502 Influence of Obstacles on the Properties of a Medium. 



given at once by (67). The equation may also be written in 

 the form given by Lorentz, 



~fjy>- =^To= constant; . . . (79) 

 /r + 2 p v + z 



and, indeed, it appears io have been by the above argument 

 that (79) was originally discovered. 



The above formula applies in strictness only when the 

 spheres are arranged in cubic order*, and, further, when p 

 is moderate. The next approximation is 



v— 1 A v + 4/3' 



If the obstacles be cylindrical, and arranged in squnre 

 order, the compound medium is doubly refracting, as in the 

 usual electric theory of light, in which the medium is sup- 

 posed to have an inductive capacity variable with the direction 

 of displacement, independently of any discontinuity in its 

 structure. The double refraction is of course of the uniaxal 

 kind, and the wave-surface is the sphere and ellipsoid of 

 Huygens. 



For displacements parallel to the cylinders the resultant 

 inductive capacity (analogous to conductivity in the conduc- 

 tion problem) is clearly 1— p-¥vpi so that the value of /x" 

 for the principal extraordinary index is 



,,' = l + (v-l)p, (81) 



giving Newton's law for the relation between index and 

 density. 



For the ordinary index we have 



^ = (2(5), 



in which v'= (1 + v)/(l — v), while S 4 , S 8 . , . have the values 

 given by (49). If we omit p* &c. we get 



*-7& W 



or 



H 2 -l 1 _ 1 



fi 2 +lp v' v + 1 



(83) 



The general conclusion as regards the optical application 

 is that, even if we may neglect dispersion, we must not ex- 

 pect such formulao as (79) to be more than approximately 

 correct in the case of dense fluid and solid bodies. 



* An irregular isotropic arrangement would, doubtless, give the same 

 result. 



