Page:On Faraday's Lines of Force.pdf/51

Rh It appears from experiment that the expression $$\frac{d\alpha_0}{dt}$$ refers to the change of electro-tonic state of a given particle of the conductor, whether due to change in the electro-tonic functions themselves or to the motion of the particle.

If $$\alpha_0$$ be expressed as a function of $$x, y, z$$ and $$t$$, and if $$x, y, z$$ be the co-ordinates of a moving particle, then the electro-motive force measured in the direction of $$x$$ is

$\alpha_2= -\frac{1}{4\pi}\left(\frac{d\alpha_0}{dx}\frac{dx}{dt}+\frac{d\alpha_0}{dy}\frac{dy}{dt}+\frac{d\alpha_0}{dz}\frac{dz}{dt}+\frac{d\alpha_0}{dt}\right)$

The expressions for the electro-motive forces in $$y$$ and $$z$$ are similar. The distribution of currents due to these forces depends on the form and arrangement of the conducting media and on the resultant electric tension at any point.

The discussion of these functions would involve us in mathematical formulae, of which this paper is already too full. It is only on account of their physical importance as the mathematical expression of one of Faraday's conjectures that I have been induced to exhibit them at all in their present form. By a more patient consideration of their relations, and with the help of those who are engaged in physical inquiries both in this subject and in others not obviously connected with it, I hope to exhibit the theory of the electro-tonic state in a form in which all its relations may be distinctly conceived without reference to analytical calculations.

Summary of the Theory of the Electro-tonic State.

We may conceive of the electro-tonic state at any point of space as a quantity determinate in magnitude and direction, and we may represent the electro-tonic condition of a portion of space by any mechanical system which has at every point some quantity, which may be a velocity, a displacement, or a force, whose direction and magnitude correspond to those of the supposed electro-tonic state. This representation involves no physical theory, it is only a kind of artiﬁcial notation. In analytical investigations we make use of the three components of the electro-tonic state, and call them electro-tonic functions. We take the resolved part of the electro-tonic intensity at every point of a