Thus we have obtained the Lorentz transformation for events on the
_x_-axis. It satisfies the condition
_x _2 _c_2_t _2 = _x_2 _c_2_t_2 . . . . . . (8a).
The extension of this result, to include events which take place
outside the _x_-axis, is obtained by retaining equations (8) and
supplementing them by the relations
image048
In this way we satisfy the postulate of the constancy of the velocity
of light _in vacuo_ for rays of light of arbitrary direction, both for
the system _K_ and for the system _K _. This may be shown in the
following manner.
We suppose a light-signal sent out from the origin of _K_ at the time
_t_ = 0. It will be propagated according to the equation
image049
or, if we square this equation, according to the equation
_x_2 + _y_2 + _z_2 _c_2_t_2 = 0 . . . . . (10).
It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place as judged from _K _ in accordance with the
corresponding formula
_r _ = _ct _
or,
_x _2 + _y _2 + _z _2 _c_2_t _2 = 0 . . . . . . (10_a_).
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