I was somewhat alarmed to find that [Schreiber 1943] had not only started at a very
different point but had come to a very different conclusion. As this reference is one of
the basic defining papers of the South African system I will take the time to resolve the
differences here. He defined the meridian arc of the ellipse to be the integral over the
radius of curvature of the earth with respect to the geodetic latitude. Thus
Where phi is the geodetic latitude. (ie. the angle between the normal to the surface
and the equatorial plane). And M is the radius of curvature at that point. The definition
of the radius of curvature is the rate of change of arc length with respect to change in
angle of the normal. Thus :-
To render this palatable we introduce the geocentric latitude into the middle of the
equation.
At this point I should have realised the two approaches are in fact identical, but I
plunged on to obtain Schreiber's result. Now by pythagoras's theorem ...
Parametrizing the meridian by t...
Using the parametric equations of the spheroid (4)
we obtain...
Now the geocentric latitude in terms of the geodetic latitude is..
From which we obtain...
Substituting for b in the above and t in (58) and
simplifying we get...
Thus obtaining at last Schreiber's result...
Which, despite the fact that the geocentric and the geodetic latitude are very similar in form and numeric value, this equation is so different in form from (55) that I believed it to be different. But a bit of work at this stage or a bit of intelligence at an earlier stage will show that these two equations are identical.
On a spherical earth this equation simplifies greatly to L(t)=at.
Given that the original references work with the geodetic latitude instead of the geocentric latitude, and they work in degrees instead of radians, comparing my results on a term for term basis with the original references is very difficult. Such a task is certainly made much more difficult by the way in which rational formulas involving square roots can be transformed to equivalent, but very different in appearance, alternative forms.