we erect a normal whose direction is expressed by the point of the auxiliary sphere. By moving along the curved line, and will therefore change their positions, while remains constant, and and are always equal to Therefore describes the great circle one of whose poles is The element of this great circle will be equal to if denotes the radius of curvature of the curve. And again, if we denote the direction of this element upon the sphere by then will evidently lie in the same great circle and be from as well as from If we now set
then we shall have
since, in fact, are merely the coordinates of the point referred to the centre of the sphere.
Since by the solution of the equation the coordinate may be expressed in the form of a function of we shall, for greater simplicity, assume that this has been done and that we have found
We can then write as the equation of the surface
or
From this follows, if we set
where are merely functions of and We set also
Therefore upon the whole surface we have
and therefore, on the curve,
Hence differentiation gives, on substituting the above values for