R= 



470 Messrs. G. C. Foster and 0, J, Lodge on the Flow 



be very small in comparison with any of the distances between 

 the poles ; and this can no longer be true when any two of them 

 coincide. By slightly modifying the notation, we can obtain an 

 expression which does not fail in the same way. Thus, let 

 the points to which the values of r lf r 2 , . . . refer be respec- 

 tively a'point F where the straight line AB cuts the equipoten- 

 tial circle round A, and a point G where the same line cuts the 

 equipotential circle round B. Then we have AF = BG=p, and 

 the expression for the resistance of the sheet to the current from 

 all four poles becomes 



1 / BE . B'F AG . A'G \ 



4™S ' ° S UF . A'F ' BG . B'G> * [ ™ a) 

 while the resistance of the disk bounded by the circular flow- 

 line may be written 



-, 1 , AG.BF.A' G . B'F , on . 



R = WT l0g p*.A'Y.WG - ' ' ( 20fl) 

 In this form the expression for B/ admits of extension to some 

 special cases. For example, let the pole A pass to the edge of the 

 disk, while B goes to the centre ; then A' will coincide with A, 

 and B' will go to an infinite distance. We have then sensibly 

 AG=BF = A'G== the radius of the disk( = P), also A'F=p,and 

 the resistance to the flow between the pole at the edge and the 

 pole at the centre of the disk becomes 



i- l0 4- ( 21 ) 



If both poles go to the edge, A' coincides with A, and B' with B, 

 the system being reduced to two equal opposite poles at a distance 

 AG=BF=2« from each other. In accordance with this, the 

 resistance of the whole sheet becomes 



1 , 2a 



TTfcS ' p' 



which 1 is the value already given for the same case in equa- 

 tion (8). 



40. If, all else remaining as before, the signs of the poles A' 

 and B' are interchanged (that is, if A' becomes a sink and B' a 

 source), the circle drawn with the centre C and radius v'CA . CA ; 

 becomes an equipotential line instead of a line of flow, the circle 

 with centre D and radius \^DB . DA remains an equipotential 

 line, and the only circular lines of flow are the circle through the 

 four poles and the imaginary circle (§ 36) whose centre is the 

 intersection of AB' and A'B. The resistance of the sheet to the 

 flow of electricity between this new arrangement of poles is given 

 by interchanging in (19a) the signs of the radii vectores drawn 



