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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.
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a' for the instant, at which the local time in P is t', we must take q and q u', such as they are in the element clS at the instant at
which the local time of that clement is
§ 6. It will suffice fot‘ our purpose to consider two special cases. The first is that of an electrostatic system, i, e. a system having no other motion but the translation with the velocity w. In this case ti/ = 0, and therefore, by (12), a' =0. Also, rp' is independent of t', so that the equations (11), (13) and (14) reduce to
. . (19)
After having determined the vector i>' by means of these equations, we know also the electric force acting on electrons that belong to the system. For these the formulae (10) become, since u' = 0,
.....(20)
i
The result may be put in a simple form if we compare the moving system 2 with which we are concerned, to another electrostatic system 2' which remains at rest and into which 2 is changed, if the dimensions parallel to the axis of x are multiplied by hi, and the dimensions which have the direction of y or that of z, by I, a deformation for which (kl, I, I) is an appropriate symbol. In this new system, which we may suppose to be placed in the above mentioned space S', we shall give to the density the value q’, determined by (7), so that the charges of corresponding elements of volume and of corresponding electrons are the same in 2 and 2’. Then we shall obtain the forces acting on the electrons of the moving system 2, if we first determine the corresponding forces in 2', and next multiply their components in the direction of the axis of <b by
I1, and their components perpendicular to that axis byThis is
conveniently expressed by the formula
It is further to be remarked that, after having found i>' by (19), we can easily calculate the electromagnetic momentum in the moving system, or rather its component in the direction of the motion. Indeed, the formula
(21)
