and since we always choose the point
so that
then for the shortest line
or
and
must coincide. Therefore
and we have here, instead of
curved lines upon the auxiliary sphere, only
to consider. Every element of the second line is therefore to be regarded as lying in the great circle
And the positive or negative value of
refers to the concavity or the convexity of the curve in the direction of the normal.
We shall now investigate the spherical angle upon the auxiliary sphere, which the great circle going from
toward
makes with that one going from
toward one of the fixed points
e.g., toward
In order to have something definite here, we shall consider the sense from
to
the same as that in which
and
lie. If we call this angle
then it follows from the theorem of Art. 7 that
or, since
and
we have
Furthermore,
or
and
![{\displaystyle \tan \phi ={\frac {Y\xi -X\eta }{\zeta }}={\frac {\zeta '}{\zeta }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/837db024c372e56b3b354a67608c4508aeb3716d)