The equation of a planetary orbit in the axes of Pseudo-Euclidean Space-Time has, in [2] been derived, in the form of a first order equation, from a Lagrangian analysis of the metric of the General Theory. To obtain that form here, the simplest process is to obtain the first integral of (C.8) from which the desired relationship can be obtained directly. The easiest manner to obtain the first integral of (C.8) is firstly, via a re-arrangement of (5.4), thus

From (5.3) note that

which with (3.20) gives

Inserting this into (B.1) then gives

which incidentally can be shown to be the first integral of (3.18), the equation of planar motion in D₁.

Now, from (2.15)

Also from (C.7) and (B.4)

which when inserted into (B6) yields

as the first integral of (C.8).

Transformation to the Axes of D₀ and Derivation of the Equation of the Orbit.

Transformation of (B.8) to the axes of D₀ via (4.7), (4.12) and (4.13) gives

For simplicity write this as

where (5.5) has also been inserted

The equation of the orbit, (expressed in the axes D₀), can now be derived in the conventional manner as follows. Put

so that

Inserting this and (B.11) into (B.10) yields

Expanding, this finally reduces to the desired expression, thus

as derived in [2], pp 198, Eq[58.35].

Finally, in (B.10) the simplifying identity

was inserted. To shown that this is identical to the same parameter in [2], pp197, Eq(58.26), insert (4.7) and (4.18) thus

which from (3.6) and (3.20) becomes

and which with (2.15) and (2.19) then gives

so that insertion of (4.7) again into this finally gives

As derived in [2].

Also from (5.5) it can be seen that the constant h in this paper is identical to the parameter m in [2], pp197, Eq(58.27). These results provide additional proof that a central orbit in D₁ is identical to that in the General Theory.

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