390 Messrs. G. C. Foster and 0. J. Lodge on the Flow of 



will bound spaces of equal resistance if 



»a- r i __. r 8 -r a } 

 irfo + rj 7r(r 3 + r 2 ) 

 or 



r x r 3 =r\. 



Hence the radii of successive equipotential circles form a geo- 

 metrical progression whose common ratio may be called fi: 



6. To find the resistance of a belt bounded by concentric 

 circles whose radii are r x and r 2 respectively, let 



so that there are n equipotential circles contained in the belt, 

 where 



* i r <2 



n=-. log—- 



log fi ° r. 



Denote the thickness of the conducting sheet by 8 and its con- 

 ductivity by k ; then, by the last section, the resistance of a nar- 

 row ring whose bounding circles have the radii 1 and 1 + Ar(=yu.) 

 will be 



TC 1 Ar _ 1 Ar 



°~ ^S * i(2 + Ar)2w ~ ^cB ' 2 + Ar' 



But the whole belt we are considering is made up of n rings, 

 each of the resistance R . So the resistance of the whole belt is 



t> tj _ 1 Ar , r% 



a -*m - (2 + Ar)7r/c3 - log (1 + Ar) iog. rj S 



or, letting Ar decrease without limit, 



*»^xi « 



7. Putting this value of R into the expression for the differ- 

 ence of potential of two circles of radii r, and r 2 , we have 



v-- y *=<&sMd> 



whence the potential at any point of the sheet is at once given 

 in terms of the strength of the source and the potential at unit 

 distance from the source. Thus let Y Y be the potential at unit 

 distance, then the potential V at any distance r becomes 



8. The same result may be obtained without previously cal- 



