where is the frequency, is "Rydberg's constant," [Pg 33] and are small integers, and are what are called "terms." After the formula had been discovered, new lines agreeing with it were sought and found. Certain lines formerly attributed to hydrogen, and not agreeing with the above formula, were attributed by Bohr to ionized helium; they are given by the formula: Bohr's theoretical grounds for attributing these lines to helium were afterwards confirmed experimentally by Fowler. It will be seen that they fit into the formula (1) when is substituted for , a fact which Bohr's theory explains, as well as the more delicate fact that, to make the formula exact, we have to substitute, not exactly , but a slightly smaller quantity. The form of the equation (1) suggested to Bohr that a line of the hydrogen spectrum is not to be regarded as something which the atom emits when it is in a state of periodic vibration, but as produced by a change from a state connected with one integer to a state connected with another. This would be explained if the orbit of the electron were not just any orbit possible on Newtonian principles, but only an orbit connected with an integral "quantum number"—i.e. with a multiple of . The way in which Bohr achieved a theory on these lines is as follows. He supposed that the electron can only revolve round the nucleus in certain circles, these being such that, if is the moment of momentum in any orbit, we shall have: where is, as always, Planck's constant, and is a small whole number. (In theory might be any whole number, but in[Pg 34] practice it is never found to be much larger than 30, and that only in certain very tenuous nebulæ.) The reason why the quantum principle assumes just this form will be explained presently. Now if is the mass of the electron, the radius of its orbit, and its angular velocity, we have: But, on grounds of the usual theory, since the radial acceleration of the electron is and the force attracting it to the nucleus is we have: From equations (3) and (4) we obtain: The possible orbits for the electron are obtained by putting = 1, 2, 3, 4, ... in the above formulæ for . Thus the smallest possible orbit is: and the other possible orbits are , , , etc. For the energy in an orbit of radius we have, since the potential energy is double the kinetic energy with its sign changed:[11] in virtue of (5). Thus when the electron falls from an orbit whose radius is to one whose radius is , there is a loss of energy: [Pg 35] It is assumed that this energy is radiated out in a light-wave whose energy is one quantum of energy , where is its frequency. Hence we obtain the frequency of the emitted light by the equation: This agrees exactly with the observed lines if [see equation (1)]: where is Rydberg's constant. On inserting numerical values, it is found that this equation is verified. This striking success was, from the first, a powerful argument in favour of Bohr's theory. [Pg 36]