It is proposed in this paper to consider the spectral lines which would arise in the case of an electron describing an orbit about two centres of attracting force. The assumptions of the Quantum Theory as generalised by Wilson and Sommerfeld are made. The centres of attracting force are assumed, as in the simpler applications of the Quantum Theory, to be at rest. Such a spectrum would arise in the case of a hydrogen molecule which had lost one electron and in which the remaining electron described an orbit about the positive nuclei. Novikoff has shown that this configuration would be stable (vide Whittaker's Analytical Dynamics, 2nd ed., p. 407, ex. 1). It follows that, if at any stage an electron were withdrawn from a molecule, the remaining electron would describe a stable orbit. Much experimental work has been done on the secondary spectrum of hydrogen, but I am not aware that any theoretical explanation has been given of the immense number of lines found by experiment. Dufour in Ann. de Chimie et de Physique, 1906, gives a very complete account of the work done on this spectrum and the various controversies which it has evoked. Dufour's conclusion is that the spectrum of the atom is the primary spectrum, and that the secondary spectrum is due to the molecule. H. E. Watson in Proc. Roy. Soc., vol. lxxxii, 1909, states the main points at issue, and gives the most modern measurements of the lines and compares his results with wavelengths found by Hasselberg (Mém. Acad. Imp. St Petersburg, vol. xxxi, 1883), Ames (Phil. Mag., vol. xxx, 1890), and Frost (Astrophysical Journal, vol. xvi, 1902).