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



and if s=p, they reduce to It= — k (neglecting the small 



term ~ log 7r), which is right for the resistance of a wire 



or thin strip of sectional area ph and length c. The presence of 

 the term neglected in comparison with a term which has p in its 

 denominator, can be accounted for by remembering that the elec- 

 trodes are not exactly straight bars at a distance c, but quadrants 

 of small circles whose centres are distant c from one another. 



§ 11. We may also consider one special case of the infinite 

 "wedge " or irregular two-sided polygon. Let the two poles 

 be one on each side of the wedge at the same distance r from its 



rrr 



angle 0, and call this angle 0= — (n integer). The images of A 



will lie on a circle with centre (fig. 2), as in the kaleidoscope. 



Fig. 2. 







/ 



V 



! 



r 



V 







/ v « 



\ 



\ 



\ 



I 



| 











\ 









?Aa 







ien 







">*~- 





Ar, 









AB: 



=AiB 



= 2r sin 







L 2' 





AiA= 



=A 2 A= 



--2r 



. 20 



sin I' 



A 2 B= 



=A 3 B= 



= 2r sin 



30 





A 3 A= 



=A 4 A= 



=2r 



. ±0 



so the product which occurs in (/3) equals 



T~ U . n0 

 . 2 30 . 2 50 f . 2 n, Sm 2 

 2r_ sm V m V sm V- | sm 2 



p ' . 2 20 . 2 4<9 . 2 60 1 . .n-2,. 



r sm 2 -^- sm 2 -^ sm 2 -5- , . • I sm 2 ~ ir - a 



2 2 2 2 . nd , . ,1 



^ _. sin^r- when n is odd. 



. 2 n-l, A 



2 n— 1^ 

 3m 2 — ^-0 



^ sin^- when n is even, 



