164 Dr. L. Silberstein on Radiation 



of light in vacuo. If o) = ^(KE 2 + M 2 ) be the density, and 

 F = c VEM the flux, of electromagnetic energy, then, where- 

 ever these equations are valid, we have the relation 



'dco/'dt + divF = 0, (a') 



meaning that the loss of energy of each volume-element 

 equals the amount of energy radiated outwards. Thus the 

 energy belonging to any initial field E, M will almost instan- 

 taneously spread out beyond all limits, leaving behind either 

 no field at all or a purely statical one. Genuine sources of 

 energy are usually treated as singular points of integrals 

 of the unmodified equations (a), as for instance in the well- 

 known case of the Hertzian dipole, and in those derivable 

 from it by axial differentiation. All of these are ultimately 

 point-sources, and lead essentially to the \~ 4 -type of the 

 law of radiation. Another method of treatment consists in 

 adapting particular solutions of (a) to the surface of a perfect 

 conductor, usually a sphere, which plays the part of an elec- 

 tric oscillator ; but here again the supply of energy is not 

 explicitly taken into account. A third method consists in 

 treating the sources as elaborate electronic mechanisms. 

 The latter method, which has already yielded some inter- 

 esting results, is, undoubtedly, very tempting; but it has 

 the disadvantage that it obliges us to enter into all the 

 minute details of the hypothetical structure. Again, in 

 the existing investigations of this type even the continuous 

 supply of energy is not explicitly stated. 



So much to justify a return to Heaviside's method of 

 dealing with sources, which is free from the above objec- 

 tions and which recommends itself by its very breadth. It 

 seems that it has not received from modern physicists the 

 attention it deserves. It may be useful to recall shortly 

 Heaviside's general concepts and equations *. 



Let s be the productivity of the source, per unit time and 

 unit volume ; then, instead of (a'), 



s = dft>/d£ + divF', (b') 



F' being the flux of energy in this more general case. 

 Heaviside splits s into its electric and magnetic parts ; each 

 of these parts is represented as the scalar product of the 

 corresponding current into the auxiliary vectors e, m, the 

 impressed electric and magnetic force, respectively. Thus, 



* For a full exposition see Oliver Heaviside's ' Electromagnetic 

 Theory,' vol. i. 1893. 



