Green can correlate his t_{2} with
the t_{1}
given in Red's letters,
and Red can correlate his t_{3} with
the t_{2}
given in Green's letters.
t_{2}
= f(
t_{1}
)
and
t_{3}
= g(t_{2}
).
In general f(w) and g(w) will not be the same function.
We therefore "recalibrate" both Red's and Green's measurement of time, so as to have the two functions the same.
We find the "functional square root" of f(w) and g(w),
that is a function h( ) such that h(h(w)) = g(f(w)).
We cannot do this directly, but we can find functions h( ) and k( ), such that for any variable x,
k^{1}[h(h(k(x)))] = g(f(x)).
Red and Green agree to recalibrate their clocks, so that
henceforward, a message dispatched by Red at
t
will be received by Green at
t
where
t = h(t)
and vice versa.
Red now will not be able to say
My messages travel faster than
yours do, Green.
Nor Green to say
My messages travel faster than
yours do,
Red.
I
Radar Rules
(i)
t_{2}
= 1/2 (t_{1}
+
t_{3}
)
(ii)
l_{2}
= 1/2 c (t_{1}

t_{3}
).
II
Radio Rules
(i)
t = h(t)
(ii)
t = h(t).
By radar rule I (i)
t_{2}
= 1/2 (
t_{1}
+
t_{3})
By radio rule II (i)
t_{1} =
h^{1}(
t_{2})
By radio rule II (ii)
t_{3}
= h(
t_{2})
Therefore
Q(
t_{2})
= t_{2} =
1/2 [h^{1}
(t_{2}) +
h(t_{2})].
(1)
Also
l_{2} =
vt_{2}.
t_{2} =
l_{2}/v =
1/2 c (t_{3}

t_{1})/v
and hence