We shall have
where , and is always to be taken with the positive sign.
Let us now write
then the substitution 1) takes the form
the coefficients being essentially real.
If now in the above-mentioned rotation round the z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and by , we at once perceive that simultaneously, new magnitudes , where
and , where
must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A'), and (B'), the new equations being obtained by simply dashing the old set.
All these equations can be written in purely real figures, and we can then formulate the last result as follows.
If the real transformations 4) are taken, and x', y', z', t' be taken as a new frame of reference, then we shall have