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XLIII. On a reciprocal relation between the Electrostatic 

 Fields of certain Distributions of Electricity and the 

 Magnetic Fields of corresponding Uniform Currents. 

 By Professor A. Gray, F.R.S* 



1. .1 uniform circular linear distribution of electricity and 

 a uniform circular current. 



THE heavy circle in fig. 1 represents the circular electrical 

 distribution of line density p and radius a. P is a 

 point external to the circle and its plane, at a distance a' 

 from its centre, and A is the intersection of CP with the 

 circle. 



Rff.l. 



Consider an element of the circle at E of length ds. 

 Project this element radially to E' on the concentric circle 

 (of radius a') described through P, and denote by ds' the 

 length of the projection. Then ds /ds = a'/a. Denote EP 

 •by r, ^CPE by ; we have then also AE' = r, and 

 Z.OE'A=0. 



The repulsion of the charge on ds exerted on a unit charge 

 at P, that is the electric field intensity at P, is 



-_ cos 6 ds a cos 6 ds' /1X 



V*=P—?-=P3—js- W 



Now the second of these forms is, by the diagram, and 

 * Communicated by the Author. 



