X. Chi the Induction of Electric Currents in Conducting Shells of Small Thickness. 



By S. H. BUIIBURY, M. A. .formerly Fellow of St. John's College, Cambrulge. 



Communicated by H. W. WATSON, D.Sc., F.R.S. 







Received March 22, Read May 3, 1888. 



IK a conductor be placed in a magnetic field, and the field be made to vary, closed 

 electric currents will be induced on the surface or within the substance of the 

 conductor. If the conductor be a wire forming a closed curve in which the electric 

 current is regarded as having only one degree of freedom, the laws of induction in it 

 are well known. 



The problem of induction of electric currents in solids or hollow shells has been 

 treated by several writers, generally with reference only to conductors of particular 

 shape, as spheres, infinite planes, cylinders, or ellipsoids, and with reference to special 

 variations of the external magnetic field.* 



The object of this paper is to investigate a general theory applicable to surfaces of 

 any shape in presence of an external magnetic field varying in any manner. 



It will be necessary to make use of one conclusion at which Professor LAMB arrives 

 in the second of the memoirs above referred to, namely, that the displacement currents, 

 which according to MAXWELL'S theory exist in the dielectric, have in all cases subject 

 to experiment no sensible influence in modifying the currents of conduction which 

 may be induced in metallic conductors. We may, therefore, treat the currents cf 

 conduction as the only possible currents. 



Tlie Condition of Continuity. 



Let u, v, w denote the components of electric current referred to unit of area. 

 Then, if there be no time variation of free electricity, the condition of continuity 

 requires that 



du dv dw _ 



te + dy + 'd: = 



* Professor NIVEK, ' Pbil. Trans.,' 1881, p. 307; Professor H. LAMB, 'Phil. Trans.,' 1883, p. 519, 1887, 

 p. 131; Professor LARMOE, 'Phil. Mag.,' January, 1884; Mr. O. HKAVISIDK, 'Phil. Mag.,' August, 

 September, 1886. 



MDCOCLXXXVIII. A. 2 Q 30.8.88 



