Before going further, it will be well to consider how far periodicity retains this importance in the newer quantum mechanics inaugurated by Heisenberg. For this purpose, we may concentrate attention upon the one fundamental equation involving h in the new system. This equation takes the form:[70] [Pg 344] where and are matrices, being a Hamiltonian co-ordinate in the new sense, and the corresponding "impulse," also in the new sense; while is the unit matrix. This equation is asserted to hold for all motions, not only for such as are periodic. But in the case of motions which are not periodic, it gives a result which approximates to that of classical mechanics. Thus it remains the case that the new mechanics is only necessitated by periodic motions, although it is technically possible to find a quantum principle which is also applicable to non-periodic motions. Hence the importance of periodicity remains intact from an empirical point of view, though somewhat diminished from the point of view of a statement of fundamental laws. In any case, it remains sufficiently important to demand a separate discussion. Traditionally, periodicity in physics was a question of motion: a body described the same path in space over and over again. With the coming of relativity, it has become necessary to modify this account somewhat. In space-time, every point has a date, and cannot be occupied twice; neither the earth nor an electron can describe again the orbit it described on a former occasion. And periodicity will be relative to a given system of co-ordinates: if, in one system, a co-ordinate runs through a given range of values repeatedly, and always in equal times, it may happen that, in another system, even if there is an oscillating co-ordinate, its periods are not all equal. A change of axes may even take away all trace of periodic character from a process. Since, however, the quantum principle compels us to treat periodicity as physically important, it would seem that we must regard it as a character belonging to a process when referred to axes which move with it, since this would overcome the difficulties connected with relativity. If, in certain cases, this method is not open to us, some other must be found which equally avoids these difficulties. But where processes connected with[Pg 345] matter (as opposed to electromagnetic processes) are concerned we shall, I think, find no other possibility except to take axes which move with the matter concerned. But this makes it impossible to treat periodicity as fundamentally a character exhibited in a motion, since we have reduced to rest the body in which the periodic process is taking place. The suggestion I have to make is that, fundamentally, periodicity is constituted by the recurrence of qualities. In the present chapter, I wish to consider what can be meant by the "quality" of an event; I wish also to investigate the connection of quality with causality and motion and periodicity.