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 -∂a/∂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^[1] for the electrotonic intensity due to a current i′ flowing in a circuit s′ the value which results from Neumann's theory, namely,

$\mathbf {a} =i'\int {\frac {\mathbf {ds'} }{r}}$.

It may, however, be remarked that the equation

$\mathrm {curl} \ \mathbf {a} =\mathbf {B}$,

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

$\mathrm {div} \ \mathbf {a} =\psi$,

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^[2]; for, in addition to those applications which have been mentioned, Maxwell showed that

1. Cf. p. 224.
2. These operators had, however, occurred frequently in the writings of Stokes, especially in his memoir of 1849 on the Dynamical Theory of Diffraction.

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