The weakly nonlinear boundary value problem of wave propagation in an optical fibre (for the transverse electric mode, for example) is formulated and a modified linear solution is obtained. It is shown that a self-consistent theory of fibre optics should be weakly nonlinear. The mode of critical refraction that does not exist in the linear theory is obtained, showing that it is a mode consisting of resonance modes. It is shown that the signal carriers in a long fibre are of resonance modes, not normal modes. Some experimental data are given for comparison with the theoretical predictions, and the agreement seems satisfactory.