( 1 1 ) ν = K(--- - -------), ( 4 _n_² ) 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/_n_² approaches zero, and can be made as close to zero as desired by letting _n_ increase indefinitely. In mathematical terminology, as _n_ = ∞, 1/_n_² = 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. [Illustration: 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/_n_²) = 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 “ | --------+----------------------------+-------------+-------------+