of Electricity in a uniform plane conducting Surface* 469 



a distribution of poles as that referred to at the beginning of 

 this paragraph. Then, taking points on the circumference of 

 the circles round A and B as the points to which the values of 

 r and r 1 in equation (18) respectively refer, we have approximately 



A r'\ , (AB) 2 .AB'.A'B 

 (bg- r j=log 2>AA/>BB , ; 



and since for the case supposed k = k' = 2, the resistance of the 

 part of the sheet extending between the pair of circles round A 

 and A' and the pair round B and B' is represented with similar 

 approximation by 



' 1 . (A B)«.AB'.A 'B .... 



R= i^8- l0g V.AA'.BB' < 19) 



If a circle be drawn with the centre C and radius 



CP= */CA . CA'= a/CB . CB', 



this circle will coincide, as already pointed out (§ 35), with a 

 flow-line of the system due to the combined action of the four 

 poles. Consequently no electricity passes into or out of this 

 circle ; and therefore the whole of the electricity supplied by 

 the source A flows to the sink B inside the circle (fig. 9), while the 

 whole of what is supplied by the source A' flows to B' outside 

 the circle. It follows, since the sources A and A' were assumed 

 of the same strength, that the resistance offered by the part of 

 the sheet lying within the circle of radius CP to the flow of elec- 

 tricity between the electrodes A and B is the same as the resist- 

 ance of the part of the sheet lying outside this circle to the flow 

 of electricity between the electrodes A' and B'. Hence also the 

 resistance of a disk bounded by the circle in question and con- 

 taining the two electrodes A and B is equal to twice the resist- 

 ance of the entire sheet to all four poles, and is therefore repre- 

 sented by 



J_ (AB)*.AB'.A'B 



n -2Tr K 8- l ° S ^.AA'.BB' ' ' * * ( M > 



which is the formula referred to in § 1 (page 387) as having been 

 given by Kirchhoff for the resistance of a circular disk with two 

 small circular electrodes anywhere upon it. 



39. As already stated, this formula is only approximate, and 

 in certain special cases it entirely fails. For instance, if one (or 

 both) of the poles passes to the edge of the disk, then, in order 

 that the circumference may still continue to be a line of flow, 

 the second pole of the same sign must coincide with it ; conse- 

 quently in such a case AA' or BB', or both, will vanish, and the 

 expression for the resistance fails by becoming infinite. The 

 reason evidently is that equation (20) was got by assuming p to 



