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22 currents to the new ideas; and there still remains the same ambiguity in respect to mechanical forces on portions of flexible or extensible current-circuits. There is however no ultimate ambiguity as regards electromotive phenomena in bodies at rest, the equations of the theory sufficing to eliminate the arbitrary element that initially must be introduced, and thus to give a definite determination of the electric force at each point of space. New difficulties, of practical importance only in the theory of radiation, occur when the material medium which carries the current is in motion: and the theory for that case was left in the form of a first approximation, which assumed that the aethereal disturbance was simply convected by the moving matter, that being amply sufficient for ordinary electrodynamic applications. The dynamical methods were however sketched by Maxwell which would have to be employed to work out a more definite scheme of the relation of aether to the matter at rest in it or moving through it: and quantities of dynamical origin or suggestion, such as the vector potential of electric currents, which have sometimes been considered so great a complication by subsequent writers as to justify their summary abolition, turn out in fact to be of the essence of a more thorough analysis.

16. The dynamical scheme which thus in Maxwell's hands furnished a formulation of the electrodynamics of material systems at rest in the aether, completely effective except as regard the material mechanical forces acting on the matter carrying the currents, was one of continuous differential analysis: the matter was taken as simply modifying, where it existed, the effective constants in the formula for the spacial distribution of electric energy: when the aether did not move with any finite speed or the matter move across it, there was no pressing occasion to separate the energy into a part belonging to and propagated by the aether and a part attached to the molecules of matter. The theory, at the stage at which it was left by Maxwell, being a theory of complete electric circuits,