Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/305

 in this the architecture of his system was displayed, stripped of the scaffolding by aid of which it had been first erected.

As the equations employed were for the most part the same as had been set forth in the previous investigation, they need only be briefly recapitulated. The magnetic induction μH, being a circuital vector, may be expressed in terms of a vector-potential A by the equation

The electric displacement D is connected with the volume-density ρ of free electric charge by tho electrostatic equation

The principle of conservation of electricity yields the equation

where ι denotes the conduction-current.

The law of induction of currents—namely, that the total electromotive force in any circuit is proportional to the rate of decrease of the number of lines of magnetic induction which pass through it—may be written

from which it follows that the electric force E must be expressible in the form

where ψ denotes some scalar function. The quantities A and ψ which occur in this equation are not as yet completely determinate; for the equation by which A is defined in terms of the magnetic induction specifies only the circuital part of A; and as the irrotational part of A is thus indeterminate, it is evident that ψ also must be indeterminate, Maxwell decided the matter by assuming A to be a circuital vector; thus

and therefore