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 (Mt ), i.e.,

[1] Mean Sidereal Period of Mercury = Mt phi 0
[2] Mean Sidereal Period of Venus = Mt phi 2
[3] Mean Sidereal Period of Uranus = Mt 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]: ViVenus - ViMercury : Mean Velocity of Uranus (Vr) may expressed in terms of phi and exponential Relations [1] through [6], i.e.,

[7] (Mt phi2)1/3 - Mt 1/3 (Mt phi12)-1/3

[7a] Mt 1/3phi2/3 - Mt 1/3 Mt-1/3phi-4

and then restated in terms of the inverse velocity of Mercury since: Mvi = Mt 1/3 ( Relation [5] ):

[8]  Mvi(phi2/3 - 1) Mvi -1phi-4

[9] Mvi2(phi2/3 - 1) phi-4

[10] Mvi2 phi-4( phi2/3 - 1)-1

But from Relation [6], Mvi2 is equivalent to the mean distance of Mercury (Ma ), thus:

[11] Mean Distance of Mercury = Mvi2 = Ma phi-4(phi2/3 - 1)-1

therefore a new phi-based value for the mean sidereal period of Mercury (Mt3) may be obtained from Relation [11] and the application of Kepler's Third Law:

[12] Mean Sidereal Period of Mercury = Mt3 = phi-6(phi2/3-1)-3/2 = 0.2395640495... Years