In a preliminary note read before the Society on July 4, 1904, I drew attention to the fact that a number of line-series, forming a group which includes the first series of Hydrogen, can be represented by an equation of the form

where v denotes the wave-frequency of any line of the series, v∞ that of the so-called “tail” of the series (m = ∞), and a1, b1 constants depending on the nature of the emitting substance; the frequencies of successive lines being obtained by substituting successive integers for m. We see at once that this equation is a generalisation of Balmek's formula, into which it is transformed by equating b1 to zero. In the same note I also pointed out the existence of another group represented by an equation of the same form, if (m + ½) is substituted for m. As a special case (b1 = 0) this group contains the second Hydrogen series discovered by Professor Pickering in the spectrum of ζ Puppis. Subsequent investigations convinced me, however, that, although a considerable number of line-series may be classified into these two groups, there are numerous instances where the more general formula

must be employed, in which μ represents various fractional numbers.