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