218 PROCEEDINGS OP THE AMERICAN ACADEMY 



XIV. 



ON SOME SIMPLE CASES OF ELECTRIC FLOW 

 IN FLAT CIRCULAR PLATES. 



By B. O. Peirce. 



Presented May 13, 1891. 



1. It is well known * that, if in a thin conducting plate of indefinite 

 extent there are two sources and a sink, each of strength numerically 

 equal to m, situated respectively at points A, B, C, which lie in order 

 upon a straight line, one of the lines of flow consists in part of a cir- 

 cumference of radius \/0A ■ CB drawn around C as a centre; so 

 that the flow inside the circumference would be unchanged if the part 

 of the plate outside it were cut away. In other words, if a circum- 

 ference be drawn in a thin conducting plane plate of indefinite extent, 

 the " image " in this circumference of a source, of strength m, situated 

 at a point P in the plane, is made up of a sink, of strength m, at the 

 centre of the circle, and a source of the same strength at Q, the inverse 

 point of P with respect to the circumference. 



If a sink be regarded as a negative source, it follows from this that 

 if inside a circumference drawn in a thin plane conducting plate of 



indefinite extent there are sources at the points Ai, A^, As, A^, 



of strengths algebraically equal to mi, m^, m^, .... m^, respectively, 

 and sources of strengths algebraically equal to — Wj, — m.2, — m^, 



— wjj, at the corresponding inverse points, then, if m^ -\- m., 



+ m3 + — -\- m^ = 0, there is no flow of electricity across the 

 circumference. 



2. If, at a fixed point P in a thin plane plate there is a sink of 

 strength numerically equal to m, and at another point P' in the plate 

 an equal source, and if P' be made to approach P as a limit and the 

 product m ■ PP' be kept always equal to a given constant, /a, we have 



* Kirclihoff, Popg. Ann. 1845, p. 497 ; W. R. Smith, Proc. Ed. Roy. Soc, 

 1869-70; Foster and Lodge, Phil. Mag., 1875; Minchln's " Uniplanar Kinemat- 

 ics," p. 213; etc., etc. 



