For the subordinate series of the ionized atom, [Pg 108] This equation “must be used wherever the ionization potential of the stage in question is closely preceded by the ionization potential of the preceding stage. The corrections ... result in making the maxima for the lines of the two stages occur farther apart in the temperature scale. If we express the correction in the form of a factor () ... then is of the order . Since varies roughly as or , we see that the importance of the correction is determined by the closeness of to 1.” The values of , the fractional concentration of the atom in question, are obtained through the application of the ordinary methods of statistical mechanics to the equilibrium between atoms and electrons in the reversing layer. The values of at the maximum are obtained by differentiating the expression for with respect to , and equating to zero, since the maximum of absorption will occur when is at a maximum. The analytical treatment calls for no comment. Its basis has been fully discussed by R. H. Fowler[374] in a series of papers. The weights () of the atomic states employed were based on the work of Bohr[375] on the relative values of the a priori probabilities of the different stationary states for hydrogen. On this view, for all atoms excepting those of H and He+, for which it is equal to 2. The convergence of the series for was not established by Fowler and Milne, but the authors regard the subsequent investigation by Urey[376] as justifying their assumption that “for physical reasons one must suppose the series effectively cut off after a certain number of terms. Usually the series then reduces (as regards its numerical value) practically to its first term.” PHYSICAL CONSTANTS REQUIRED IN THE FORMULAE The application of the equations will of course depend upon an accurate knowledge of the constants involved. The quantities , , , and require no comment. The symmetry number [Pg 109] of the neutral atom is in effect the number of spectroscopic valency electrons given in Bohr’s table,[377] for the atoms for which it is known. In all the applications made by Fowler and Milne the quantity was equated to 1 or 2, and it is very probable that this number is not in any case exceeded. For carbon, where the chemical valency is equal to 4, the value of is still 2, as has been shown by Fowler’s analysis of the spectrum of ionized carbon.[378] The value of is not known for atoms in the long periods, but in the present work it is assumed to be 1 and 2 for atoms with arc spectra which show even and odd multiplicities, respectively. The uncertainty in the value of introduces only a relatively small error into the result, since depends on the first power of , and in no case considered can exceed five.