Proposes a general set-up for the descent method of J.L. Colliot-Thelene and J.J. Sansuc applied to varieties fibred over the projective line over a number field. Proofs of the Manin obstruction to the Hasse principle and weak approximation; Difficulties in the application of the descent method brought about by the geometry of the generic fibre; Introduction of a new concept of a `degenerate' fibre.