Balmer discovered that wave-lengths of the red and of the green hydrogen line are to each other exactly as two integers, namely, as 27 to 20, and that the wave-lengths of the green and violet lines are to each other as 28 to 25. Continued reflection on this correspondence led him to enunciate a rule which can be expressed by a simple formula. When frequency is substituted for wave-length Balmer’s formula is written as

ν = K 1 - 1 ,
4n²

where ν is the frequency of a hydrogen line, K a constant equal to 3·29 × 10¹⁵ and n an integer. If n takes on different values, ν becomes the frequency for the different hydrogen lines. If n = 1 ν is negative, for n = 2 ν is zero. These values of n therefore have no meaning with regard to ν. But if n = 3, then ν gives the frequency for the red hydrogen line Hα; n = 4 gives the frequency of the green line Hᵦ and n = 5 that of the violet line Hᵧ. Gradually more than thirty hydrogen lines have been found, agreeing accurately with the formula for different values of n. Some of these lines were not found in experiment, but were discovered in the spectrum of certain stars; the exact agreement of these lines with Balmer’s formula was strong evidence for the belief that they are due to hydrogen. The formula thus proved itself valuable in revealing the secrets of the heavens.

As n increases 1/n² approaches zero, and can be made as close to zero as desired by letting n increase indefinitely. In mathematical terminology, as n = ∞, 1/n² = 0 and ν = K/4 = 823 × 10¹², corresponding to a wave-length of 365 μμ. Physically this means that the line spectrum of hydrogen in the ultra-violet is limited by a line corresponding to that frequency. Near this limit the hydrogen lines corresponding to Balmer’s formula are tightly packed together. For n = 20 ν differs but little from K/4, and the distance between two successive lines corresponding to an increase of 1 in n becomes more and more insignificant. [Fig. 13], where the numbers indicate the wave-lengths in the Ångström unit (0·1 μμ), shows the crowding of the hydrogen lines towards a definite boundary. [The following table], where K has the accurate value of 3·290364 × 10¹⁵, shows how exactly the values calculated from the formula agree with experiment.

Fig. 13.—Lines in the hydrogen spectrum corresponding to the Balmer series.

Table of some of the Lines of the Balmer Series

ν = K(1/4 - 1/n²) = ν
(calculated).		ν (found).λ (found).
n = 3 K(¼ - ¹/₉ ) = 456,995 bills  456,996 bills  656·460 μμ Hα
n = 4 K(¼ - ¹/₁₆ ) = 616,943 “616,943  “486·268 “ Hᵦ
n = 5 K(¼ - ¹/₂₅ ) = 690,976 “690,976  “434·168 “ Hᵧ
n = 6 K(¼ - ¹/₃₆ ) = 731,192 “731,193  “410·288 “ Hδ
n = 7 K(¼ - ¹/₄₉ ) = 755,440 “755,441  “397·119 “ Hε
n = 20 K(¼ - ¹/₄₀₀) = 814,365 “814,361   “368·307 “

From arguments in connection with the work of the Swedish scientist, Rydberg, in the spectra of other elements, Ritz, a fellow countryman of Balmer’s, has made it seem probable that the hydrogen spectrum contains other lines besides those corresponding to Balmer’s formula. He assumed that the hydrogen spectrum, like other spectra, contains several series of lines and that Balmer’s formula corresponds to only one series. Ritz then enunciated a more comprehensive formula, the Balmer-Ritz formula:

ν = K 1 - 1 ,
n″ ²n′ ²