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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

x2 + y2 + z2 - c2t2 = 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 - c2t′2 = 0 . . . . . . (10a).

In order that equation (10a) may be a consequence of equation

(10), we must have

x′2 + y′2 + z′2 - c2t′2 = σ (x2 + y2 + z2 - c2t2) (11).