In order that equation (10_a_) may be a consequence of equation (10),
we must have
_x _2 + _y _2 + _z _2 _c_2_t _2 = (_x_2 + _y_2 + _z_2 _c_2_t_2)
(11).
Since equation (8_a_) must hold for points on the _x_-axis, we thus
have = 1. It is easily seen that the Lorentz transformation really
satisfies equation (11) for = 1; for (11) is a consequence of (8_a_)
and (9), and hence also of (8) and (9). We have thus derived the
Lorentz transformation.
The Lorentz transformation represented by (8) and (9) still requires to
be generalised. Obviously it is immaterial whether the axes of _K _ be
chosen so that they are spatially parallel to those of _K_. It is also
not essential that the velocity of translation of _K _ with respect to
_K_ should be in the direction of the _x_-axis. A simple consideration
shows that we are able to construct the Lorentz transformation in this
general sense from two kinds of transformations, viz. from Lorentz
transformations in the special sense and from purely spatial
transformations. which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing
in other directions.
Mathematically, we can characterise the generalised Lorentz
transformation thus:
It expresses _x , y , x , t _, in terms of linear homogeneous functions
of _x, y, x, t_, of such a kind that the relation
_x _2 + _y _2 + _z _2 _c_2_t _2 = _x_2 + _y_2 + _z_2 _c_2_t_2
(11_a_).
is satisficd identically. That is to say: If we substitute their
expressions in _x, y, x, t_, in place of _x , y , x , t _, on the
left-hand side, then the left-hand side of (11_a_) agrees with the
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