500 Lord Rayleigh on the Influence of Obstacles 



desired to push the application as far as possible, we must 

 employ closer approximations to (26) &c. It may be re- 

 marked that however far we may go in this direction, the 

 final formula will always give /jl 2 explicitly as a function of p. 

 For example, in the case of rigid cylindrical obstacles, we 

 have from (35) 



, , 1+P--305V+.. . n 



, *»' = (1-f) 1-^-8058^+ . ■ • • • (77) 



It will be evident that results such as these afford no 

 foundation for a theory by which the refractive properties of 

 a mixture are to be deduced by addition from the corre- 

 sponding properties of the components. Such theories re- 

 quire formulae in. which p occurs in the first power only, as 

 in (76). 



If the obstacles are themselves elongated, or, even though 

 their form be spherical, if they are disposed in a rectangular 

 order which is not cubic, the velocity of wave-propagation 

 becomes a function of the direction of the wave-normal. As 

 in Optics, we may regard the character of the refraction as 

 determined by the form of the wave-surface. 



The aeolotropy of the structure will not introduce any cor- 

 responding property into the potential energy, which depends 

 only upon the volumes and compressibilities concerned. The 

 present question, therefore, reduces itself to the consideration 

 of the kinetic energy as influenced by the direction of wave- 

 propagation. And this, as we have seen, is a matter of the 

 electrical resistance of certain compound conductors, on the 

 supposition, which we continue to make, that the wave- 

 length is very large in comparison with the periods of the 

 structure. The theory of electrical conduction in general 

 has been treated by Maxwell ('Electricity/ §297). A 

 parallel treatment of the present question shows that in all 

 cases it is possible to assign a system of principal axes, 

 having the property that if the wave-normal coincide with 

 any one of them the direction of flow will also lie in the 

 same direction, whereas in general there would be a diver- 

 gence. To each principal axis corresponds an efficient " den- 

 sity," and the equations of motion, applicable to the medium 

 in the gross, take the form 



d*g dS d\ dS d 2 £ d8 



Sz? =mi ^' ^ =mi -%' ^ =Wl S> 



where f, 77, f are the displacements parallel to the axes, m l is 

 the compressibility, and 



&=# + *? + #. 



dx dy dz' 



