**THE DETERMINATION OF THE CONSTANT***
Mt3*

Once a value for the constant *k*
has been established,
the mean sidereal periods of the three planets associated with
inverse-velocity
relation [**3b**] may be expressed as integral parts of an
exponential
function based on *k = phi*, the respective heliocentric
positions
of the planets in question, and the known sidereal period of Mercury (*M*_{t}^{
}), i.e.,

**[1] Mean Sidereal Period of Mercury
= ***M*_{t} phi^{ 0
}[2] Mean Sidereal Period
of
Venus = *M*_{t} phi ^{2
}[3] Mean Sidereal Period
of
Uranus = *M*_{t} phi ^{12
}

Because of the following velocity
expansions of Kepler's third law of
planetary motion:^{4}

**[4] Mean Velocity (Vr) = Mean
Distance (A)**^{-1/2} = Mean Period (T)^{-1/3
}[5] Inverse Velocity (Vi) =
Mean
Distance (A)^{1/2} = Mean Period (T)^{1/3
}[6] Mean Distance (A) =
Inverse
Velocity(Vi)^{2} = Mean Period (T)^{2/3}

inverse-velocity relation [3b]: *Vi*_{Venus}
- Vi_{Mercury
}: Mean Velocity of Uranus (Vr) may expressed in terms of phi
and exponential Relations [1] through [6], i.e.,

**[7]*** (M*_{t }phi^{2})^{1/3}
- M_{t }^{1/3} (M_{t }phi^{12})^{-1/3}

**[7a]*** M*_{t }^{1/3}phi^{2/3}
- M_{t }^{1/3} M_{t}^{-1/3}phi^{-4}

and then restated in terms of the
inverse velocity of Mercury since:
*Mvi = M*_{t }^{1/3} ( Relation [5] ):

**[8]*** Mvi(phi*^{2/3}
- 1) Mvi^{ -1}phi^{-4}

**[9]*** Mvi*^{2}(phi^{2/3}
- 1) phi^{-4}

**[10] ***Mvi*^{2} phi^{-4}(
phi^{2/3} - 1)^{-1}

But from Relation [6], *Mvi*^{2}
is equivalent to the mean
distance of Mercury (*M*_{a}_{ }), thus:

**[11] *** Mean Distance of Mercury
= Mvi*^{2} = M_{a} phi^{-4}(phi^{2/3} -
1)^{-1}

therefore a new *phi*-based value
for the mean sidereal period
of Mercury (*M*_{t3}) may be obtained from Relation [11]
and
the application of Kepler's Third Law:

**[12] ***Mean Sidereal Period
of Mercury = M*_{t3} = phi^{-6}(phi^{2/3}-1)^{-3/2}
= __0.2395640495... Years__

RETURN
TO SECTION III