with the Floiv of Electricity in a Plane, 381 



§ 10. One more special case of the infinite strip I will take, 

 because a constantly occurring product occurs here in its sim- 

 plest form. Let the two poles be both on the same side of the 

 strip and at a distance c from one another. The images of A 



are just as before, and the product • -t^-t- • -r~^ . . . equals 



S G 1 2 2 s 2 + J 2V + c 2 4V + c 2 4V + c 2 \i 

 \p*' 2 V v 2 V " 4V ' 4V '") 



=K 1+ A)( 1+ AX 1+ <&)••• 



. , ire 

 o Slnh 2i- 



p TTC 



Therefore 



*=^(|^S < 2 > 



Similarly, if we had taken the poles one on each side of the 

 strip, but, instead of opposite each other as at first, at a distance 

 c from one another measured parallel to the sides of the strip, 

 we should have obtained 



E =iMl cosh E>-- 2 < 3 > 



If * be made infinite, (2) becomes — k log -, which is the 



correct expression for the case of two poles on the edge of half 

 an infinite sheet. If s is small, (2) and (3) become equal ; 



* Here and elsewhere sinh # may be considered merely an abbreviation 

 for — o — i or as equal to j sin ix. Similarly 



, e*+e-* 

 cosh x=s — — — , or = cos«.r, 



and 



tanh #= - — - — = - tan ix. 

 e*+e-* i ' 



where *= tj — J. It is useful to remember that 



cosh 2 x— sinh 2 a: =1, 



sinh0= tanh0= sin 0=0, 



cosh0= tanh oo =1, 



sinh oo a= cosh oo =$e°° 



that 

 that 

 and that 



