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.

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.