on the Undulatory Hypothesis of Light. 479 



law will appear by considering that when a series of waves enters 

 perpendicularly into a refracting medium, the ratio of the breadth 

 occupied by a given number (n) of waves without the medium to 

 that occupied by the same number within, is equal to p, ; so that 

 n\ = n\fj,. Hence the rays of given colour propagated in differ- 

 ent directions in a crystalline medium will all have the same 

 value of Xfx,. Taking this into account, and substituting for q 



(a 2 e 2 \ 

 1 — £-g-J 3 it will be found that the 



equation (a) may be put under the form 



/* 2 -l=A + Be 2 , 



terms involving powers of e 2 above the first being omitted on 

 account of the small ratio of this quantity to a 2 . The factors A 

 and B are altogether independent of p, and e 2 . 



The apparent elasticity of the aether within the crystal will 

 depend both on the obstacles presented to its free motion by the 

 atoms of the crystal, and, as we have shown above, on the elas- 

 ticity of the crystal itself. Let a 12 , b n , d 2 be the coefficients of 

 the apparent elasticity of the sether in the directions of the axes 

 of coordinates, and r 2 that for the given direction. Then for 

 rays of given colour we shall have the three equations 



'J-1=A+B»A |!-1=A + B e2 2 , f-l = A + Be s \ 



together with the equation 



^_l = A-fBe 2 . 

 r 4 



If the three equations be respectively multiplied by cos 2 a, cos 2 /3, 

 cos 2 y, and the sum of the results be compared with the last equa- 

 tion, account being taken of the equation (/3), it will readily be 

 seen that i ^ cos 2 a cos a^ C os 2 7 



J3--5T- +— pT + -72- 



This equation we shall afterwards cite by its usual appellation, 

 the equation of elasticity. 



We come now to a critical part of our theory, by which it is 

 distinguished from all other attempts to solve the same problem. 

 Although the above equation gives the effective elasticity of the 

 sether in any direction in the crystal, we cannot immediately 

 infer from it the velocity of propagation of a plane wave, because, 

 as I have shown from hydrodynamical principles, the composite 

 character of the wave must be taken into account. The effect 

 this circumstance has on the velocity of propagation may be de- 

 termined by the following considerations. First, it is clear that 

 the wave cannot be composed of ray-undulations the transverse 

 motions of which are the same in all directions from the axis, 



