The direction of the normal to the surface toward that side which we regard as the upper side is represented upon the auxiliary sphere by the point
Let
Also let
denote an infinitely small line upon the surface; and, as its direction is denoted by the point
on the sphere, let
We then have
therefore
and, since
must be equal to
we have also
Since
depend only on the position of the surface on which we take the element, and since these equations hold for every direction of the element on the surface, it is easily seen that
must be proportional to
Therefore
Therefore, since
and
or
If we go out from the surface, in the direction of the normal, a distance equal to the element
then we shall have
and
We see, therefore, how the sign of
depends on the change of sign of the value of
in passing from the lower to the upper side.
Let us cut the curved surface by a plane through the point to which our notation refers; then we obtain a plane curve of which
is an element, in connection with which we shall retain the above notation. We shall regard as the upper side of the plane that one on which the normal to the curved surface lies. Upon this plane