See Ritz, Die iiltere Geschichte des Veste and der Stadt Recklinghausen (Erzen, 1904).
All these forms are put into the shade by that which was introduced by Ritz, led thereto apparently by theoretical considerations.
The fact that the addition of the term introduced by Ritz not only gives a more satisfactory representation of each series, but verifies the above relationship with a much closer degree of approximation, proves that Ritz's equation forms a marked step in the right direction.
Trunk series: t N = [s +al +b/s 1 [1 5 +a1 +b'/(I.5)2}2 Main Branch Series: t ytr' - I I N [2 + al + 6/29 2 [r+al Side Branch Series: t nT = N [2 +al+6,/22]2 [s+c+d,s92 Here s stands for an integer number beginning with 2 for the trunk and 3 for the main branch, and r represents the succession of numbers 1 5, 5, 3 5, &c. As Ritz points out, the first two equations appear only to be particular cases of the form n I I N +1)2 in which s and r have the form given above.
Hicks 1 has modified Rydberg's equation in a way similar to that of Ritz as shown by (5) above.
If we compare Balmer's formula with the general equation of Ritz, we find that the two can be made to agree if the ordinary hydrogen spectrum is that of the side branch series and the constants a', b, c and d are all put equal to zero.'
Ritz in the paper already mentioned follows in the footsteps of Riecke and elaborates the argument.
Now it follows from Rydberg's second law put on a more accurate basis by Ritz that in one case at any rate a negative period has reality and must be interpreted just as if it were positive.