454 Messrs. G. C. Eoster and 0. J. Lodge on the Flow 

 for circles round the source, and 



M-J-l and M-Jjs 



a-\-d v r a + d% r 9 



for circles round the sink. Substituting these values, we get 



2ttk8' 1 ° E Pl {a + dz) 

 if both circles surround the source, and 



2tt K 8' * p^a + d,) 



if they both surround the sink. In the former case we have 

 Pi<p 2 , and in the latter case pi>/? 2 ; but using ^ for the radius 

 of the smaller circle and p 2 for that of the larger, we may write 



^•%-f^ • • • • • .« 



for the resistance between two equipotential circles surrounding 

 the same pole, whether that pole be a source or a sink. 



If the circles surround opposite poles, the resistance becomes 



B // = * .log (* + *!)(* + <*.) . . . . (7) 

 2itk8 ° Pl p 2 



In the case of circles round the same pole, if the other pole 

 is infinitely distant, the value of R' becomes 



which is identical with (1), the value found for the resistance of 

 an annular belt of internal radius p x and external radius p 2 ; and 

 in fact the case supposed (an infinite distance between source 

 and sink) is physically identical with the case of a single pole in 

 an infinite sheet. 



When the circles surround opposite poles, if the radii are equal, 

 the resistance becomes 



1 ' a + d 1 , a + \/a 2 + p' 2 

 — Aog or — .log Z-4 



7TK0 ? p 7T/C0 ° p 



or when the common radius is small as compared with a, 



1 , 2a 



-^log — 



777CO " p 



23. The following modified forms of (6) and (7) obtained by 

 means of the relations (§ 20) 



R=^i°s^ (8) 



p a 



