with the Flow of Electricity in a Plane, 379 



Resistance of some rectilinear figures to the current flowing 

 hetween two small circular electrodes. 



§ 8. Let us first remind ourselves how the general resistance- 

 expression is obtained. Given an infinite sheet with any num- 

 ber k of equal sources A 1? A 2 , * . * As and the same number of equal 

 sinks Bi, B 2 , ... B^ in it, the resistance between the two equipo- 

 tential lines which pass through any two fixed points P and Q 

 is given by Ohm's law as 



B _v P -v Q 



'PQ- 



kq 



where Y P stands for the actual potential at the point P, and 

 kq stands for the whole strength of current flowing in the 

 sheet — that is to say, for the quantity of electricity emitted by 

 each source per second, multiplied by the number of sources. 

 Now (A) gives as the potential at any point P 



Vp ~ 27r f cS l0g B 1 P.B 2 F... W 



hence 



And these expressions are perfectly general ; for unequal poles 

 may be regarded as aggregations of equal ones, and the num- 

 ber of sinks must always be equal to the number of sources, 

 although some of them may often be at infinity. 



Now we know that the equipotential lines of high potential 

 break up into k portions which ultimately become little circles, 

 one surrounding each source. Similarly the lines of low po- 

 tential surround the sinks. Take the point P on the circum- 

 ference of one of these little circles (radius /?i), that surround- 

 ing the source A say (omitting its suffix); and take the point 

 Q on a circle (radius p 2 ) surrounding the sink B. Then the 

 resistance between the two small circular electrodes A and B 

 is k times as much as that offered by the sheet to the 2k poles, 

 or 



Bab =7^ ^lo£ 



1 , fAB.AxB.A2B... BA.BiA.B2A... 



, 1 XXJL» . X3Li-L» . X3l2-»-» • • • JUX1 . JJ-^jOk. . XJ 2 X\. • 



8 g I pi.AjA.AjA... ' p2.BjB.B2B.. 



}»(-) 



where the higher powers of p ± and p 2 are neglected. Since 

 they are both small there is no necessity to distinguish them, 

 and p 2 may be written for their product. Moreover in very 

 many cases the sources and sinks will be similarly arranged ; so 

 that the two fractions which are multiplied together inside the 

 brackets of (a) are equal to one another, and we may then use 



