Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/293

 through that surface." The electromotive force of induction at the place (x, y, z) is -&part;a/&part;t: as Maxwell said, "the electromotive force on any element of a conductor is measured by the instantaneous rate of change of the electrotonic intensity on that element." From this it is evident that a is no other than the vector-potential which had been employed by Neumann, Weber, and Kirchhoff, in the calculation of induced currents; and we may take for the electrotonic intensity due to a current i&prime; flowing in a circuit s&prime; the value which results from Neumann's theory, namely,

It may, however, be remarked that the equation

taken alone, is insufficient to determine a uniquely; for we can choose a so as to satisfy this, and also to satisfy the equation

where ψ denotes any arbitrary scalar. There are, therefore, an infinite number of possible functions a. With the particular value of a which has been adopted, we have
 * $$\begin{matrix}

\mathrm{div}\ \mathbf{a} & = & \frac{\partial}{\partial x}i'\int_{s'}\frac{dx'}{r} + \frac{\partial}{\partial y}i'\int_{s'}\frac{dy'}{r} + \frac{\partial}{\partial z}i'\int_{s'}\frac{dz'}{r} \\ \ & = & -i'\int_{s'}\left(dx' . \frac{\partial}{\partial x'} + dy' . \frac{\partial}{\partial y'} + dz' . \frac{\partial}{\partial z'}\right)\left(\frac{1}{r}\right) \\ \ & = & -i'\int_{s'}d\left(\frac{1}{r}\right) \\ \ & = & 0; \\

\end{matrix}$$ so the vector-potential a which we have chosen is circuital.

In this memoir the physical importance of the operators curl and div first became evident ; for, in addition to those applications which have been mentioned, Maxwell showed that