A new analysis of the numbers and types of critical points in the phonon spectra of silicon and germanium is presented. The existence of some of these critical points can be deduced from symmetry arguments alone, but a complete scheme can be set up only with the aid of a lattice dynamics calculation, and we make extensive use of the results of a shell model calculation for germanium. The analysis is extended to determine the critical points on the two-phonon bands, i.e. the bands constructed by adding pairs of phonons with equal and opposite wave vector. With the aid of selection rules, the two-phonon critical-point scheme, and neutron scattering results, we are able to predict the number, shape and approximate positions of the slope discontinuities which should be observed in the infra-red lattice band spectra of silicon and germanium. Comparison with detailed experimental results confirms our scheme. The slope discontinuities show up as fine structure on the main absorption spectrum with the predicted shapes and approximate energies. By making use of the actual energies determined by the infra-red measurements we calculate the energies of phonons at $\Gamma$, X and L in the Brillouin zone to an accuracy of two wave numbers, which is a substantial improvement on the accuracy of the neutron results. For the case of diamond, for which there are no neutron results, we compare the infra-red lattice band spectrum with the second-order Raman spectrum.