The space charge in the screen-anode space of a beam tetrode is examined, taking into account the effect of electrons returned from the virtual cathode into the region between the grid and cathode. As with previous investigations which do not take into account the effect of the returned electrons in the cathode-grid space, it is found that over a range of anode voltages just below the knee, three solutions of the problem satisfy Newton's laws of motion for the electrons and also Poisson's equation for the charge distributions. Two of these solutions give rise to discontinuities in the characteristic. The third, however, is continuous for all anode voltages from zero up to the knee, and corresponds closely to experiments on actual valves. In it, a virtual cathode which remains at a fixed point between the screen and the anode is set up. The question arises, What property of this solution determines its correspondence with experiment? It is found that Maupertuis's principle of least action, as an integral of momentum with respect to distance, can be applied. The principle of least action as it stands cannot, however, be applied to a single arbitrarily chosen electron, but by calculating the action per unit of time, it can be applied to all three solutions, whether electrons are reflected or not. This leads to the conclusion that the action per unit of time is least for the third solution. Since this corresponds to the conditions observed experimentally, it is asserted that, in problems concerned with the motion of charged particles in which more than one consistent solution can be found, the actual state set up will be that for which the action per second is least.A simple theory of the effect of perturbations of the space-charge density, due, for example, to the random fluctuations of the cathode emission, is developed. The rate of growth of a perturbation depends on its size, and if it is small, it is unlikely to grow to such an extent, before being swept to the anode, as to cause instability of a particular charge distribution. It is concluded that such perturbations do not form an alternative explanation of the fact that the state containing the virtual cathode is always that set up.The growth of the perturbations is shown to depend on whether the anode potential remains constant or can vary during the passage of a perturbation. It is concluded that the theory of the fluctuations of a space-charge-limited current cannot be dealt with accurately on the assumption that the fluctuation current is independent of the anode load.Finally, the fluctuations of the space-charge-limited current in diodes and triodes is described on the basis of the given theory of perturbations. For identical space-charge conditions near the cathode, the diode has a planar electric field which exercises no electron-optical action, while the triode has a strong focusing action in the space between grid and cathode. The conclusion, that the perturbations can grow to a greater extent in diodes than in triodes, agrees with the experimental results on the noise of diode and triode valves.