288 Mr. GL H. Livens on the 



The first integral in this last expression is obviously- 

 equal to 



2c 2 ( a s r ss > J 



whilst in the second it is easy to verify that 



( (v s K s >)(i> s ,E s )dv = | (*JE.)(ivE,)<fo ; 



so that it is equal on the whole to 



^ 2 f { 2*) } - 8 -Lj'(divA)^. 



Thus, finally, we have 



i- i H',/r = f , { 2 «^+ 22 ^^ 1 - JL f(di v A)' dv f 



87T J 2o 2 I a s r ss < J 8tt J v 



which is equivalent to the formula obtained by Mr. Bieder- 

 mann, although the present deduction is somewhat simpler. 



This result is now applied to the case of two linear 

 conducting circuits when the current in each is the same 

 all round — that is, when the conditions are those of an 

 equilibrium theory. In this case we must assume that 

 div A = at all points of the field ; so that the outstanding 

 integral in the general reduction vanishes, and the dis- 

 tribution of the kinetic energy with the density H 2 /87r in 

 the surrounding field is equivalent to the expression 



t — A / s e * Vs2 + gv e * e *'(y* v »'} X 



i) 2 1 ^^ ^ - ' 



AC \^ CL S T ss i 



as it must be in order that the correct value for the mutual 

 potential of the two circuits may be obtained. 



In the more general case, however, there is no general 

 warrant for the assumption that divA = ; but then the 

 use of the expression T for the kinetic energy is not 

 legitimate. By assuming that it is, Mr. Biedermann tacitly 

 neglects the interaction between the electric and magnetic 

 fields in the aether ; for it is just this action which gives 

 rise to the outstanding term in the ordinary expression 

 for the integral in the special case under investigation. 

 This may best be seen by transforming the integral for the 

 kinetic energy into the form 



y (Ac) 



dv, 



