The extremities of all shortest lines of equal lengths
correspond to a curved line whose length we may call
We can evidently consider
as a function of
and
and if the direction of the element of
corresponds upon the sphere to the point
whose coordinates are
we shall have
Consequently
This magnitude we shall denote by
which itself, therefore, will be a function of
and
We find, then, if we differentiate with respect to
because
and therefore its differential is equal to zero.
But since all points [belonging] to one constant value of
lie on a shortest line, if we denote by
the zenith of the point to which
correspond and by
the coordinates of
[from the last formulæ of Art. 13],
if
is the radius of curvature. We have, therefore,
But
because, evidently,
lies on the great circle whose pole is
Therefore we have
![{\displaystyle {\frac {\partial u}{\partial s}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86bebb646d40d43f38f52b9aa358734ce88e0fe7)