However this may be, it is clear that what we know is the changes of energy when an atom emits light, and we know that in the case of hydrogen or ionized helium these changes are measured by . It seems almost unavoidable to infer that the previous state of the atom was characterized by the integer and the later one by the integer . But to assume orbits and so on, though proper as a help to the imagination, is hardly sufficiently justified by the analogy of large-scale processes, since the quantum principle itself shows the danger of relying upon this analogy. In large-scale occurrences there is nothing to suggest the quantum, and perhaps other familiar features of such occurrences may result merely from statistical averaging. [Pg 39] It may be worth while to consider briefly the elliptical orbits which are possible.[15] This will also illustrate the application of the quantum principle to systems with more than one co-ordinate. Taking polar co-ordinates, the kinetic energy is: The two generalized momenta are therefore: We have thus two quantum conditions: By Kepler's second law, is constant; call it . Thus: The other integration is more troublesome, but we arrive at the result that, if and are the major and minor axes of the ellipse, A little further calculation leads to the result that the energy in the orbit which has the quantum numbers , is: This is exactly the same as in the case of circular orbits, except that replaces . If this were all, the line spectrum of hydrogen would be exactly the same whether elliptic orbits occurred or not, and there would be no empirical means of deciding the question. However, by introducing considerations derived from the special theory of relativity we are able to distinguish between the results to be expected from circular and elliptic orbits[Pg 40] respectively, and to show that the latter must occur to account for observed facts. The crucial point is the variation of mass with velocity: the faster a body is moving, the greater is its mass. Therefore in an elliptic orbit the electron will have a greater mass at the perihelion than at the aphelion. From this it is found to follow that an elliptic orbit will not be accurately elliptic, but that the perihelion will advance slightly with each revolution.[16] That is to say, taking polar co-ordinates , , the co-ordinate increases by slightly more than between one minimum of and the next. The system is thus "conditionally periodic"—i.e. each separate co-ordinate changes periodically, but the periods of the two do not coincide. The result[17] is that the equation is replaced by: being the velocity of light, and , as before, the angular momentum. It will be seen that is very nearly 1, because is large.