30 . THE THEORY OF RELATIVITY
There are two different sets of what are commonly called Max-wellian equations for moving media: i° a system of equations which * * may be gathered together from different chapters of Maxwell's
' Treatise,' and which we shall call shortly the equations of Maxwell, though it can be reasonably doubted whether Maxwell himself would consent to attribute to them general validity, especially with the inclusion of optics; and 2° a system of equations which Hertz obtained by a certain, apparently the most obvious, extension of the meaning of the form i., IL, and which Heaviside, independently, constructed by introducing into Maxwell's equations a supplementary term dictated by reasons of electro-magnetic symmetry ; these are widely known as the Hertz-Heaviside equations for moving bodies.
We shall use for i° and 2° the abbreviations (Mx), (HH). Neither has been able to stand the test of experience. Though contrary to the historical order, it will be more instructive to consider first the latter and then the former system of equations.
Let us return to the semi-integral form of electromagnetic laws i. and ]i., given, in words and symbols, on pp. 22-23. These are valid for a ponderable dielectric medium or body, stationary with respect to our frame S, and for any surface <r which, together with its bounding circuit s, is fixed in the body. Thus the surface tr, through- which the 'current' is to be taken, is itself fixed in S. Now, what Hertz did in order to obtain the required extension, was simply to suppose that i. and n. are still valid for a body, I f rigid or deformable, moving with respect to *$" in any arbitrary
manner, provided that the currents on the left-hand side of these equations are taken through a surface composed always of the same particles of the body, or—to put it shortly—through an individual cr, together with its s. This gives for the current per unit area of <r, instead of the local time-rate of change "d^fdf, if v be the velocity of a particle relatively to S,
|§ + vdiv<B + curlV©v, (6)
Of
and a similar expression for the magnetic current,* while the right-hand sides of i., n., containing only the instantaneous values of line integrals, remain obviously unaffected by the Hertzian require-ment. The distribution of Jft being supposed solenoidal, as
*See Note 2.