It will be seen that, by our rule, we obtain various series of terms. The first series is obtained by subtracting from Rydberg’s constant successively a quarter, a ninth, a sixteenth ... of itself, so that the wave-numbers of its lines are respectively ¾, ⁸⁄₉, ¹⁵⁄₁₆ ... of Rydberg’s constant. These wave-numbers correspond to lines in the [Pg 47] ultra-violet, which can be photographed but not seen; this series of lines is called, after its discoverer, the Lyman series. Then there is a series of lines obtained by subtracting from a quarter of Rydberg’s constant successively a ninth, a sixteenth, a twenty-fifth ... of Rydberg’s constant, so that the wave-numbers of this series are ⁵⁄₃₆, ³⁄₁₆, ²¹⁄₁₀₀ ... of Rydberg’s constant. This series of lines is in the visible part of the spectrum; the formula for this series was discovered as long ago as 1885 by Balmer. Then there is a series obtained by taking a ninth of Rydberg’s constant, and subtracting successively a sixteenth, twenty-fifth, etc. of Rydberg’s constant. This series is not visible, because its wave-numbers are so small that it is in the infra-red, but it was discovered by Paschen, after whom it is called. Thus so far as the conditions of observation admit, we may lay down this simple rule: the lines of the hydrogen spectrum are obtained from Rydberg’s constant, by dividing it by any two square numbers, and subtracting the smaller resulting number from the larger. This gives the wave-number of some line in the hydrogen spectrum, if observation of a line with that wave-number is possible, and if there are not too many other lines in the immediate neighbourhood. (A square [Pg 48] number is a number multiplied by itself: one times one, twice two, three times three, and so on; that is to say, the square numbers are 1, 4, 9, 16, 25, 36, etc.). All this, so far, is purely empirical. Rydberg’s constant, and the formulæ for the lines of the hydrogen spectrum, were discovered merely by observation, and by hunting for some arithmetical formula which would make it possible to collect the different lines under some rule. For a long time the search failed because people employed wave-lengths instead of wave-numbers; the formulæ are more complicated in wave-lengths, and therefore more difficult to discover empirically. Balmer, who discovered the formula for the visible lines in the hydrogen spectrum, expressed it in wave-lengths. But when expressed in this form it did not suggest Ritz’s Principle of Combination, which led to the complete rule. Even after the rule was discovered, no one knew why there was such a rule, or what was the reason for the appearance of Rydberg’s constant. The explanation of the rule, and the connection of Rydberg’s constant with other known physical constants, was effected by Niels Bohr, whose theory will be explained in the next chapter. [Pg 49] V. POSSIBLE STATES OF THE HYDROGEN ATOM