22 I Dr. C. H. Lees on the Conductivities of Heterogeneous 



the flux between any two points of the area A BCD will be 

 identical with the quantities for the corresponding points of 

 the area A'JB'C'D f . 



If V is the difference of potential between the points A, C 

 of fig, 3 and AC = a, the flux bounded by the stream-lines DC 

 and AB 



= h a C0S e J = k,Y cot 0, where = /L DAC. 

 a sin 6 



Hence the flux bounded by the stream-lines D'O'.and A'B f in 

 fig. 2 under the potential-difference V is equal to £ t V cot#. 



If k is the apparent conductivity of the square, the flux 

 bounded by the stream-lines D'C and A'B' is equal to kV; 

 hence 



^Vcottf^V, or k = k 1 Gote. 



Now in fig. o, since the flux is continuous across AC, 

 fe cot DAC = £ 2 cot BCA. Also L BAC= L DAC. 



.•. ^cot = A- 2 tan ; 



L <?, cot 6 =■ i / _? . 



V k Y 



Therefore the apparent conductivity k of the square 



If now this medium, consisting half of material k i} half 

 of material k 2 , be supposed homogeneous, and be combined, in 

 the way indicated in fig. 1, with an equal volume of the k L 

 medium, we have k f the conductivity of the new medium, 



= **"*!*, 



= k^ k£. 



Similarly, by making other combinations of the new 

 media and the old, we deduce that if a volume pi of a 

 medium of conductivity k x be combined in this manner with 

 a volume 1— p ± of a second medium of conductivity k 2 , the 

 conductivity k of the combination will satisfy the equation 



*=* 1 ftifc a ( 1 -ft>j (1) 



or 



log k = Pl log k x + (1 — p L ) log £ 2 , 



k 

 -logk 2 + Pl logj (2) 



That is, for the mixture considered, the logarithms of the 

 conductivities for fluxes parallel to the diagonals in fig. 1 

 follow the law of means. 



