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, (13)
. (14)

The symbol is an abbreviation for and denotes for a vector whose components are . The expresson has a similar meaning.

In order to obtain the solution of (11) and (12) in a simple form, we may take x', y', z' as the coordinates of a point P' in a space S', and ascribe to this point, for each value of t' , the values of , belonging to the corresponding point P (x, y, z) of the electromagnetic system. For a definite value t' of the fourth independent variable, the potentials φ' and in the point of the system or in the corresponding point P' of the space S' , are given by[1]

(15)
. (16)

Here dS' is an element of the space S', r' its distance form P' and the brackets serve to denote the quantity and the vector , such as they are in the element dS' , for the value of the fourth independent variable.

Instead of (15) and (16) we may also write, taking into account (4) and (7),

(17)
(18)

the integrations now extending over the electromagnetic system itself. It should be kept in mind that in these formulae r' does not denote the distance between the element dS and the point (x, y, z) for which the calculation is to be performed. If the element lies at the point (x1,y1},z1), we must take

.

It is also to be remembered that, if we wish to determine φ and

  1. M. E., §§ 5 and 10.