must be proportional to the quantities
respectively. If
is an element of the curve;
the point upon the auxiliary sphere, which represents the direction of this element;
the point giving the direction of the normal as above; and
the coordinates of the points
referred to the centre of the auxiliary sphere, then we have
Therefore we see that the above differentials will be equal to
And since
are proportional to the quantities
the character of the shortest line is such that
To every point of a curved line upon a curved surface there correspond two points on the sphere, according to our point of view; namely, the point
which represents the direction of the linear element, and the point
which represents the direction of the normal to the surface. The two are evidently
apart. In our former investigation (Art. 9), where [we] supposed the curved line to lie in a plane, we had two other points upon the sphere; namely,
which represents the direction of the normal to the plane, and
which represents the direction of the normal to the element of the curve in the plane. In this case, therefore,
was a fixed point and
were always in a great circle whose pole was
In generalizing these considerations, we shall retain the notation
but we must define the meaning of these symbols from a more general point of view. When the curve
is described, the points
also describe curved lines upon the auxiliary sphere, which, generally speaking, are no longer great circles. Parallel to the element of the second line,