we draw a radius of the auxiliary sphere to the point
but instead of this point we take the point opposite when
is more than
from
In the first case, we regard the element at
as positive, and in the other as negative. Finally, let
be the point on the auxiliary sphere, which is
from both
and
and which is so taken that
lie in the same order as
The coordinates of the four points of the auxiliary sphere, referred to its centre, are for
Hence each of these
points describes a line upon the auxiliary sphere, whose elements we shall express by
We have, therefore,
In an analogous way we now call
the measure of curvature of the curved line upon the curved surface, and its reciprocal
the radius of curvature. If we denote the latter by
then
If, therefore, our line be a shortest line,
must be proportional to the quantities
But, since at the same time
we have
and since, further,
![{\displaystyle {\begin{aligned}\xi 'X+\eta 'Y+\zeta 'Z&=\cos \lambda 'L\\&=\pm (X^{2}+Y^{2}+Z^{2})\\&=\pm 1,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3095cf49ee599e9cb4d2b197be5a273f55c82ef)