Then the radiation actually emitted should, according to the correspondence principle, be such that the intensities of the lines corresponding to the double and triple jumps, which start from a given stationary state, are respectively one-half and one-third of the intensity corresponding to a single jump from the same state. By these examples we can obtain an idea of how the correspondence principle may in certain cases account for various facts, as to what spectral lines cannot be expected to appear at all, although they would be given by a particular transition, and concerning the distribution of intensities in those which really appear. The illustration given above, however, has really nothing much to do with actual problems, and objections may be raised to the rough way in which the illustration has been handled. The correspondence principle has its particular province in more complicated electron motions than those which appear in the unperturbed hydrogen atom—motions which, unlike the simple elliptical motion, are not composed of a series of harmonic oscillations (ω, 2ω, 3ω ...) but may be considered as compounded of oscillations whose frequencies have other ratios. The correspondence principle has, in such cases, given rise to important discoveries and predictions which agree completely with the observations. We have dwelt thus long upon the difficult correspondence principle, because it is one of Bohr’s deepest thoughts and chief guides. It has made possible a more consistent presentation of the whole theory, and it bids fair to remain the keystone of its future development. But from these general considerations we shall now proceed to more special phases of the problem and examine one of the first great triumphs in which the theory showed its ability to lead the way where previously there had been no path. The False Hydrogen Spectrum. In 1897 the American astronomer, Pickering, discovered in the spectrum of a star, in addition to the usual lines given by the Balmer series, a series of lines each of which lay about midway between two lines of the Balmer series; the frequencies of these lines could be represented by a formula which was very similar to the Balmer formula; it was necessary merely to substitute _n_ = 3½, 4½, 5½, etc., in the formula on p. 57 instead of _n_ = 3, 4, 5, etc. It was later discovered that in many stars there was a line corresponding to _n″_ = ³/₂ or _n′_ = 2 in the usual Balmer-Ritz formula (p. 59). It was considered that these must be hydrogen lines, and that the spectral formula for this element should properly be written 1 1 ν = K ---------- - --------- ( _n″_/2 )² ( _n′_/2 )²