resolved into two, which are respectively parallel to the two 

 axes of the diametral section of the surface of elasticity by a 

 plane parallel to this wave-front ; and it follows from what has 

 been said that each of these component displacements will be 

 propagated as an independent plane-polarized wave, the velocities 

 of propagation being proportional to the axes of the section,* 

 and therefore inversely proportional to the axes of the section of 

 the inverse surface of this with respect to the origin, which is 

 the ellipsoid 



* + + *-i. 



i 2 3 



But this is precisely the result to which, as we have seen, 

 Fresnel had been led by purely geometrical considerations ; and 

 thus his geometrical conjecture could now be regarded as 

 substantiated by a study of the dynamics of the medium. 



It is easy to determine the wave-surface or locus at any 

 instant say, t = 1 of a disturbance originated at some previous 

 instant say, = at some particular point say, the origin. For 

 this wave-surface will evidently be the envelope of plane waves 

 emitted from the origin at the instant t = that is, it will be 

 the envelope of planes 



Ix + my + nz - v = 0, 



where the constants /, m, n, v are connected by the identical 

 equation I 2 + m* + n z = 1, 



* It is evident from this that the optic axes, or lines of single wave-velocity, 

 along which there is no double refraction, will be perpendicular to the two 

 circular sections of the surface of elasticity. 



K 



