Bohr's theory has been generalized by Wilson[12] and Sommerfeld so as to allow also elliptic orbits: these have two quantum numbers, one corresponding, as before, to angular momentum or the moment of momentum (which is constant, by Kepler's second law), the other depending upon the eccentricity. Only certain eccentricities are possible; in fact, the ratio of the minor to the major axis is always rational, and has as its denominator the quantum number corresponding to the moment of momentum. In order to explain the Zeeman effect (which arises in a magnetic field) we used a third quantum number, corresponding to the angle between the plane of the magnetic field and the plane of the electron's orbit. In all cases, however, there is a general principle, which must now be explained. This will show, also, why, in Bohr's theory, the quantum equation (2) takes the form it does.[13] The first thing to observe is that the quantum principle is really concerned with atoms of action, not of energy: action is energy multiplied by time. Suppose now that we have a system depending upon several co-ordinates, and periodic in respect of each. It is not necessary to suppose that each co-ordinate has the same period: it is only necessary to suppose that the system is "conditionally periodic"—i.e. that each co-ordinate separately is periodic. We must further assume that our co-ordinates are so chosen as to allow "separation of variables" (as to which, see Sommerfeld, op. cit., pp. 559-60). We then define the "momentum" (in a generalized sense) associated with the co-ordinate as the partial differential of the kinetic energy with respect to —i.e. calling the generalized momentum , we put: where is the kinetic energy. The quantum condition is to apply to the integral of over a complete period of —i.e. we are to have: where the integration is taken through one complete period of . Here will be the quantum number associated with the co-ordinate . The above is a general formula of which all known cases of quantum phenomena are special cases. This is its sole justification. The above principle is exceedingly complicated—more so, even, than it appears in our summary account, which has omitted various difficulties. It is possible that its complication may be due to the fact that quantum dynamics has had to force its way through the obstacles which the classical system put in its way; it is possible also that quantum phenomena may turn out to be deducible from classical principles. But before pursuing this line of thought, it may be well to say[Pg 37] a few words about the developments of Bohr's theory by Sommerfeld and others.