It is first noted for future reference that substitution for , derived from (3.18), into (3.7) gives

(C.1)

First the second order variation of radial position with respect to the proper time of the gravitating mass is computed thus

(C.2)

which from (2.15) becomes

(C.3)

working this out yields

(C.4)

Substitution for from (C.1) then gives after reduction

(C.5)

Now substitution for from (3.17) gives

(C.6)

But

(C.7)

and insertion of this into (C.6) finally gives

(C.8)

which is the required relationship, expressed in the axes of D₁ and the parameter u. Substitution for u, (from (4.7)) and its spatial gradient, reduces (C.8) to

(C.9)

being the equation of motion as a function of the proper time expressed fully in the axes of D₁.

Note that because the time dilatation effect is embodied in the proper time t _p, the reactive acceleration term due to this in (3.17), is not present in (C.9).

On to the next Section: Appendix D

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