in Rectangular Order upon a Medium. 485 



By (3) each B can be expressed in terms of the correspond- 

 ing A. For brevity, we will write 



A„=^- 2 "B n , ...... (12) 



where 



• = (l + v)/(l-v) (13) 



We are now prepared to find the approximate value of the 

 conductivity. From (6) the conductivity of the rectangle is 



C_£j- 1 _2wBi'l _£T i^Bil . 



V,~«t; /gVxJ-aV afiKf 



so that the specific conductivity of the actual medium for 

 currents parallel to a is 



1 -§Bk ™ 



and the ratio of H to B! is given approximately by (10) 

 and (12). 



In the first approximation we neglect X 4 , 2 6 . . . ., so that 

 A 3 , A 5 . . . B 3 , B 5 . . . . vanish. In this case 



H=A 1 + B l 2 2 =B 1 (v'a- 2 + 2 2 ), . . . (15) 



and the conductivity is 



1 "a i 8(v / + a 2 2 2 ) ' (16) 



The second approximation gives 



^=/ + a^ 2 -^-a 8 S/, .... (17) 



and the series may be continued as far as desired. 



The problem is thus reduced to the evaluation of the quan- 

 tities S 2 , 2 4 , . . . We will consider first the important parti- 

 cular case which arises when the cylinders are in square 

 order, that is when /3 = a. f and rj in (11) are then both 

 multiples of a, and we may write 



2„ = «-"S„. ....... (18) 



where 



& n = t(m' + im)-»; (19) 



the summation being extended to all integral values of m, m\ 

 positive or negative, except the pair ra = 0, m! = 0. The 

 quantities S are thus purely numerical, and real. 



The next thing to be remarked is that, since m f m' are as 



