222 Dr. C. H. Lees on the Conductivities of Heterogeneous 



the plane eqni potential surfaces in the media ; in the second 

 the parts of the two media consist of right prisms of rectan- 

 gular cross-section, one dimension of the cross-section being 

 infinitely great compared to the other, and the length of the 

 prism being again infinitely great compared to this greatest 

 cross-sectional dimension, placed with their edges either 

 parallel or perpendicular to the equipotential surfaces. The 

 difference of order of magnitude of the three dimensions of 

 the elementary parts of the two media in this last case is 

 necessary in order that the disturbance of the flow near the 

 edges of the prism may be neglected in the calculation. 



These three cases, although of considerable interest, have- 

 only a limited application on account of the restrictions in- 

 troduced to enable the calculations to be carried out ; and I 

 propose here to consider the case of a medium formed of an 

 equal number of infinitely long prisms of square cross-section, 

 of two media having conductivities k x and k 2 , arranged as 

 shown in fig. 1, and bo-inded by two parallel equipotential 



Ffe. li 



A L' _L B_ _ 



planes, AB, CD, drawn through the diagonals of the cross- 

 sections of the prisms. 



I assume that the flux and potential are continuous at the 

 surfaces of separation of the two media. If, as in the case of 

 the electrical potentials of copper and zinc, a difference of 

 potential may exist between the media on the two sides of 

 the surface of separation without a flux resulting, the potential 

 which is continuous through the surface must be understood 

 to be that to which the flux is due. 



From symmetry it is evident in the first place that A'B', G'D\ 

 and in the second place that EF, E'F', &c, 

 are equipotential planes. Also that AC, BD, 

 LM, I/M', &c.j are lines of flow. Hence 

 the problem is reduced to finding the 

 conductivity of a prism of square cross- 

 section A'B'C'D' (fig. 2), of which A'D' 

 and B'C are equipotential surfaces, A / B / 

 and D'C lines of flow, and A'B'C consists 

 of a medium of conductivity k 2 , A'D'C 

 of a medium of conductivity k v 



