The A B C of Atoms

would give 15 lines, 7 terms would give 21, 8 would give 28, 100 would give 4950. This shows that, the more lines and terms there are, the more surprising it becomes that the Principle of Combination should be true, and the less possible it becomes to attribute its truth to chance. The number of lines in the spectrum of hydrogen is very large. The terms of the hydrogen spectrum can all be expressed very simply. There is a certain fundamental wave-number, called Rydberg’s constant after its discoverer. Rydberg discovered that this constant was always occurring in formulae for series of spectral lines, and it has been found that it is very nearly the same for all elements. Its value is about 109,700 waves per centimetre. This may be taken as the fundamental term in the hydrogen spectrum. The others are obtained from it by dividing it by 4 (twice two), 9 (three times three), 16 (four times four), and so on. This gives all the terms; the lines are obtained by subtracting one term from another. Theoretically, this rule gives an infinite number of terms, and therefore of lines; but in practice the lines grow fainter as higher terms are involved, and also so close together that they can no longer be distinguished. For this reason, it is not necessary, in practice, to take account of more than about 30 terms; and even this number is only necessary in the case of certain nebulæ. 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 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 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.