Influence of Obstacles on the Properties of a Medium. 481 



limited to two particles only. Given the proper temperature 

 with corresponding conditions of mass, shape, and distribu- 

 tion of charge on the particles, and, as it seems to me, almost 

 any amount of molecular complexity is possible. That I have 

 not taken this possibility into account does not, however, 

 vitiate the results here brought forward, as they do not pre- 

 tend to greater accuracy than that of their order of magnitude. 

 It is the cumulative value of these results which will, I 

 hope, be regarded as sufficient reason for the publication of 

 what is at best an incomplete piece of theory. 



Univ. Coll. Bristol. 



LVT. On the Influence of Obstacles arranged in Rectangular 

 Order upon the Properties of a Medium. By Lord Rayleigh, 

 Sec. R.S* 



THE remarkable formula, arrived at almost simultaneously 

 byL. Lorenzf and H. A. LorentzJ, and expressing the 

 relation between refractive index and density, is well known ; 

 but the demonstrations are rather difficult to follow, and the 

 limits of application are far from obvious. Indeed, in some 

 discussions the necessity for any limitation at all is ignored. 

 I have thought that it might be worth while to consider the 

 problem in the more definite form which it assumes w r hen the 

 obstacles are supposed to be arranged in rectangular or square 

 order, and to show how the approximation may be pursued 

 when the dimensions of the obstacles are no longer very small 

 in comparison with the distances between them. 



Taking, first, the case of two dimensions, let us investigate 

 the conductivity for heat, or electricity, of an otherwise uniform 

 medium interrupted by cylindrical obstacles which are ar- 

 ranged in rectangular order. The sides of the rectangle will 

 be denoted by a, /3, and the radius of the cylinders by a. The 

 simplest cases would be obtained by supposing the material 

 composing the cylinders to be either non-conducting or per- 

 fectly conducting ; but it will be sufficient to suppose that it 

 has a definite conductivity different from that of the remainder 

 of the medium. 



By the principle of superposition the conductivity of the 

 interrupted medium for a current in any direction can be de- 

 duced from its conductivities in the three principal directions. 



* Communicated by the Author. 

 t Wied. Ann. xi. p. 70 (1880). 

 \ Wied. Ann. ix. p. 641 (1880). 



