Rydberg favours the former view, but he does not attempt to obtain any very close approximation between the observed and calculated values of the frequencies.
Halm subsequently showed that if N may differ in different cases, the equation is a considerable improvement on Rydberg's.
According to an important law discovered by Rydberg and shortly afterwards independently by the writer, the frequency of the common root of the two branches is obtained by subtracting the frequency of the root of the trunk from that of its least refrangible and strongest member.
Rydberg discovered a second relationship, which, however, involving the assumed equation connecting the different lines, cannot be tested directly as long as these equations are only approximate.
These forms all agree in making the frequency negative when s falls below a certain value sp. Rydberg's second law states that if the main branch series is taken, the numerical value of np_I corresponding to sp_ I is equal.