H. A. Lorentz. Electromagnetic phenomena in a system moving with any velocity smaller than that of light. // Proceedings Royal Acad., Amsterdam. Vol. VI., 1904

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$ 4. We shall further transform these formulae by a change of variables. Putting

........(3)

and understanding by I another numerical quantity, to be determined further on, I take as new independent variables

.....(4)

.....(5)

and I define two new vectors b' and I)' by the formulae

tor which, on account of (3), we may also write

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As to the coefficient I, it is to be considered as a function of w, whose value is 1 for 10 = 0, and which, for small values of to, differs from unity no more than by an amount of the second order.

The variable t‘ may be called the “local time⁵’; indeed, for h = 1, Z = 1 it becomes identical with what I have formerly understood by this name.

If, finally, we put

... (7)

... (8)

these latter quantities being considered as the components of a new vector «', the equations take the following form :

. . (9) . (10)

The meaning of the symbols div' and rot' in (9) ivS similar to that of div and rot in (2); only, the differentations with respect to x, y, z are to be replaced by the corresponding ones with respect to y', z'.

§ 5. The equations (9) lead to the conclusion that the vectors b' and I)¹ may be represented by means of a scalar potential <p' and a vector potential These potentials satisfy the equations^a)

.....(11)

.... (12)

and in terms of them i>' and &' are given by

1) M. E., §§ 4 and 10.