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XIX. On the Principle of Least Actum in the Theory of 

 Electrodynamics. — I. By Gr. H. Livens *. 



IT is a fundamental problem in the general theory of 

 electrodynamics to formulate a scheme for the description 

 of the general phenomena of the electromagnetic field on an 

 analogy with the behaviour of: some hypothetical system 

 obeying the ordinary laws of generalized mechanics. Now, 

 any dynamical problem can be enunciated in a single formula 

 as a variation problem, and thus if the laws of electro- 

 dynamics can be derived in a minimum or variational 

 theorem, the formulation is virtually complete ; there remain 

 only such interpretations, explanations, and developments as 

 will correlate the corresponding integrals relating to known 

 dynamical systems. 



In this form the problem has been solved in slightly 

 different ways by Larmor |, Lorentz J, Macdonald §, and 

 others. The integral which is the subject of variation in 

 each case is virtually the same as 



~w\dv]dt 9 



07T J J 



wherein L is the part of the Lagrangian function of the 

 system not depending on the conditions of the field ; A is 

 the vector potential of the magnetic field ; C is the total 

 current of Maxwell's theory and includes both the true 

 electric and fictitious sethereal fluxes, and E is the electric 

 force intensity at the typical field-point. It is assumed that 

 the system consists entirely of free sether and electrons, and 

 the integration with respect to v is taken over the whole 

 field. 



On analogy with the known properties of electrostatic 

 fields it is assumed that the term in E 2 represents the true 

 potential energy of elastic strain in the sether, both as 

 regards its distribution and total amount; it then follows 

 that the integral 



J>Jti 



(AC) 



¥ 



(AC) dv 



represents the totality and distribution of the kinetic energy 

 of electric origin in the sethereal field. 



* Communicated by the Author. 

 t ' vEther and Matter' (Cambridge, 1900), ch. iv. 

 | ' La theorie electrornagnetique de Maxwell, &c.' (Leiden, 1892), 

 ■j §55-61. 



§ 'Electric Waves' (Cambridge, 1902), App. C. 



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