Electricity in a uniform plane conducting Surface. 391 



culating the resistance of a belt of finite breadth as follows. The 

 radii of successive equipotential circles, beginning at the pole 

 itself, are (§ 5) 



Let the common difference of potential between consecutive 

 circles be Av ; then, if r=/jb n , the potential at distance r is less 

 than the potential at unit distance by n times Av, or 



V^V.— nAv=Y 1 — ; logr. 



But -. is constant : and putting 1 + dr for a. where dr is the 



log/z, 



breadth of the infinitely narrow belt whose inner boundary is 



the circle of unit radius, we have 



Av dv dv 



log p log (1 + dr) dr 



Multiplying this by the circumference of the belt (2ir) and by 

 the conductivity and thickness of the sheet, we get the strength 

 of the current across the whole belt, or 



2lTKh~ =Q, 



dr 



Av Q 



log fju 2ttkB 

 whence 



as before 



Two equal Opposite Poles in an Infinite Sheet. 



9. If there are two or more electrodes in an infinite uniform 

 conducting sheet, the strength and direction of the current and 

 the potential at any part of the sheet are obtained by the simple 

 superposition of the effects which each electrode would produce 

 at that part if it were the only electrode in the sheet. This 

 might be regarded as probable a priori ; and it is proved by 

 experiment to be the case. Hence the effect of any number of 

 poles in a sheet may be deduced by properly extending the con- 

 clusions already arrived at with regard to a single pole. We will 

 first discuss the case of a single source and a single sink of equal 

 strength. 



10. Lines of Flow. — Let A (Plate IX. fig. 1) be the source, 

 and B the sink. The flow-lines due to these, taken separately, 

 would be two equiangular pencils of straight lines drawn out- 



