Equations in a Homogeneous Isotropic Medium. 33 



The above applies to a homogeneous medium. But, if in 



curl (H-li)= (477/; + cp)E, \. . . (la) 



curl (e-E)=(47r^ + At/>)H, / . . . (2 a) 



differing from (1), (2) only in the introduction of impressed 

 forces e and h, we write 



(H,li,E,e)=(Hi,lii, El,el)e-P^ 

 we reduce them to 



curl(Hi-li,)=c(<7+^)Ei -1 



curl (ei-Ei)=/*(-o-+79)HiJ ' ' ' \ i 



and these, if o- = 0, are the equations of a nonconducting 

 dielectric. That is, 



p = iirhlc = 4:'7rgfiJ, = constant 



is the required condition. Therefore c and fi may vary any- 

 how, independently, provided k and g vary similarly*. The 

 impressed forces should subside according to e"''^, in order to 

 preserve similarity to the phenomena in a nonconducting 

 dielectric. 



Observe that there will be taihng now, on account of the 

 variability of (/^/c)^ or (jlv. That is, there are reflexions and 

 refractions due to change of medium. The peculiarity is that 

 they are of the same nature with as without conductivity. 



6. First Special Case. — A special case of (11) is given by 

 taking /i = and ^ = ; that is, a real conducting dielectric 

 possessing no magnetic inductivity, in which k/c is constant. 

 If the initial field be polar, then 



E = Eoe-p', H=0 (12) 



This extension of Maxwell's before-mentioned solution I have 

 given before, and also the extension to any initial field, 

 and the inclusion of impressed forcesf. The theory of the 

 result has considerable light now thrown upon it. 



If the initial field be arbitrary, the solenoidal part of the flux 

 displacement disappears instantly, therefore (12) is the solu- 

 tion, provided Eq means the polar part of the initial field; that 

 is, Eq must have no curl, and the flux cEo/47r must have the 

 same divergence as the arbitrarily given displacement. 



Now an impressed force e produces a solenoidal flux only. 

 Therefore it produces its full effect and sets up the appropriate 



* In § 4 of the article referred to in the last footnote the property 

 was described only in reference to a homogeneous medium. 



t "Electromagnetic Induction," 'Electrician,' December 18, 1885, and 

 January 1, 1886. 



Phil. Mag. S. 5. Vol. 27. No. 164. Jan. 1889. D 



