Ill 



ON THE FLOW OF ELECTRICITY IN 

 CONDUCTING SURFACES * 



THE conditions of a steady flow of electricity in a con 

 ducting surface are completely determined, if we know 

 either the nature of the electrical distribution throughout 

 the surface, or the direction and intensity of the flow at 

 every point. On the first of these ways of considering the 

 question, the problem is solved if we can express the 

 potential v at any point as a function of the co-ordinates, 

 and the nature of the distribution will be indicated to the 

 eye by forming the equipotential curves 



v = const. . . . (i). 



From the second point of view, we should endeavour 

 to determine the lines of flow by equations of the form 



= const. . . . (2). 



The curves determined by equations (i) and (2) are 

 obviously orthogonal, and since 



d-v d 2 v _ 



dtf + W~&quot; 

 we know, by a theorem of Lame&quot; and Stokes, 2 that 



&amp;lt;Pu &u_ 



dx* + dy*~ 

 Kirchhoff, in the year 1845, took up the problem for plane 



1 Communicated by Professor Tait to the Royal Society of Edin 

 burgh on February 21, 1870, and published in its Proceedings, vol. vii. 

 pp. 79-99. 



2 See Thomson and Tail s Natural Philosophy, i. 542. 



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