We may use our knowledge of the locus of the intersection
of corresponding rays of two pencils to establish the theorem :

If the points of two curves have «1:1 correspondence, the
curves have the same deficiency.
This means that, if to each point P with Cartesian coordinates
(x, y) on one curve corresponds a point P' with coordinates* (x\ y')
on the other so that x', y' may be expressed rationally in terms
of x, y and vice versa, then the curves have the same deficiency.
It is at once seen that, if points P, P', P" are taken on
three curves, guch that the coordinates of P can be expressed
rationally in terms of P' and vice versa, while the coordinates
of P' can be expressed rationally in terms of P" and vice
versa, then the coordinates of P can be expressed rationally
in terms of P" and vice versa.
