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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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.

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. . . (13)

• • • (14)

The symbol A' is an abbreviation for, and gmdUp

denotes a vector whose components areThe expression

grail' has a similar meaning;.

In order to obtain the solution of (11) and (12) in a simple form, we may take &, y‘, z' as the coordinates of a point P¹ in a space S', and ascribe to this point, for each value of ?, the values of q', u', <p', belonging to the corresponding point P (.v, ;/, s) of the electromagnetic system. For a definite value f of the fourth independent variable, the potentials <p' and a' in the point P of the system or in the corresponding point P' of the space >$', are given by ')

......(15) ......(16)

Here clS' is an element of the space S', r' its distance from P' and the brackets serve to denote the quantity </ and thevector

q' u', such as they are in the element dS', for the valueof

the fourth independent variable.

Instead of (J5) and (16) we may also write, taking into account (4) and (7),

.....(17)

.......(IS)

the integrations now extending over the electromagnetic system itself. It should be kept in mind that in these formulae /•' does not denote the distance between the element dS and the point (ji, y, z) for which the calculation is to be performed. If the element lies at the point (*i> ;'A> ²i)i ^we must take

It is also lo be remembered that, if we wish to determine <p' and

i) M. E., §§ 5 and 10.