Self-induction of Wires. 335 



on the second is 



conv. YE^/^ir— (E, curl H 2 — H 2 curl E^tt, 



= E 1 F 2 + H 2 Grj, 



= EiCa + E^a + HA^Tr. . . . (92; 



Similarly, by the action of the second system on the first, 



conv. YE 2 H 1 /47r=E 2 C 1 + E 2 D 1 + H 1 B 2 /47r. . . (93) 



Addition gives the equation of mutual activity. And, sub- 

 tracting (93) from (92), we find 



conv. (YE 1 H 2 -YE 2 H 1 )/47r = (E 1 D 2 -E 2 D 1 ) 



-(H^-H.BOMtt; . (94) 



since E 1 C 2 = E 1 Z:E 2 = E 2 &Ei = E 2 C 1? if there be no rotatory 

 power, or C be a symmetrical linear function of E. Similarly 

 for D and E, and B and H. Hence, if the systems are normal, 

 making d/dt=p l in one, and^? 2 in the other, (94) becomes 



conv. (YE 1 H 2 -YE 2 H 1 )/47r=( i > 2 -p 1 )( E i D 2-H 1 B 2 /47r). (95) 



Therefore, by the well-known theorem of Convergence, if 

 we integrate through any region, and U 12 , T 12 be the mutual 

 electric energy and the mutual magnetic energy of the two 

 systems in that region, we obtain 



where N is the unit normal drawn inward from the boundary 

 of the region, over which the summation extends. And if the 

 region include the whole space through which the systems 

 extend, the right member will vanish, giving U 12 = T 12 , when 

 these are complete. 



From (96) we obtain, by differentiation, the value of twice 

 the excess of the electric over the magnetic energy of a single 

 normal system in any region ; thus 



2(U-T) = SN(vEg-V^H)/47r. . (97) 



This formula, or special representatives of the same, is very 

 useful in saving labour in investigations relating to normal 

 systems of subsidence. 



The quantity that appears in the numerator in (96) is the 

 excess of the energy entering the region through its boundary 

 per second by the action of the second system on the first, 

 over that similarly entering due to the action of the first on 



