472 Prof. Challis on the Theory of Double Refraction 



dx 

 the velocity -j- y it is evident that a portion of the velocity of the 



incident waves equal to this velocity, and the corresponding 

 portion of the condensation, will be transmitted through the 

 atom (supposed extremely small) just as if its place were occu- 

 pied by the fluid itself. Consequently the expression for the 

 accelerative force of the atom due to the waves will in this case 

 be 



k(msm^(bt-x + c)- d ^) 



But, in addition to this force, there will necessarily be brought 

 into play, by the displacement of the atoms from their positions 

 of equilibrium, a force due to the elasticity of the refracting 

 medium. Now from the foregoing considerations it may be seen 

 that we are not here concerned with the absolute displacement 

 of a given atom, but with the relative displacement of adjacent 

 atoms, because the accelerative force of elasticity depends only 

 on relative displacement. In fact, on account of the large value 

 of X compared with the mean interval between the atoms, all the 

 atoms included between two planes at the distances x — Ax and 

 x-\- Ax from the origin, will have movements very little different 

 from that of the atom at the distance x, 2A# being supposed 

 very small compared to X. That being the case, we shall suffi- 

 ciently take into account the effect of the elasticity of the medium 

 on the motion of a given atom, by treating the medium as if it 

 were a continuous elastic substance ; and whatever may be the 

 relation between its pressure (p) and its density (o), for the 

 very small movements that will have to be considered, the varia- 

 tions of pressure may be assumed to be proportional to the va- 

 riations of density. Hence, if the elasticity of the medium be 

 such that dp = e <2 dS, we shall have for the accelerative force of the 



atom due to the elasticity — e 2 . -7-. But since the movements of 



ax 



the atoms are governed by the action of the setherial waves, it 

 follows that these movements and the accompanying condensa- 

 tions will be propagated uniformly through the medium with the 

 same velocity as the waves of light — that is, with the velocity b. 

 Motion of that kind, as I have shown elsewhere, and as, in fact, 

 it is easy to verify, must satisfy, without reference to the action 

 of forces, the equations 



v = b8=f{x-bt), 

 v being the velocity of a particle of the medium. Hence it 

 follows that 



B 'dx- e 'bdx~ b' J[X bt) -tfdt' 



