130 Mr. 0. Heaviside on the 



system after the first moment, yet at the first moment (when 

 the previously acting impressed forces finally cease) the elec- 

 tric field and the magnetic field are independent. The energy 

 which is dissipated according to Joule's law has two sources, 

 the electric and the magnetic energies. Now we may, by 

 longitudinal impressed force, set up a certain distribution of 

 magnetic energy, but no electric energy. Or, having set up 

 a certain magnetic and a certain electric field by a particular 

 distribution of impressed force, we may alter it in various 

 ways, so as to keep the magnetic field the same whilst we vary 

 the electric field. So both fields require to be known, or 

 equivalent information given. 



We may then decompose them into the proper normal 

 systems by means of the universal conjugate property derived 

 from the equation of activity, that of the equality of the mutual 

 electric energy of two complete normal systems to their mutual 

 magnetic energy {' Electrician/ November 27, 1885). Thus, 

 if U n and T u are the doubles of the complete electric and 

 magnetic energies of any normal system, and U 01 is the mutual 

 electric energy of the initial electric field and the normal 

 electric field in question, and T 01 is the mutual magnetic 

 energy of the initial magnetic field and the normal magnetic 

 field, we shall have tt m 



A^fcio! (23) 



as the expression for the value of the coefficient A v which 

 settles the actual size of the normal system in question. Equal 

 roots require further investigation. This would complete the 

 theoretical treatment. It is best to use the electric and mag- 

 netic forces as initial data in the general case. As regards 

 potentials, we cannot express the electric energy in terms of 

 merely the electric potential and the electrification, but require 

 to use also the vector potential Z and the magnetic current. 



Now there are several important practical simplifications. 

 Suppose, first, that the thickness of the sheath is only a 

 small fraction of its distance from the axis. Then it may be 

 treated as if it were infinitely thin, making the sheath a linear 

 conductor ; of course its resistance may remain the same as if 

 of finite thickness. Let « 4 be the very small thickness of the 

 sheath, then the big fraction on the left side of (22) will 

 become 



(Jo + ^JQKi— (Kp + gs^KQJi 



(J x + s 3 a 4 J 2 ) K : — (Kx + s 3 a 4 K 2 ) J, 



= _J_J 1 |orJoK 1 ^ 



s 3 a 4 JgK!— J l K 2 v s 3 a 4 



