so that the potential falls at the electrodes are proportional to the squares of the velocities of the ions. The change in potential across the layers is proportional to the square of the current, while the potential change between the layers is proportional to the current, the total potential difference between the plates is the sum of these changes, hence the relation between V and i will be of the form V = Ai + Bi². Mie (_Ann. der. Phys._, 1904, 13, P. 857) has by the method of successive approximations obtained solutions of equation (8) (i.) when the current is only a small fraction of the saturation current, (ii.) when the current is nearly saturated. The results of his investigations are represented in fig. 12, which represents the distribution of electric force along the path of the current for various values of the current expressed as fractions of the saturation current. It will be seen that until the current amounts to about one-fifth of the maximum current, the type of solution is the one just indicated, i.e. the electric force is constant except in the neighbourhood of the electrodes when it increases rapidly. Though we are unable to obtain a general solution of the equation (8), there are some very important special cases in which that equation can be solved without difficulty. We shall consider two of these, the first being that when the current is saturated. In this case there is no loss of ions by recombination, so that using the same notation as before we have d --(n1k1X) = q, dx d --(n2k2X) = -q. dx The solutions of which if q is constant are n1k1X = qx, n2k2X = I/e - qx = q(l - x), if l is the distance between the plates, and x = 0 at the positive electrode. Since dX/dx = 4[pi](n1 - n2)e, we get 1 dX² / 1 1 \ l ----- --- = qx ( -- + -- ) - q --, 8[pi] d²x \k1 k2 / k2 or X² x² / 1 1 \ lx ----- = q -- ( -- + -- ) - q -- + C, 8[pi] 2 \k1 k2 / k2 where C is a quantity to be determined by the condition that _ / l | Xdx = V, _/0 where V is the given potential difference between the plates. When the force is a minimum dX/dx = 0, hence at this point lk1 lk2 x = -------, l - x = -------. k1 + k2 k1 + k2 Hence the ratio of the distances of this point from the positive and negative plates respectively is equal to the ratio of the velocities of the positive and negative ions.