380 Mr. 0. J. Lodge on some Problems connected 



the simplified expression 



^ 1 . /AB.A 1 B.A 2 B...\ 



(« 



p . A X A . A 2 A 



§ 9. Before proceeding to apply the image theory and this 

 expression to the general case of a two-sided figure, it will be 

 well to consider a special case or two, — and first the very 

 simple case of a parallel-sided strip of breadth s, with the two 

 poles, A and B, one on each side of the strip and opposite to one 

 another (fig. 1). 



Fig. 1. 



h 



B L 



A will be reflected in the^opposite side, its image Ai being 

 at a distance s from that side and from B ; then A a will be 

 reflected, its image A 2 being 2s distant from A and 3s from B ; 

 then A 2 will be reflected, and so on. The images of B will 

 evidently be similarly arranged to those of A mutatis mutandis; 

 so we may use the second of the resistance-formulae 

 p 2 . AB.A 1 B... 



doubling it because the poles are only halfway on the sheet 

 (see below, § 22) 



Now 

 AB =A 1 B = s, 

 A 2 B = A 3 B = 3s, 

 A 4 B=A 5 B=5s, 



AiA=A 2 A=2s, 

 A 3 A=A 4 A=4s ; 



Hence the above product becomes 



1.3.3.5.5.7... 

 2.2.4.4.6.6... 



which equals 

 Therefore 



R=— lo — 



7Tk8 



(1) 



