in Rectangular Order upon a Medium. 487 



integral. Hence 



the limits for y being + v, and those for x being v and co . 

 Ultimately i? is to be made infinite. 

 We have 



J-' 1 



d v - 



[x + iyf 



I 

 X + IV 





i 2v 







^ 



—iv ~~ x* + v 2 ' 





and 















r 



2vdx 

 x 2 + v*~ 



2 tan -1 



OO - 



-2 tan" 1 1=4tt. 







Accordingly 

















S 2 = 



7T. 



• 







• • 



(22) 



In the case 



of square 



order, 



equations (10) (12) 



give 





Ha 2 



= • + <*% 



r a 8 . 



V 



u- 



-£«-... 



• 









3 a 8 



i/« 8 



S 4 2 



7 a" 



--,^16^8 .. 



• J • 



(23) 



and by (14) 















Conductivity = l %- . ^A (24) 



If p denote the proportional space occupied by the 

 cylinders, 



J p = 7ra 2 /a 2 ; (25) 



and 



Conductivity = 1 ^ wp—... ■ ■ (26) 



Of the double summation indicated in (19) one part can 

 be effected without difficulty. Consider the roots of 



sin (f — im7r) = 0. 

 They are all included in the form 



f = mfir + 2?W7T, 

 where m 1 is any integer, positive, negative, or zero. Hence 

 we see that sin (jj—irnir) may be written in the form 



A(i-J-Yi-_i-Yi- T - J -Yi--4=-\... 



\ xmir/X imir-\-irJ\ imTr — 7rJ\ wiir + zir J 



