PROFESSOR CHALLIS, ON THE TRANSMISSION OF LIGHT, ETC. 525 



Consequently by the foregoing equations, 



Comparing this expression for R with tlie former, we have 



iiiV pdw li- pcLv 



2. Hitherto we have supposed the atoms of the medium to be absohitely fixed. If, as it is 

 reasonable to suppose, they are moveable by the mechanical action of the aetiierial vibrations, the 

 retardation produced by them will differ from that obtained above. Assuming the mean effect of 

 the presence of the atoms in this case also to be an apparent diminution of the elasticity of tiie 

 sether, the accclerative force of the retardation will vary as the density of the medium and the 

 difference of the effective accclerative forces of the tether and the atoms at a given position. That 

 is, if v be the velocity of an atom, where the velocity of the vibrating .-Bther is v, we shall have 

 dv dv \ I \ \ da dv 



R = - Kd {—- , very nearly. And, as before, R = aM 1 I — i- = - (n? - i) 



\dt df I ■ ■' V 1^1 pdx dt 



Hence, putting q for the ratio of -— to — , it follows that ^/ - 1 = KS{1 — r/). 



3. Since the retardation will be less and the velocity of propagation greater when the atoms 



are moved than when they are fixed, /x will be less in the former case than in the latter, and 



consequently (jr is a positive quantity. As it is known from experience that the rate of propagation 



of light in a given direction in a medium, is uniform and independent of the intensity of the light, 



dv dv 



the ratio of to — must be the same at different points of the same wave, and the same also 



dt df ' 



for vibrations of different magnitudes, if the breadths of the waves be given. But to account for 



the pha;nomenon of dispersion, q must be a function of \ the breadth of the wave. For our 



present enquiry it is not necessary to ascertain the form of this function. It is only necessary to 



assume that in crystallized media q is different for different directions. The theoretical reason for 



this probably is, that the retardation depends on the elasticity of the medium, and that the elasticity 



of crystallized media, and consequently the mobility of their particles, depends on the direction. 



4. What has been said above respecting the transmission of light through transparent media, 

 will suffice for the consideration of the theory of Double Refraction, on which I am now about to 

 enter. It will be assumed that in any medium which does not retard the progression of the 

 luminous rays equally in all directions, there are at least three directions at right angle-; 

 to each other, in which the retardation will take place in the manner hitherto supposed. Let 

 n', h^, c'' be the constants of elasticity for plane waves in these three directions, and let a be the 

 velocity of the waves in free space. Then, (/,, q^, q^, being the values of g- for the same directions, 

 the time of vibration being given, we have, 



4=1+^5(1-7.)- ^., = \ +Ki(l-q,), 4 = I + ^<U' - 7:.)- 

 a I)' c " 



a. When an atom of tlie medium is displaced in one of tlic three rectangular directions above 

 mentioned, the direction of (lisplaceiiient coincides by hypothesis with the line of jjropagation of the 

 waves. Although in general tills will not l)c the case, waves may still be |)n)pagated in all tlireetions 

 in the medium. For supposing plane waves of given breadth to be jiropagated simultaneously in 

 the three rectangular directions, (which may be called the axes of elasticity,) the resulting ell'oet on a 



