of Electricity in a uniform plane conducting Surface, 

 may also be noted : 



R' = 



R": 



2ttk8 

 1 



PA 

 log^, 



R' = 



R" = 



27r*8"°A l A 2 



1 . (v// , 1 -x// 1 )(v / ^ + v / / 2 ) 

 2tt*;8 0S (v^W/i)(VVVw' 



~r~ (A+A)(^+^) 



2™S ^'(^-^(v/^-N/y . 



455 



(9) 



(10) 



24. If two equipotential circles alone are given and it is re- 

 quired to find the resistance between them, the position of the 

 poles A and B being unknown, the expressions given above for 

 R' and R" are inapplicable (unless the circles are so small that 

 their centres sensibly coincide with the poles) ; for in such cases 

 the values of r, r 7 , a, d, h, and / are all unknown. The distance, 

 however, between the centres of the circles is directly measurable ; 

 and the resistance can be expressed in terms of this distance and 

 of the known radii as follows. 



Calling the distance between the centres D, we have for circles 

 surrounding the same pole, 



D = d a -rf l = v / «* + ri-\/« 8 + tf, 



which gives 

 d l = \/ 



a 2 + tf = 



2D 



^v^+pj^ei 



•A+w 



2D 



and 



_\/(pl+Po + V)(pl+p2-V)(pl-p* + WPl-p2-V) 



Hence (6) may be written * 

 1 



2D 



R' = 



— loo 



P^lPi + Pt + ty fa+Pz-Wpi-pi + tyipi-p ^^ + Pl-pl-W 



Pl^(Pl+p2 + ^)(pl + P^^)^-p2 + ^)(pl-p2-^)+P 2 l -pl + ^ 



When the equipotential circles surround different poles, we have 

 D=^ 1 + ^=v/« 2 + p2 + x / a 2 +/0 2 j 



* This expression was given in a slightly different form by Gaugain on 

 the authority of Blavier (Ann. de Chim. et de Phys. Ser. 3. vol. lxvi. p. 203. 

 1862); and a demonstration has since been published by the latter 

 (Journal de Physique, vol. iii. p. 115, April 1874). 



•(H) 



