The confined propagation of slow neutrons along waveguides of a small cross section (fibres), discussed theoretically some time ago and demonstrated experimentally by other authors later, is analysed further. Motivated by those experiments, a three-dimensional quantum-mechanical treatment of the associated bending losses is presented. An approximate (two-parameter) formula is derived for the transmission coefficient Υ of the curved fibre, which displays its explicit dependence on the curvature radius R_cu, for 0<R_cu< + ∞. By adjusting one of the parameters (b), consistently with the theoretical analysis, and by fitting the other, the approximate formula describes all measured data for Υ. The parameter b (which accounts for the experimental data for Υ at large R_cu) allows the estimation of the number of effectively excited modes in the fibre.