392 Mr. G. H. Livens on the Flux of 



and by Ampere's relation this is again equal to 



so that now we have 



T = 1 fyv | (AdC ! )-(dv f (IdB), 

 with 



Again, by transforming the integral in I, a variant of 

 this form is got in which 



and 



T = I (dv [ f (AdC) -({c Curl I . dA)] 



There is another transformation derivable on the same 

 basis as these last two, but it reduces eventually to the 

 more general form deduced above and need not further 

 detain us. 



5. So far we have discussed merely the mathematically 

 possible transformations of which the theory is capable, 

 without stopping to examine for each case the physical 

 significance of the results therein obtained. A consi- 

 deration of this other side of the matter will, however, 

 soon show that certain difficulties are involved in Mac- 

 donald's form of the theorj- and the alternative one 

 succeeding it in the above discussion. Let us first consider 

 the question of the magnetic energy of the system: the 

 expression 



T = 1 ( dv j (AdC) -idv\ (MB) 



obtained at the end of the last paragraph cannot possibly 

 represent the energy properly available in the system in 

 the general case. In fact it represents the total energy 

 of the system arising on account of the magnetic forces, 

 and this total includes the magnetic energy in the aether 

 of which the part 



~ | dv j (CrfA) 



is alone available, together with all the intrinsic energy 



